title:
Solitary-waves of the Nonpolynomial Schrodinger Equation: Bright
  Solitons in Bose-Einstein Condensates

abstract:
  Recently we have derived an effective one-dimensional nonpolynomial
Schr\"odinger equation (NPSE) [Phys. Rev. A {\bf 65}, 043614 (2002)] that
accurately describes atomic Bose-Einstein condensates under transverse harmonic
confinement. In this paper we analyze the stability of the solitary-wave
solutions of NPSE by means of the Vakhitov-Kolokolov criterion. Stable
self-focusing solutions of NPSE are precisely the 3D bright solitons
experimentally found in Bose-Einstein condensates [Nature {\bf 417}, 150
(2002); Science {\bf 296}, 1290 (2002)]. These self-focusing solutions
generalize the ones obtained with the standard nonlinear Schr\"odinger equation
(NLSE) and take into account the formation of instabilities in the transverse
direction. We prove that neither cubic nor cubic-quintic approximations of NPSE
are able to reproduce the unstable branch of the solitary-wave solutions.
Moreover, we show that the analitically found critical point of NPSE is in
remarkable good agreement with recent numerical computations of 3D GPE. We also
discuss the formation of multiple solitons by a sudden change of the sign of
the nonlinear strength. This formation can be interpreted in terms of the
modulational instability of the time-dependent macroscopic wave function of the
Bose condensate. In particular, we derive an accurate analytical formula for
the number of multiple bright solitons.


title:
Large-Eddy Simulation closures of passive scalar turbulence: a
  systematic approach

abstract:
  The issue of the parameterization of small scale (``subgrid'') turbulence is
addressed in the context of passive scalar transport. We focus on the Kraichnan
advection model which lends itself to the analytical investigation of the
closure problem. We derive systematically the dynamical equations which rule
the evolution of the coarse-grained scalar field. At the lowest-order
approximation in $l/r$, $l$ being the characteristic scale of the filter
defining the coarse-grained scalar field and $r$ the inertial range separation,
we recover the classical eddy-diffusivity parameterization of small scales. At
the next-leading order a dynamical closure is obtained. The latter outperforms
the classical model and is therefore a natural candidate for subgrid modelling
of scalar transport in generic turbulent flows.


title: Stochastic stabilization of cosmological photons

abstract:
  The stability of photon trajectories in models of the Universe that have
constant spatial curvature is determined by the sign of the curvature: they are
exponentially unstable if the curvature is negative and stable if it is
positive or zero. We demonstrate that random fluctuations in the curvature
provide an additional stabilizing mechanism. This mechanism is analogous to the
one responsible for stabilizing the stochastic Kapitsa pendulum. When the mean
curvature is negative it is capable of stabilizing the photon trajectories;
when the mean curvature is zero or positive it determines the characteristic
frequency with which neighbouring trajectories oscillate about each other. In
constant negative curvature models of the Universe that have compact topology,
exponential instability implies chaos (e.g. mixing) in the photon dynamics. We
discuss some consequences of stochastic stabilization in this context.


title: An Explicit Characterization of Calogero--Moser Systems

abstract:
  Combining theorems of Halphen, Floquet, and Picard and a Frobenius type
analysis, we characterize rational, meromorphic simply periodic, and elliptic
KdV potentials. In particular, we explicitly describe the proper extension of
the Calogero--Moser locus associated with these three classes of
algebro-geometric solutions of the KdV hierarchy with special emphasis on the
case of multiple collisions between the poles of solutions.


title: On a Theorem of Halphen and its Application to Integrable Systems

abstract:
  We extend Halphen's theorem which characterizes the solutions of certain
$n$th-order differential equations with rational coefficients and meromorphic
fundamental systems to a first-order $n \times n$ system of differential
equations. As an application of this circle of ideas we consider stationary
rational algebro-geometric solutions of the $\kdv$ hierarchy and illustrate
some of the connections with completely integrable models of the
Calogero-Moser-type. In particular, our treatment recovers the complete
characterization of the isospectral class of such rational KdV solutions in
terms of a precise description of the Airault-McKean-Moser locus of their
poles.


title:
Onset of Collective Oscillation in Chemical Turbulence under Global
  Feedback

abstract:
  Preceding the complete suppression of chemical turbulence by means of global
feedback, a different universal type of transition, which is characterized by
the emergence of small-amplitude collective oscillation with strong turbulent
background, is shown to occur at much weaker feedback intensity. We illustrate
this fact numerically in combination with a phenomenological argument based on
the complex Ginzburg-Landau equation with global feedback.


title: On the canonically invariant calculation of Maslov indices

abstract:
  After a short review of various ways to calculate the Maslov index appearing
in semiclassical Gutzwiller type trace formulae, we discuss a
coordinate-independent and canonically invariant formulation recently proposed
by A Sugita (2000, 2001). We give explicit formulae for its ingredients and
test them numerically for periodic orbits in several Hamiltonian systems with
mixed dynamics. We demonstrate how the Maslov indices and their ingredients can
be useful in the classification of periodic orbits in complicated bifurcation
scenarios, for instance in a novel sequence of seven orbits born out of a
tangent bifurcation in the H\'enon-Heiles system.

categories: nlin.CD math-ph math.MP math.SG quant-ph


title: Scattering of dipole-mode vector solitons: Theory and experiment

abstract:
  We study, both theoretically and experimentally, the scattering properties of
optical dipole-mode vector solitons - radially asymmetric composite
self-trapped optical beams. First, we analyze the soliton collisions in an
isotropic two-component model with a saturable nonlinearity and demonstrate
that in many cases the scattering dynamics of the dipole-mode solitons allows
us to classify them as ``molecules of light'' - extremely robust spatially
localized objects which survive a wide range of interactions and display many
properties of composite states with a rotational degree of freedom. Next, we
study the composite solitons in an anisotropic nonlinear model that describes
photorefractive nonlinearities, and also present a number of experimental
verifications of our analysis.


title:
The Rolling Body Motion Of a Rigid Body on a Plane and a Sphere.
  Hierarchy of Dynamics

abstract:
  In this paper we consider cases of existence of invariant measure, additional
first integrals, and Poisson structure in a problem of rigid body's rolling
without sliding on plane and sphere. The problem of rigid body's motion on
plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that
the equations of motion are reduced to a second-order linear differential
equation in the case when the surface of dynamically symmetric body is a
surface of revolution. These results were partially generalized by P. Woronetz,
who studied the motion of body of revolution and the motion of round disk with
sharp edge on the surface of sphere. In both cases the systems are Euler-Jacobi
integrable and have additional integrals and invariant measure. It turns out
that after some change of time defined by reducing multiplier, the reduced
system is a Hamiltonian system. Here we consider different cases when the
integrals and invariant measure can be presented as finite algebraic
expressions.
  We also consider the generalized problem of rolling of dynamically
nonsymmetric Chaplygin ball. The results of studies are presented as tables
that describe the hierarchy of existence of various tensor invariants:
invariant measure, integrals, and Poisson structure in the considered problems.


title:
On a family of solutions of the KP equation which also satisfy the Toda
  lattice hierarchy

abstract:
  We describe the interaction pattern in the $x$-$y$ plane for a family of
soliton solutions of the Kadomtsev-Petviashvili (KP) equation,
$(-4u_{t}+u_{xxx}+6uu_x)_{x}+3u_{yy}=0$. Those solutions also satisfy the
finite Toda lattice hierarchy. We determine completely their asymptotic
patterns for $y\to \pm\infty$, and we show that all the solutions (except the
one-soliton solution) are of {\it resonant} type, consisting of arbitrary
numbers of line solitons in both aymptotics; that is, arbitrary $N_-$ incoming
solitons for $y\to -\infty$ interact to form arbitrary $N_+$ outgoing solitons
for $y\to\infty$. We also discuss the interaction process of those solitons,
and show that the resonant interaction creates a {\it web-like} structure
having $(N_--1)(N_+-1)$ holes.


title: Resonant phenomena in slowly perturbed elliptic billiards

abstract:
  We consider an elliptic billiard whose shape slowly changes. During slow
evolution of the billiard certain resonance conditions can be fulfilled. We
study the phenomena of capture into a resonance and scattering on resonances
which lead to the destruction of the adiabatic invariance in the system.


title: Gaudin Models and Bending Flows: a Geometrical Point of View

abstract:
  In this paper we discuss the bihamiltonian formulation of the (rational XXX)
Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued
spins. In particular, we focus on the homogeneous models. We find a pencil of
Poisson brackets that recursively define a complete set of integrals of the
motion, alternative to the set of integrals associated with the 'standard' Lax
representation of the Gaudin model. These integrals, in the case of su(2),
coincide wih the Hamiltonians of the 'bending flows' in the moduli space of
polygons in Euclidean space introduced by Kapovich and Millson. We finally
address the problem of separability of these flows and explicitly find
separation coordinates and separation relations for the r=2 case.


title:
Reducibility of zero curvature representations with application to
  recursion operators

abstract:
  We present a criterion of reducibility of a zero curvature representation to
a solvable subalgebra, hence to a chain of conservation laws. Namely, we show
that reducibility is equivalent to the existence of a section of the
generalized Riccati covering. Results are applied to conversion between
Guthrie's and Olver's form of recursion operators.


title: Role of pressure in turbulence

abstract:
  There is very limited knowledge of the kinematical relations for the velocity
structure functions higher than three. Instead, the dynamical equations for the
structure functions of the velocity increment are obtained from the Navier
Stokes equation under the assumption of the local homogeneity and isotropy.
These equations contain the correlation between the velocity and pressure
gradient increments, which is very difficult to know theoretically and
experimentally. We have examined these dynamical relations by using direct
numerical simulation data at very high resolution at large Reynolds numbers,
and found that the contribution of the pressure term is important to the
dynamics of the longitudinal velocity with large amplitudes. The pressure term
is examined from the view point of the conditional average and the role of the
pressure term in the turbulence dynamics is discussed.


title:
Bi-hamiltonian Geometry and Separation of Variables for Gaudin Models: a
  case study

abstract:
  We address the study of the classical Gaudin spin model from the
bi-Hamiltonian point of view. We describe in details the sl(2) three particle
case.


title: Wigner--Dyson statistics for a class of integrable models

abstract:
  We construct an ensemble of second--quantized Hamiltonians with two bosonic
degrees of freedom, whose members display with probability one GOE or GUE
statistics. Nevertheless, these Hamiltonians have a second integral of motion,
namely the boson number, and thus are integrable. To construct this ensemble we
use some ``reverse engineering'' starting from the fact that $n$--bosons in a
two--level system with random interactions have an integrable classical limit
by the old Heisenberg association of boson operators to actions and angles. By
choosing an $n$--body random interaction and degenerate levels we end up with
GOE or GUE Hamiltonians. Ergodicity of these ensembles completes the example.


title: Generalized hierarchy of matrix Burgers type and $n$-wave equations

abstract:
  This article concerns the dressing method for solving of multidimensional
nonlinear Partial Differential Equations. In particular, we join hierarchy of
matrix Burgers type equation with hierarchies of equations integrable by the
Inverse Spectral Transform (IST). Example of resonance interaction of wave
packets in (3+1)-dimensions is given.


title: Localization and Coherence in Nonintegrable Systems

abstract:
  We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian
oscillator chains approaching their statistical asympotic states. In systems
constrained by more than one conserved quantity, the partitioning of the
conserved quantities leads naturally to localized and coherent structures. If
the phase space is compact, the final equilibrium state is governed by entropy
maximization and the final coherent structures are stable lumps. In systems
where the phase space is not compact, the coherent structures can be collapses
represented in phase space by a heteroclinic connection to infinity.


title:
Two New Integrable Lattice Hierarchies Associated With A Discrete
  Schrodinger Nonisospectral Problem and Their Infinitely Many Conservation
  Laws

abstract:
  In this article, by means of using discrete zero curvature representation and
constructing opportune time evolution problems, two new discrete integrable
lattice hierarchies with n-dependent coefficients are proposed, which related
to a new discrete Schr\"{o}dinger nonisospectral operator equation. The
relation of the two new lattice hierarchies with the Volterra hierarchy is
discussed. It has been shown that one lattice hierarchy is equivalent to the
positive Volterra hierarchy with n-dependent coefficients and another lattice
hierarchy with isospectral problem is equivalent to the negative Volterra
hierarchy. We demonstrate the existence of infinitely many conservation laws
for the two lattice hierarchies and give the corresponding conserved densities
and the associated fluxes formulaically. Thus their integrability is confirmed.


title: Effects of a nonlinear perturbation on dynamical tunneling in cold atoms

abstract:
  We perform a numerical analysis of the effects of a nonlinear perturbation on
the quantum dynamics of two models describing non-interacting cold atoms in a
standing wave of light with a periodical modulated amplitude $A(t)$. One model
is the driven pendulum, considered in ref.\cite{raiz1}, and the other is a
variant of the well-known Kicked Rotator Model. In absence of the nonlinear
perturbation, the system is invariant under some discrete symmetries and
quantum dynamical tunnelling between symmetric classical islands is found. The
presence of nonlinearity destroys tunnelling, breaking the symmetries of the
system. Finally, further consequences of nonlinearity in the kicked rotator
case are considered.


title: Non-linear fractal interpolating functions of one and two variables

abstract:
  We consider non-linear generalizations of fractal interpolating functions
applied to functions of one and two variables. The use of such interpolating
functions in resizing images is illustrated.


title:
Learning a world model and planning with a self-organizing, dynamic
  neural system

abstract:
  We present a connectionist architecture that can learn a model of the
relations between perceptions and actions and use this model for behavior
planning. State representations are learned with a growing self-organizing
layer which is directly coupled to a perception and a motor layer. Knowledge
about possible state transitions is encoded in the lateral connectivity. Motor
signals modulate this lateral connectivity and a dynamic field on the layer
organizes a planning process. All mechanisms are local and adaptation is based
on Hebbian ideas. The model is continuous in the action, perception, and time
domain.


title: Noise Correlations in Shear Flows

abstract:
  We consider the effects of a shear on velocity fluctuations in a flow. The
shear gives rise to a transient amplification that not only influences the
amplitude of perturbations but also their time correlations. We show that, in
the presence of white noise, time correlations of transversal velocity
components are exponential and that correlations of the longitudinal components
are exponential with an algebraic prefactor. Cross correlations between
transversal and downstream components are strongly asymmetric and provide a
clear indication of non-normal amplification. We suggest experimental tests of
our predictions.


title: Irreversible quantum graphs

abstract:
  Irreversibility is introduced to quantum graphs by coupling the graphs to a
bath of harmonic oscillators. The interaction which is linear in the harmonic
oscillator amplitudes is localized at the vertices. It is shown that for
sufficiently strong coupling, the spectrum of the system admits a new continuum
mode which exists even if the graph is compact, and a {\it single} harmonic
oscillator is coupled to it. This mechanism is shown to imply that the quantum
dynamics is irreversible. Moreover, it demonstrates the surprising result that
irreversibility can be introduced by a "bath" which consists of a {\it single}
harmonic oscillator.


title: Control of Integrable Hamiltonian Systems and Degenerate Bifurcations

abstract:
  We discuss control of low-dimensional systems which, when uncontrolled, are
integrable in the Hamiltonian sense. The controller targets an exact solution
of the system in a region where the uncontrolled dynamics has invariant tori.
Both dissipative and conservative controllers are considered. We show that the
shear flow structure of the undriven system causes a Takens-Bogdanov
birfurcation to occur when control is applied. This implies extreme noise
sensitivity. We then consider an example of these results using the driven
nonlinear Schrodinger equation.


title: Stretching and relaxation dynamics in double stranded DNA

abstract:
  We study numerically the mechanical stability and elasticity properties of
duplex DNA molecules within the frame of a network model incorporating
microscopic degrees of freedom related with the arrangement of the base pairs.
We pay special attention to the opening-closing dynamics of double-stranded DNA
molecules which are forced into non-equilibrium conformations. Mechanical
stress imposed at one terminal end of the DNA molecule brings it into a
partially opened configuration. We examine the subsequent relaxation dynamics
connected with energy exchange processes between the various degrees of freedom
and structural rearrangements leading to complete recombination to the
double-stranded conformation. The similarities and differences between the
relaxation dynamics for a planar ladder-like DNA molecule and a twisted one are
discussed in detail. In this way we show that the attainment of a
quasi-equilibrium regime proceeds faster in the case of the twisted DNA form
than for its thus less flexible ladder counterpart. Furthermore we find that
the velocity of the complete recombination of the DNA molecule is lower than
the velocity imposed by the forcing unit which is in compliance with the
experimental observations for the opening-closing cycle of DNA molecules.


title:
Irregular isomonodromic deformations for Garnier systems and Okamoto's
  canonical transformations

abstract:
  In this paper we describe the Garnier systems as isomonodromic deformation
equations of a linear system with a simple pole at zero and a Poincar\'e rank
two singularity at infinity. We discuss the extension of Okamoto's birational
canonical transformations to the Garnier systems in more than one variable and
to the Schlesinger systems.


title: The scattering approach for the Camassa-Holm equation

abstract:
  We present an approach proving the integrability of the Camassa--Holm
equation for initial data of small amplitude.


title: Complexity of Langton's Ant

abstract:
  The virtual ant introduced by C. Langton has an interesting behavior, which
has been studied in several contexts. Here we give a construction to calculate
any boolean circuit with the trajectory of a single ant. This proves the
P-hardness of the system and implies, through the simulation of one dimensional
cellular automata and Turing machines, the universality of the ant and the
undecidability of some problems associated to it.


title:
Punctuated equilibria and 1/f noise in a biological coevolution model
  with individual-based dynamics

abstract:
  We present a study by linear stability analysis and large-scale Monte Carlo
simulations of a simple model of biological coevolution. Selection is provided
through a reproduction probability that contains quenched, random interspecies
interactions, while genetic variation is provided through a low mutation rate.
Both selection and mutation act on individual organisms. Consistent with some
current theories of macroevolutionary dynamics, the model displays
intermittent, statistically self-similar behavior with punctuated equilibria.
The probability density for the lifetimes of ecological communities is well
approximated by a power law with exponent near -2, and the corresponding power
spectral densities show 1/f noise (flicker noise) over several decades. The
long-lived communities (quasi-steady states) consist of a relatively small
number of mutualistically interacting species, and they are surrounded by a
``protection zone'' of closely related genotypes that have a very low
probability of invading the resident community. The extent of the protection
zone affects the stability of the community in a way analogous to the height of
the free-energy barrier surrounding a metastable state in a physical system.
Measures of biological diversity are on average stationary with no discernible
trends, even over our very long simulation runs of approximately 3.4x10^7
generations.

categories: nlin.AO cond-mat.stat-mech physics.bio-ph q-bio.PE


title: On the influence of noise on chaos in nearly Hamiltonian systems

abstract:
  The simultaneous influence of small damping and white noise on Hamiltonian
systems with chaotic motion is studied on the model of periodically kicked
rotor. In the region of parameters where damping alone turns the motion into
regular, the level of noise that can restore the chaos is studied. This
restoration is created by two mechanisms: by fluctuation induced transfer of
the phase trajectory to domains of local instability, that can be described by
the averaging of the local instability index, and by destabilization of motion
within the islands of stability by fluctuation induced parametric modulation of
the stability matrix, that can be described by the methods developed in the
theory of Anderson localization in one-dimensional systems.


title: Bright and dark breathers in Fermi-Pasta-Ulam lattices

abstract:
  In this paper we study the existence and linear stability of bright and dark
breathers in one-dimensional FPU lattices. On the one hand, we test the range
of validity of a recent breathers existence proof [G. James, {\em C. R. Acad.
Sci. Paris}, 332, Ser. 1, pp. 581 (2001)] using numerical computations.
Approximate analytical expressions for small amplitude bright and dark
breathers are found to fit very well exact numerical solutions even far from
the top of the phonon band. On the other hand, we study numerically large
amplitude breathers non predicted in the above cited reference. In particular,
for a class of asymmetric FPU potentials we find an energy threshold for the
existence of exact discrete breathers, which is a relatively unexplored
phenomenon in one-dimensional lattices. Bright and dark breathers superposed on
a uniformly stressed static configuration are also investigated.


title: Tri-Hamiltonian Structures of The Egorov Systems

abstract:
  The description of the Egorov hydrodynamic type systems is presented in
language of conservation laws. Under extra conditions of semisimplicity and
homogeneity tri-Hamiltonian structures of such systems are described. Even and
odd cases are related with flat and metric of constant curvature, respectively.
Some important and well-known examples like averaged by Whitham approach and
dispersionless limits of vector NLS and vector LSR are investigated.


title:
General soliton matrices in the Riemann-Hilbert problem for integrable
  nonlinear equations

abstract:
  We derive the soliton matrices corresponding to an arbitrary number of
higher-order normal zeros for the matrix Riemann-Hilbert problem of arbitrary
matrix dimension, thus giving the complete solution to the problem of
higher-order solitons. Our soliton matrices explicitly give all higher-order
multi-soliton solutions to the nonlinear partial differential equations
integrable through the matrix Riemann-Hilbert problem. We have applied these
general results to the three-wave interaction system, and derived new classes
of higher-order soliton and two-soliton solutions, in complement to those from
our previous publication [Stud. Appl. Math. \textbf{110}, 297 (2003)], where
only the elementary higher-order zeros were considered. The higher-order
solitons corresponding to non-elementary zeros generically describe the
simultaneous breakup of a pumping wave $(u_3)$ into the other two components
($u_1$ and $u_2$) and merger of $u_1$ and $u_2$ waves into the pumping $u_3$
wave. The two-soliton solutions corresponding to two simple zeros generically
describe the breakup of the pumping $u_3$ wave into the $u_1$ and $u_2$
components, and the reverse process. In the non-generic cases, these
two-soliton solutions could describe the elastic interaction of the $u_1$ and
$u_2$ waves, thus reproducing previous results obtained by Zakharov and Manakov
[Zh. Eksp. Teor. Fiz. \textbf{69}, 1654 (1975)] and Kaup [Stud. Appl. Math.
\textbf{55}, 9 (1976)].


title: Steady State Multiplicity in a Polymer Electrolyte Membrane Fuel Cell

abstract:
  A simplified differential reactor model that embodies the essential physics
controlling PEM fuel cell (PEM-FC) dynamics is presented. A remarkable analogy
exists between water management in the differential PEM-FC and energy balance
in the classical exothermic stirred tank reactor. Water, the reaction product
in the PEM-FC autocatalytically accelerates the reaction rate by enhancing
proton transport through the PEM. Established analyses of heat autocatalyticity
in a CSTR are modified to present water management autocatalyticity in a
stirred tank reactor PEM-FC.


title: Chaos-based cryptograph incorporated with S-box algebraic operation

abstract:
  Error function analysis is an effective attack against chaotic cryptograph
[PRE 66, 065202(R) (2002)]. The basin structure of the error function is
crucial for determining the security of chaotic cryptosystems. In the present
paper the basin behavior of the system used in [21] is analyzed in relation
with the estimation of its practical security. A S-box algebraic operation is
included in the chaotic cryptosystem, which considerably shrinks the basin of
the error function and thus greatly enhances the practical security of the
system with a little computational expense. PACS numbers: 05.45.Vx, 05.45.Ra


title: Swarming dynamics of a model for biological groups in two dimensions

abstract:
  We investigate a class of continuum models for the motion of a
two-dimensional biological group under the influence of nonlocal social
interactions. The dynamics may be uniquely decomposed into incompressible
motion and potential motion. When the motion is purely incompressible, the
model possesses solutions which have constant population density and sharp
boundaries for all time. Numerical simulations of these "swarm patches'' reveal
rotating mill-like swarms with circular cores and spiral arms. The sign of the
social interaction term determines the direction of the rotation, and the
interaction length scale affects the degree of spiral formation. When the
motion is purely potential, the social interaction term has the meaning of
repulsion or attraction depending on its sign. For the repulsive case, the
population spreads and the density profile is smoothed. With increasing
interaction length scale, the motion becomes more convective and experiences
slower diffusive smoothing. For the attractive case, the population
self-organizes into regions of high and low density. The characteristic length
scale of the density pattern is predicted and confirmed by numerical
simulations.


title: Interaction of moving breathers with an impurity

abstract:
  We analyze the influence of an impurity in the evolution of moving discrete
breathers in a Klein--Gordon chain with non-weak nonlinearity. Three different
behaviours can be observed when moving breathers interact with the impurity:
they pass through the impurity continuing their direction of movement; they are
reflected by the impurity; they are trapped by the impurity, giving rise to
chaotic breathers. Resonance with a breather centred at the impurity site is
conjectured to be a necessary condition for the appearance of the trapping
phenomenon.


title: Universality and Decidability of Number-Conserving Cellular Automata

abstract:
  Number-conserving cellular automata (NCCA) are particularly interesting, both
because of their natural appearance as models of real systems, and because of
the strong restrictions that number-conservation implies. Here we extend the
definition of the property to include cellular automata with any set of states
in $\Zset$, and show that they can be always extended to ``usual'' NCCA with
contiguous states. We show a way to simulate any one dimensional CA through a
one dimensional NCCA, proving the existence of intrinsically universal NCCA.
Finally, we give an algorithm to decide, given a CA, if its states can be
labeled with integers to produce a NCCA, and to find this relabeling if the
answer is positive.


title:
Point Mutations and Transitions Between Cellular Automata Attractor
  Basins

abstract:
  We consider transformations between attractor basins of binary cylindrical
cellular automata resulting from mutations. A t-point mutation of a state
consists in toggling t sites in that state. Results of such mutations are
described by a rule-dependent probability matrix. The structure of this matrix
is studied in relation to the structure of the state transition diagram and
several theorems relating these are proved for the case of additive rules. It
is shown that the steady state of the Markov process defined by the probability
matrix is always the uniform distribution over the state transition diagram.
Some results on eigenvalues are also obtained.


title: Periodic solutions of a system of complex ODEs. II. Higher periods

abstract:
  In a previous paper the \textit{real} evolution of the system of ODEs
\ddot{z}_{n} + z_{n}=\sum\limits_{m = 1, m \ne n}^{N} g_{nm}{(z_{n} - z_{m})}
^{- 3}, z_{n} \equiv z_{n}(t), \qquad \dot {z}_{n} \equiv \frac{d
z_{n}(t)}{dt}, \qquad n = 1,...,N is discussed in C_{N}, namely the N dependent
variables z_{n}, as well as the N(N - 1) (arbitrary!) ``coupling constants''
g_{nm}, are considered to be \textit{complex} numbers, while the independent
variable t (``time'') is \textit{real}. In that context it was proven that
there exists, in the phase space of the initial data z_{n}(0), \dot {z}_{n}
(0), an open domain having \textit{infinite} measure, such that \textit{all}
trajectories emerging from it are \textit{completely periodic} with period
2\pi, z_{n} (t + 2\pi) = z_{n}(t). In this paper we investigate, both by
analytical techniques and via the display of numerical simulations, the
remaining solutions, and in particular we show that there exist many --
emerging out of sets of initial data having nonvanishing measures in the phase
space of such data -- that are also \textit{completely periodic} but with
periods which are \textit{integer multiples} of 2\pi. We also elucidate the
mechanism that yields \textit{nonperiodic} solutions, including those
characterized by a ``chaotic'' behavior, namely those associated, in the
context of the initial-value problem, with a \textit{sensitive dependence} on
the initial data.


title:
Stable solitons in coupled Ginzburg-Landau equations describing
  Bose-Einstein condensates and nonlinear optical waveguides and cavities

abstract:
  We introduce a model of a two-core system, based on an equation of the
Ginzburg-Landau (GL) type, coupled to another GL equation, which may be linear
or nonlinear. One core is active, featuring intrinsic linear gain, while the
other one is lossy. The difference from previously studied models involving a
pair of linearly coupled active and passive cores is that the stabilization of
the system is provided not by a linear diffusion-like term, but rather by a
cubic or quintic dissipative term in the active core. Physical realizations of
the models include systems from nonlinear optics (semiconductor waveguides or
optical cavities), and a double-cigar-shaped Bose-Einstein condensate with a
negative scattering length, in which the active ``cigar'' is an atom laser. The
replacement of the diffusion term by the nonlinear loss is principally
important, as diffusion does not occur in these physical media, while nonlinear
loss is possible. A stability region for solitary pulses is found in the
system's parameter space by means of direct simulations. One border of the
region is also found in an analytical form by means of a perturbation theory.
Moving pulses are studied too. It is concluded that collisions between them are
completely elastic, provided that the relative velocity is not too small. The
pulses withstand multiple tunneling through potential barriers. Robust
quantum-rachet regimes of motion of the pulse in a time-periodic asymmetric
potential are found as well.


title:
The incompressible Navier-Stokes for the nonlinear discrete velocity
  models

abstract:
  We establish the incompressible Navier--Stokes limit for the discrete
velocity model of the Boltzmann equation in any dimension of the physical
space, for densities which remain in a suitable small neighborhood of the
global Maxwellian.
  Appropriately scaled families solutions of discrete Boltzmann equation are
shown to have fluctuations that locally in time converge strongly to a limit
governed by a solution of Incompressible Navier--Stokes provided that the
initial fluctuation is smooth, and converges to appropriate initial data.
  As applications of our results, we study the Carleman model and the
one-dimensional Broadwell model.


title:
The influence of quantum field fluctuations on chaotic dynamics of
  Yang--Mills system

abstract:
  On example of the model field system we demonstrate that quantum fluctuations
of non-abelian gauge fields leading to radiative corrections to Higgs potential
and spontaneous symmetry breaking can generate order region in phase space of
inherently chaotic classical field system. We demonstrate on the example of
another model field system that quantum fluctuations do not influence on the
chaotic dynamics of non-abelian Yang--Mills fields if the ratio of bare
coupling constants of Yang--Mills and Higgs fields is larger then some critical
value. This critical value is estimated.


title:
Pole solutions in the case of problems of flame front propagation and
  Saffman-Teylor "finger" formation without surface tension: open problems and
  possible ways of their solutions

abstract:
  Some physical problems as flame front propagation or Laplacian growth without
surface tension have nice analytical solutions which replace its complex
integro-differential motion equations by simple differential equations of poles
motion in a complex plane. Investigation of these equation was the main topic
of Kupervasser Oleg Ph.D. Thesis[7]. Some very interesting open problems were
hanged up there. Here we give these open problems and possible ways of their
solutions.


title: Influence of moving breathers on vacancies migration

abstract:
  A vacancy defect is described by a Frenkel--Kontorova model with a
discommensuration. This vacancy can migrate when interacts with a moving
breather. We establish that the width of the interaction potential must be
larger than a threshold value in order that the vacancy can move forward. This
value is related to the existence of a breather centred at the particles
adjacent to the vacancy.


title:
On Conservative and Monotone One-dimensional Cellular Automata and Their
  Particle Representation

abstract:
  Number-conserving (or {\em conservative}) cellular automata have been used in
several contexts, in particular traffic models, where it is natural to think
about them as systems of interacting particles. In this article we consider
several issues concerning one-dimensional cellular automata which are
conservative, monotone (specially ``non-increasing''), or that allow a weaker
kind of conservative dynamics. We introduce a formalism of ``particle
automata'', and discuss several properties that they may exhibit, some of
which, like anticipation and momentum preservation, happen to be intrinsic to
the conservative CA they represent. For monotone CA we give a characterization,
and then show that they too are equivalent to the corresponding class of
particle automata. Finally, we show how to determine, for a given CA and a
given integer $b$, whether its states admit a $b$-neighborhood-dependent
relabelling whose sum is conserved by the CA iteration; this can be used to
uncover conservative principles and particle-like behavior underlying the
dynamics of some CA. Complements at {\tt http://www.dim.uchile.cl/\verb'
'anmoreir/ncca}


title: Solitonic lattices in photorefractive crystals

abstract:
  Two-dimensional spatial solitonic lattices are generated and investigated
experimentally and numerically in an SBN:Ce crystal. An enhanced stability of
these lattices is achieved by exploiting the anisotropy of coherent soliton
interaction, in particular the relative phase between soliton rows.
Manipulation of individual soliton channels is achieved by use of supplementary
control beams.


title: Lyaupunov Exponents, Path-Integrals and Forms

abstract:
  In this paper we use a path-integral approach to represent the Lyapunov
exponents of both deterministic and stochastic dynamical systems. In both cases
the relevant correlation functions are obtained from a (one-dimensional)
supersymmetric field theory whose Hamiltonian, in the deterministic case,
coincides with the Lie-derivative of the associated Hamiltonian flow. The
generalized Lyapunov exponents turn out to be related to the partition
functions of the respective super-Hamiltonian restricted to the spaces of fixed
form-degree.


title: Stochastic 'fuzzy confinement' of intrinsic localised modes

abstract:
  The long time diffusive behaviour of intrinsic localised modes (discrete
breathers) in the discrete damped-driven sine-Gordon chain under Gaussian white
noise (to simulate temperature) is studied. We present a theoretical model for
an approximate description of the diffusion, derive an expression for the
diffusion constant and compare with results from simulations. It turns out that
an increase of the temperature inhibits the diffusive motion in such a way that
the breather, propagating with a well-defined velocity in the noise-free case,
is almost pinned. As all physical processes in the real world occur at
temperatures T > 0, these results also have a bearing on the experimental
detection of mobile breathers, e.g. in parallel arrays of Josephson junctions.


title:
Evolution of unidirectional random waves in shallow water (the Korteweg
  - de Vries model)

abstract:
  Numerical simulations of the unidirectional random waves are performed within
the Korteweg -de Vries equation to investigate the statistical properties of
surface gravity waves in shallow water. Nonlinear evolution shows the
relaxation of the initial state to the quasi-stationary state differs from the
Gaussian distribution. Significant nonlinear effects lead to the asymmetry in
the wave field with bigger crests amplitudes and increasing of large wave
contribution to the total distribution, what gives the rise of the amplitude
probability, exceeded the Rayleigh distribution. The spectrum shifts in low
frequencies with the almost uniform distribution. The obtained results of the
nonlinear evolution of shallow-water waves are compared with known properties
of deep-water waves in the framework of the nonlinear Schrodinger equation.


title:
On triple correlations in isotropic electron magnetohydrodynamic
  turbulence

abstract:
  The evolution of the correlation characteristics in three-dimensional
isotropic electron magnetohydrodynamic turbulence is investigated. Universal
exact relations between the longitudinal and transverse two-point triple
correlations of the components of the fluctuational magnetic fields and the
rates of dissipation of the magnetic helicity and energy are obtained in the
inertial range.


title:
Statistical Mechanics: A Possible Model for Market-based Electric Power
  Control

abstract:
  Statistical mechanics provides a useful analog for understanding the behavior
of complex adaptive systems, including electric power markets and the power
systems they intend to govern. Market-based control is founded on the
conjecture that the regulation of complex systems based on price-mediated
strategies (e.g., auctions, markets) results in an optimal allocation of
resources and emergent optimal system control. This paper discusses the
derivation and some illustrative applications of a first-principles model of
market-based system dynamics based on strict analogies to statistical
mechanics.


title: Approach to self-similarity in Smoluchowski's coagulation equations

abstract:
  We consider the approach to self-similarity (or dynamical scaling) in
Smoluchowski's equations of coagulation for the solvable kernels $K(x,y)=2$,
$x+y$ and $xy$. In addition to the known self-similar solutions with
exponential tails, there are one-parameter families of solutions with algebraic
decay, whose form is related to heavy-tailed distributions well-known in
probability theory. For K=2 the size distribution is Mittag-Leffler, and for
$K=x+y$ and $K=xy$ it is a power-law rescaling of a maximally skewed
$\alpha$-stable Levy distribution. We characterize completely the domains of
attraction of all self-similar solutions under weak convergence of measures.
Our results are analogous to the classical characterization of stable
distributions in probability theory. The proofs are simple, relying on the
Laplace transform and a fundamental rigidity lemma for scaling limits.


title:
Dynamical scaling in Smoluchowski's coagulation equations: uniform
  convergence

abstract:
  We consider the approach to self-similarity (or dynamical scaling) in
Smoluchowski's coagulation equations for the solvable kernels K(x,y)=2, x+y and
xy. We prove the uniform convergence of densities to the self-similar solution
with exponential tails under the regularity hypothesis that a suitable moment
have an integrable Fourier transform. For the discrete equations we prove
uniform convergence under optimal moment hypotheses. Our results are completely
analogous to classical local convergence theorems for the normal law in
probability theory. The proofs rely on the Fourier inversion formula and the
solution by the method of characteristics for the Laplace transform.


title: Fractal clustering of inertial particles in random flows

abstract:
  It is shown that preferential concentrations of inertial (finite-size)
particle suspensions in turbulent flows follow from the dissipative nature of
their dynamics. In phase space, particle trajectories converge toward a
dynamical fractal attractor. Below a critical Stokes number (non-dimensional
viscous friction time), the projection on position space is a dynamical fractal
cluster; above this number, particles are space filling. Numerical simulations
and semi-heuristic theory illustrating such effects are presented for a simple
model of inertial particle dynamics.


title:
Exploring classical phase space structures of nearly integrable and
  mixed quantum systems via parametric variation

abstract:
  The correlation between overlap intensities and level velocities has been
introduced as a sensitive measure capable of revealing phase space
localization. Previously applied to chaotic quantum systems, here we extend the
theory to near-integrable and mixed quantum systems. This measure is useful in
the latter cases because it has the ability to highlight certain phase space
structures depending upon the perturbation used to parametrically vary the
Hamiltonian. A detailed semiclassical theory is presented relating the
correlation coefficient to the phase space weighted derivatives of the
classical action. In the process, we confront the question of whether the
Hannay-Ozorio de Almeida sum rules are simply extendable to mixed phase space
systems. In addition, the hbar-scalings of the correlation coefficient and
relevant quantities are derived for nearly integrable systems. Excellent
agreement is found between the theory and the results for integrable billiards
as well as for the standard map.


title:
Real Jacobian Elliptic Function Parametrizations for a Genuinely
  Asymmetric Biquadratic Curve

abstract:
  We give real Jacobian elliptic function parametrizations for a genuinely
asymmetric biquadratic curve where the variables and parameters are real.


title: Three Dimensional Integrable Mappings

abstract: We derive three-dimensional integrable mappings which have two invariants.


title: On Whitham's Averaging Method

abstract:
  We compare Whitham averaging method using Lagrangian and Hamiltonian
formalism.


title: Asymmetric time correlations in turbulent shear flows

abstract:
  Cross correlations between normal and downstream velocity fluctuations in a
turbulent shear flow are shown to carry information about the non-normal
amplification process. The creation of spanwise modulated streaks by downstream
vortices implies an asymmetry in temporal correlation functions. We verify this
in numerical simulations in shear flows with $Re_\lambda\approx 100$.


title: A Model for Prejudiced Learning in Noisy Environments

abstract:
  Based on the heuristics that maintaining presumptions can be beneficial in
uncertain environments, we propose a set of basic axioms for learning systems
to incorporate the concept of prejudice. The simplest, memoryless model of a
deterministic learning rule obeying the axioms is constructed, and shown to be
equivalent to the logistic map. The system's performance is analysed in an
environment in which it is subject to external randomness, weighing learning
defectiveness against stability gained. The corresponding random dynamical
system with inhomogeneous, additive noise is studied, and shown to exhibit the
phenomena of noise induced stability and stochastic bifurcations. The overall
results allow for the interpretation that prejudice in uncertain environments
entails a considerable portion of stubbornness as a secondary phenomenon.


title: Ray chaos and ray clustering in an ocean waveguide

abstract:
  We consider ray propagation in a waveguide with a designed sound-speed
profile perturbed by a range-dependent perturbation caused by internal waves in
deep ocean environments. The Hamiltonian formalism in terms of the action and
angle variables is applied to study nonlinear ray dynamics with two
sound-channel models and three perturbation models: a single-mode perturbation,
a random-like sound-speed fluctuations, and a mixed perturbation. In the
integrable limit without any perturbation, we derive analytical expressions for
ray arrival times and timefronts at a given range, the main measurable
characteristics in field experiments in the ocean. In the presence of a
single-mode perturbation, ray chaos is shown to arise as a result of
overlapping nonlinear ray-medium resonances. Poincar\'e maps, plots of
variations of the action per a ray cycle length, and plots with rays escaping
the channel reveal inhomogeneous structure of the underlying phase space with
remarkable zones of stability where stable coherent ray clusters may be formed.
We demonstrate the possibility of determining the wavelength of the
perturbation mode from the arrival time distribution under conditions of ray
chaos. It is surprising that coherent ray clusters, consisting of fans of rays
which propagate over long ranges with close dynamical characteristics, can
survive under a random-like multiplicative perturbation modelling sound-speed
fluctuations caused by a wide spectrum of internal waves.


title: Deterministic uncertainty

abstract:
  Equations of motion with delays naturally emerge in the analysis of complex
biological control systems which are organized around biochemically mediated
feedback interactions. We study the properties of a Mackey-Glass-type nonlinear
map with delay -- the deterministic part of the stochastic cerebral blood flow
map (CBFM) recently introduced to elucidate the scaling properties of cerebral
hemodynamics. We point out the existence of deterministic and nondeterministic
subspaces in the reconstructed phase space of delay-difference equations and
discuss the problem of detection of nonlinearities in time series generated by
such dynamical systems.


title: A Cantor set of tori with monodromy near a focus-focus singularity

abstract:
  We write down an asymptotic expression for action coordinates in an
integrable Hamiltonian system with a focus-focus equilibrium. From the
singularity in the actions we deduce that the Arnol'd determinant grows
infinitely large near the pinched torus. Moreover, we prove that it is possible
to globally parametrise the Liouville tori by their frequencies. If one
perturbs this integrable system, then the KAM tori form a Whitney smooth
family: they can be smoothly interpolated by a torus bundle that is
diffeomorphic to the bundle of Liouville tori of the unperturbed integrable
system. As is well-known, this bundle of Liouville tori is not trivial. Our
result implies that the KAM tori have monodromy. In semi-classical quantum
mechanics, quantisation rules select sequences of KAM tori that correspond to
quantum levels. Hence a global labeling of quantum levels by two quantum
numbers is not possible.


title: Self-similar turbulent dynamo

abstract:
  The amplification of magnetic fields in a highly conducting fluid is studied
numerically. During growth, the magnetic field is spatially intermittent: it
does not uniformly fill the volume, but is concentrated in long thin folded
structures. Contrary to a commonly held view, intermittency of the folded field
does not increase indefinitely throughout the growth stage if diffusion is
present. Instead, as we show, the probability-density function (PDF) of the
field strength becomes self-similar. The normalized moments increase with
magnetic Prandtl number in a powerlike fashion. We argue that the
self-similarity is to be expected with a finite flow scale and system size. In
the nonlinear saturated state, intermittency is reduced and the PDF is
exponential. Parallels are noted with self-similar behavior recently observed
for passive-scalar mixing and for map dynamos.


title: The Extended Toda Hierarchy

abstract:
  We present the Lax pair formalism for certain extension of the continuous
limit of the classical Toda lattice hierarchy, provide a well defined notion of
tau function for its solutions, and give an explicit formulation of the
relationship between the $CP^1$ topological sigma model and the extended Toda
hierarchy. We also establish an equivalence of the latter with certain
extension of the nonlinear Schr\"odinger hierarchy.


title:
On dissipationless shock waves in a discrete nonlinear Schr\"odinger
  equation

abstract:
  It is shown that the generalized discrete nonlinear Schr\"odinger equation
can be reduced in a small amplitude approximation to the KdV, mKdV, KdV(2) or
the fifth-order KdV equations, depending on values of the parameters. In
dispersionless limit these equations lead to wave breaking phenomenon for
general enough initial conditions, and, after taking into account small
dispersion effects, result in formation of dissipationless shock waves. The
Whitham theory of modulations of nonlinear waves is used for analytical
description of such waves.


title: Different Time Scales in Wave Function Intensity Statistics

abstract:
  Unstable periodic orbits scar wave functions in chaotic systems. This also
influences the associated spectra, that follow the otherwise universal
Porter--Thomas intensity distribution. We show here how this deviation extend
to other longer periodic orbits sharing some common dynamical characteristics.
This indicates that the quantum mechanics of the system can be described quite
simply with few orbits, up to the resolution associated to the corresponding
length.


title: Level velocity statistics of hyperbolic chaos

abstract:
  A generalized version of standard map is quantized as a model of quantum
chaos. It is shown that, in hyperbolic chaotic regime, second moment of quantum
level velocity is $\sim 1/\hbar$ as predicted by the random matrix theory.


title: Travel time stability in weakly range-dependent sound channels

abstract:
  Travel time stability is investigated in environments consisting of a
range-independent background sound-speed profile on which a highly structured
range-dependent perturbation is superimposed. The stability of both
unconstrained and constrained (eigenray) travel times are considered. Both
general theoretical arguments and analytical estimates of time spreads suggest
that travel time stability is largely controlled by a property $\omega
^{\prime}$ of the background sound speed profile. Here, $2\pi/\omega (I)$ is
the range of a ray double loop and $I$ is the ray action variable. Numerical
results for both volume scattering by internal waves in deep ocean environments
and rough surface scattering in upward refracting environments are shown to
confirm the expectation that travel time stability is largely controlled by
$\omega ^{\prime}$.


title: Network dependence of strong reciprocity

abstract:
  Experimental evidence suggests that human decisions involve a mixture of
self-interest and internalized social norms which cannot be accounted for by
the Nash equilibrium behavior of Homo Oeconomicus. This led to the notion of
strong reciprocity (or altruistic punishment) to capture the human trait
leading an individual to punish norm violators at a cost to himself. For a
population with small autonomous groups with collective monitoring, the
interplay of intra- and intergroup dynamics shows this to be an adaptive trait,
although not fully invasive of a selfish population. However, the absence of
collective monitoring in a larger society changes the evolution dynamics.
Clustering seems to be the network parameter that controls maintenance and
evolution of the reciprocator trait.


title:
Energy Trapping in Loaded String Models with Long- and Short-Range
  Couplings

abstract:
  This note illustrates the possibility in simple loaded string models of
trapping most of the system energy in a single degree of freedom for very long
times, demonstrating in particular that the robustness of the trapping is
enhanced by increasing the `connectance' of the system, i.e., the extent to
which many degrees of freedom are coupled directly by the interaction
Hamiltonian, and/or the strength of the couplings.


title: Recurrence of fidelity in near integrable systems

abstract:
  Within the framework of simple perturbation theory, recurrence time of
quantum fidelity is related to the period of the classical motion. This
indicates the possibility of recurrence in near integrable systems. We have
studied such possibility in detail with the kicked rotor as an example. In
accordance with the correspondence principle, recurrence is observed when the
underlying classical dynamics is well approximated by the harmonic oscillator.
Quantum revivals of fidelity is noted in the interior of resonances, while
classical-quantum correspondence of fidelity is seen to be very short for
states initially in the rotational KAM region.


title: Crossover Behaviour of 3-Species Systems with Mutations or Migrations

abstract:
  We study the ABC model in the cyclic competition and neutral drift versions,
with mutations and migrations introduced into the model. When stochastic
phenomena are taken into account, there are three distinct regimes in the
model. (i) In the "fixation" regime, the first extinction time scales with the
system size N and has an exponential distribution, with an exponent that
depends on the mutation/migration probability per particle. (ii) In the
"diversity" regime, the order parameter remains nonzero for very long times,
and becomes zero only rarely, almost never for large system sizes. (iii) In the
critical regime, the first extinction time has a power-law distribution with
exponent -1. The transition corresponds to a crossover from diffusive behaviour
to Gaussian fluctuations about a stable solution. The analytical results are
checked against computer simulations of the model.


title: Sound and Image Processing with Optical Biocomputers

abstract:
  This paper introduces a new method to perform signal processing using cells
or atoms to synthesize variation in digital data stored in optical format.
Interfacing small-scale biological or chemical systems with information stored
on CD, DVD and related disc media has been termed molecular media. This is
defined as micron or nanoscale interference and/or augmentation of optical data
translation with a specific focus on sound and image processing. The principle
of this approach uses cells with different morphology and size, or atom
clusters to geometrically alter the way that binary information is translated
into an output signal. These experiments demonstrate that living and
non-living, micron-sized cellular systems like bacteria, fungi or yeast and
chemical particles, clusters, emulsions or atom arrays having micron or
nanosize dimensions can be successfully interfaced with traditional computing
devices. This results in multiple hybrid, yet functional information processing
devices that can be understood as a molecular computer that demonstrates a
plurality of useful functions.


title:
Intermittency in Turbulence: computing the scaling exponents in shell
  models

abstract:
  We discuss a stochastic closure for the equation of motion satisfied by
multi-scale correlation functions in the framework of shell models of
turbulence. We give a systematic procedure to calculate the anomalous scaling
exponents of structure functions by using the exact constraints imposed by the
equation of motion. We present an explicit calculation for fifth order scaling
exponent at varying the free parameter entering in the non-linear term of the
model. The same method applied to the case of shell models for Kraichnan
passive scalar provides a connection between the concept of zero-modes and
time-dependent cascade processes.


title: Extended Toda Lattice

abstract:
  We introduce nonlocal flows that commute with those of the classical Toda
hierarchy. We define a logarithm of the difference Lax operator and use it to
obtain a Lax representation of the new flows.


title:
Communication and Synchronization in Disconnected Networks with Dynamic
  Topology: Moving Neighborhood Networks

abstract:
  We consider systems that are well modelled as a networks that evolve in time,
which we call {\it Moving Neighborhood Networks}. These models are relevant in
studying cooperative behavior of swarms and other phenomena where emergent
interactions arise from ad hoc networks. In a natural way, the time-averaged
degree distribution gives rise to a scale-free network. Simulations show that
although the network may have many noncommunicating components, the
time-averaged communication is sufficient to yield robust synchronization of
chaotic oscillators.


title: On asymptotically equivalent shallow water wave equations

abstract:
  The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at
linear order in the asymptotic expansion for unidirectional shallow water
waves. However, at quadratic order, this asymptotic expansion produces an
entire {\it family} of shallow water wave equations that are asymptotically
equivalent to each other, under a group of nonlinear, nonlocal, normal-form
transformations introduced by Kodama in combination with the application of the
Helmholtz-operator. These Kodama-Helmholtz transformations are used to present
connections between shallow water waves, the integrable 5th-order Korteweg-de
Vries equation, and a generalization of the Camassa-Holm (CH) equation that
contains an additional integrable case. The dispersion relation of the full
water wave problem and any equation in this family agree to 5th order. The
travelling wave solutions of the CH equation are shown to agree to 5th order
with the exact solution.


title:
Continuum description of finite-size particles advected by external
  flows. The effect of collisions

abstract:
  The equation of the density field of an assembly of macroscopic particles
advected by a hydrodynamic flow is derived from the microscopic description of
the system. This equation allows to recognize the role and the relative
importance of the different microscopic processes implicit in the model: the
driving of the external flow, the inertia of the particles, and the collisions
among them.
  The validity of the density description is confirmed by comparisons of
numerical studies of the continuum equation with Direct Simulation Monte Carlo
(DSMC) simulations of hard disks advected by a chaotic flow. We show that the
collisions have two competing roles: a dispersing-like effect and a clustering
effect (even for elastic collisions). An unexpected feature is also observed in
the system: the presence of collisions can reverse the effect of inertia, so
that grains with lower inertia are more clusterized.


title: Complexity Characterization of Dynamical Systems Through Predictability

abstract:
  Some aspects of the predictability problem in dynamical systems are reviewed.
The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon
entropy and algorithmic complexity is discussed. In particular, we emphasize
how a characterization of the unpredictability of a system gives a measure of
its complexity. A special attention is devoted to finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system.


title:
A propensity criterion for networking in an array of coupled chaotic
  systems

abstract:
  We examine the mutual synchronization of a one dimensional chain of chaotic
identical objects in the presence of a stimulus applied to the first site. We
first describe the characteristics of the local elements, and then the process
whereby a global nontrivial behaviour emerges. A propensity criterion for
networking is introduced, consisting in the coexistence within the attractor of
a localized chaotic region, which displays high sensitivity to external
stimuli,and an island of stability, which provides a reliable coupling signal
to the neighbors in the chain. Based on this criterion we compare homoclinic
chaos, recently explored in lasers and conjectured to be typical of a single
neuron, with Lorenz chaos.


title: Methods and Techniques of Complex Systems Science: An Overview

abstract:
  In this chapter, I review the main methods and techniques of complex systems
science. As a first step, I distinguish among the broad patterns which recur
across complex systems, the topics complex systems science commonly studies,
the tools employed, and the foundational science of complex systems. The focus
of this chapter is overwhelmingly on the third heading, that of tools. These in
turn divide, roughly, into tools for analyzing data, tools for constructing and
evaluating models, and tools for measuring complexity. I discuss the principles
of statistical learning and model selection; time series analysis; cellular
automata; agent-based models; the evaluation of complex-systems models;
information theory; and ways of measuring complexity. Throughout, I give only
rough outlines of techniques, so that readers, confronted with new problems,
will have a sense of which ones might be suitable, and which ones definitely
are not.

categories: nlin.AO cond-mat.stat-mech nlin.CD nlin.CG physics.data-an q-bio.QM


title:
Coarse Projective kMC Integration: Forward/Reverse Initial and Boundary
  Value Problems

abstract:
  In "equation-free" multiscale computation a dynamic model is given at a fine,
microscopic level; yet we believe that its coarse-grained, macroscopic dynamics
can be described by closed equations involving only coarse variables. These
variables are typically various low-order moments of the distributions evolved
through the microscopic model. We consider the problem of integrating these
unavailable equations by acting directly on kinetic Monte Carlo microscopic
simulators, thus circumventing their derivation in closed form. In particular,
we use projective multi-step integration to solve the coarse initial value
problem forward in time as well as backward in time (under certain conditions).
Macroscopic trajectories are thus traced back to unstable, source-type, and
even sometimes saddle-like stationary points, even though the microscopic
simulator only evolves forward in time. We also demonstrate the use of such
projective integrators in a shooting boundary value problem formulation for the
computation of "coarse limit cycles" of the macroscopic behavior, and the
approximation of their stability through estimates of the leading "coarse
Floquet multipliers".


title:
Asymptotic Behavior of a Model of Characteristic Earthquakes and its
  Implications for Regional Seismicity

abstract:
  The recently introduced Minimalist Model [Vazquez- Prada et al., 2002] of
characteristic earthquakes provides a simple representation of the seismicity
originated in a sin- gle fault. Here, we first characterize the properties of
this model for large systems. Then, assuming, as it has been observed, that the
size of the faults in a big enough region is fractally distributed, with a
fractal dimension D = 1.7, we describe the total seismicity produced in that
region. The resulting catalogue accounts for the addition of all the
characteristic events triggered in the faults and also includes the rest of
non-characteristic earthquakes. The global accumulated size-frequency relation
correctly describes a Gutenberg-Richter form, with a b exponent of around 0.7.


title:
Complex wave dynamics in a rugby microballs with semiconductor active
  mirrors

abstract:
  The optical implementation of complex network is considered for laser cavity
formed by active mirrors deposited on the surface of ellipsoidal semiconductor
microball having size of about 1 mm, whose eccentricity close to 0.5 and
equivalent optical circuit, containing amplifying and absorbing media in
Fourier-conjugated planes. Such rugby-ball microlaser has self-imaging cavity
with efficient diffractive interconnections between different points of the
mirrors. We show how the spatiotemporal dynamics of this nonlinear resonator is
controlled by the geometry of gain-losses distribution making possible the
analog modeling of the several generic nonlinear evolution equations.


title: Geometry and Topology of Escape I: Epistrophes

abstract:
  We consider a dynamical system given by an area-preserving map on a
two-dimensional phase plane and consider a one-dimensional line of initial
conditions within this plane. We record the number of iterates it takes a
trajectory to escape from a bounded region of the plane as a function along the
line of initial conditions, forming an ``escape-time plot''. For a chaotic
system, this plot is in general not a smooth function, but rather has many
singularities at which the escape time is infinite; these singularities form a
complicated fractal set. In this article we prove the existence of regular
repeated sequences, called ``epistrophes'', which occur at all levels of
resolution within the escape-time plot. (The word ``epistrophe'' comes from
rhetoric and means ``a repeated ending following a variable beginning''.) The
epistrophes give the escape-time plot a certain self-similarity, called
``epistrophic'' self-similarity, which need not imply either strict or
asymptotic self-similarity.


title: Geometry and Topology of Escape II: Homotopic Lobe Dynamics

abstract:
  We continue our study of the fractal structure of escape-time plots for
chaotic maps. In the preceding paper, we showed that the escape-time plot
contains regular sequences of successive escape segments, called epistrophes,
which converge geometrically upon each endpoint of every escape segment. In the
present paper, we use topological techniques to: (1) show that there exists a
minimal required set of escape segments within the escape-time plot; (2)
develop an algorithm which computes this minimal set; (3) show that the minimal
set eventually displays a recursive structure governed by an ``Epistrophe Start
Rule'': a new epistrophe is spawned Delta = D+1 iterates after the segment to
which it converges, where D is the minimum delay time of the complex.


title:
A geometric approach to the separability of the Neumann-Rosochatius
  system

abstract:
  We study the separability of the Neumann-Rosochatius system on the
n-dimensional sphere using the geometry of bi-Hamiltonian manifolds. Its
well-known separation variables are recovered by means of a separability
condition relating the Hamiltonian with a suitable (1,1) tensor field on the
sphere. This also allows us to iteratively construct the integrals of motion of
the system.


title: The structure of the 3D-vortex lattices in microchip laser resonator

abstract:
  The interaction of the optical vortices leads to formation of spatially
periodic structures of electromagnetic field which obey the scaling law typical
to Talbot self-imaging.


title: Numerical analysis of solitary waves interaction in nonlinear medium

abstract:
  Using numerical modeling investigated interaction of solitary waves
(solitons) of the regularized long wave equation. For reception the stable
model of the nonlinear medium are used methods of the linear prediction and
progressive approximation. By modeling was determined that depending on ratio
of velocities of the solitons and the form of highest derivatives balance is
possible self-organization of the medium nonequilibrium state as formation of
shock waves and stable on the form solitary waves, created as a result of full
or partial mutual penetration of the solitons. Is possible also aggregation of
the solitons in third wave. The shock waves can pass into other possible
resonance state as a wave front with stable amplitude, which precedes
developing in singularity negative front.


title: Synchronization and oscillator death in oscillatory media with stirring

abstract:
  The effect of stirring in an inhomogeneous oscillatory medium is
investigated. We show that the stirring rate can control the macroscopic
behavior of the system producing collective oscillations (synchronization) or
complete quenching of the oscillations (oscillator death). We interpret the
homogenization rate due to mixing as a measure of global coupling and compare
the phase diagrams of stirred oscillatory media and of populations of globally
coupled oscillators.


title:
Lyapunov Spreading of Semi-classical Wave Packets for the Lorentz Gas:
  Theory and Applications

abstract:
  We consider the quantum mechanical propagator for a particle moving in a
$d$-dimensional Lorentz gas, with fixed, hard sphere scatterers. To evaluate
this propagator in the semi-classical region, and for times less than the
Ehrenfest time, we express its effect on an initial Gaussian wave packet in
terms of quantities analogous to those used to describe the exponential
separation of trajectories in the classical version of this system. This result
relates the spread of the wave packet to the rate of separation of classical
trajectories, characterized by positive Lyapunov exponents. We consider
applications of these results, first to illustrate the behavior of the
wave-packet auto-correlation functions for wave packets on periodic orbits. The
auto-correlation function can be related to the fidelity, or Loschmidt echo,
for the special case that the perturbation is a small change in the mass of the
particle. An exact expression for the fidelity, appropriate for this
perturbation, leads to an analytical result valid over very long time
intervals, inversely proportional to the size of the mass perturbation. For
such perturbations, we then calculate the long-time echo for semi-classical
wave packets on periodic orbits. This paper also corrects an earlier
calculation for a quantum echo, included in a previous version of this paper.
We explain the reasons for this correction.


title:
The initial boundary value problem on the segment for the Nonlinear
  Schr\"odinger equation; the algebro-geometric approach. I

abstract:
  This is the first of a series of papers devoted to the study of classical
initial-boundary value problems of Dirichlet, Neumann and mixed type for the
Nonlinear Schr\"odinger equation on the segment. Considering proper periodic
discontinuous extensions of the profile, generated by suitable point-like
sources, we show that the above boundary value problems can be rewritten as
nonlinear dynamical systems for suitable sets of algebro-geometric spectral
data, generalizing the classical Dubrovin equations.
  In this paper we consider, as a first illustration of the above method, the
case of the Dirichlet problem on the segment with zero-boundary value at one
end, and we show that the corresponding dynamical system for the spectral data
can be written as a system of ODEs with algebraic right-hand side.

categories: nlin.SI hep-th math-ph math.AG math.AP math.MP


title:
The Phase Structure in the Evolution of Globally coupled Spatial
  Prisoner's Dilemma

abstract:
  We propose two new evolutionary rules that is not mimic evolution of
strategies based on the spatial Prisoner's Dilemma (PD). The former follows the
selfish evolutionary rule and then the coexistence phase appears with weak
phase transition with mutations. The second rule is that of globally coupled
spatial PD where the strategy of an agent is taken so that the total payoff of
the whole system increases. Then it is natulally suspected that all agents will
become cooperators in general. We, however, find that situations is rather
complerx. We find two critical points within the usual conditions in the
parameters included the payoff matrix in the PD. Further as we break the
condition without losing PD condition, we meet two more essential curitical
points. We discuss the areas between these critical points in the parameter
space and uncover the various properties by simulating on computer. We show
that in some parameter regions, strange properties, that is, like chaotic
behaveor appear and the coexistence between defectors and cooperators appears
in the large parameter region. Thus we point out that usual non-essential
condition among the parameters in the payoff matrix imposed on the PD privent
from recovering rich structures in the evolutionally spatial PS.


title:
Simulation of the weighted spatio-structured three cornered game and its
  satatistical properties

abstract:
  In this article, we study the evolution of the strategies in a sort of
weighted three cornered game, Glico Game. This game has three strategies in
each agent and three parameters in the payoff matrix.
  We divide the parameter region into 8 areas and explore its qualitative and
quantitative properties of the evolution of strategies which include Haming
distance, correlation functions and the population sequence of each strategy.
When the three payoff parameters take nearly values each other, the complex
patterns mainly appear and as the difference of parameter values are enlarged,
some peculiar strategy trends to be dominant gradually. An interesting fact is
that the strongest strategy in the payoff do not necessarily dominate the
latice world. There are even the cases where the weakest strategy in the payof
dominate. Generally many cases have some period much shoter than the dimension
of the configuration space.


title:
Synchronized clusters in coupled map networks: Self-organized and driven
  phase synchronization

abstract:
  We study the synchronization of coupled maps on a variety of networks
including regular one and two dimensional networks, scale free networks, small
world networks, tree networks, and random networks. For small coupling
strengths nodes show turbulent behavior but form phase synchronized clusters as
coupling increases. We identify two different ways of cluster formation,
self-organized clusters which have mostly intra-cluster couplings and driven
clusters which have mostly inter-cluster couplings. The synchronized clusters
may be of dominant self-organized type, dominant driven type or mixed type
depending on the type of network and the parameters of the dynamics. There are
some nodes of the floating type that show intermittent behaviour between
getting attached to some clusters and evolving independently. The residence
times of a floating node in a synchronized cluster show an exponential
distribution. We define different states of the coupled dynamics by considering
the number and type of synchronized clusters. For the local dynamics governed
by the logistic map we study the phase diagram in the plane of the coupling
constant ($\epsilon$) and the logistic map parameter ($\mu$). For large
coupling strengths and nonlinear coupling we find that the scale free networks
and the Caley tree networks lead to better cluster formation than the other
types of networks with the same average connectivity. For most of our study we
use the number of connections of the order of the number of nodes which allows
us to distinguish between the two mechanisms of cluster formation. As the
number of connections increases the number of nodes forming clusters and the
size of the clusters in general increase.


title:
Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger
  hierarchy

abstract:
  Two "elliptic analogues'' of the nonlinear Schr\"odinger hiererchy are
constructed, and their status in the Grassmannian perspective of soliton
equations is elucidated. In addition to the usual fields $u,v$, these elliptic
analogues have new dynamical variables called ``Tyurin parameters,'' which are
connected with a family of vector bundles over the elliptic curve in
consideration. The zero-curvature equations of these systems are formulated by
a sequence of $2 \times 2$ matrices $A_n(z)$, $n = 1,2,...$, of elliptic
functions. In addition to a fixed pole at $z = 0$, these matrices have several
extra poles. Tyurin parameters consist of the coordinates of those poles and
some additional parameters that describe the structure of $A_n(z)$'s. Two
distinct solutions of the auxiliary linear equations are constructed, and shown
to form a Riemann-Hilbert pair with degeneration points. The Riemann-Hilbert
pair is used to define a mapping to an infinite dimensional Grassmann variety.
The elliptic analogues of the nonlinear Schr\"odinger hierarchy are thereby
mapped to a simple dynamical system on a special subset of the Grassmann
variety.


title:
Effect of base-pair inhomogeneities on charge transport along DNA
  mediated by twist and radial polarons

abstract:
  Some recent results for a three--dimensional, semi--classical, tight--binding
model for DNA show that there are two types of polarons, named radial and twist
polarons, that can transport charge along the DNA molecule. However, the
existence of two types of base pairs in real DNA, makes it crucial to find out
if charge transport also exist in DNA chains with different base pairs. In this
paper we address this problem in its simple case, an homogeneous chain except
for a single different base pair, what we call a base-pair inhomogeneity, and
its effect on charge transport. Radial polarons experience either reflection or
trapping. However, twist polarons are good candidates for charge transport
along real DNA. This transport is also very robust with respect to weak
parametric and diagonal disorder.


title: Modulated Amplitude Waves in Bose-Einstein Condensates

abstract:
  We analyze spatio-temporal structures in the Gross-Pitaevskii equation to
study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs)
with mean-field interactions. A coherent structure ansatz yields a
parametrically forced nonlinear oscillator, to which we apply Lindstedt's
method and multiple-scale perturbation theory to determine the dependence of
the intensity of periodic orbits (``modulated amplitude waves'') on their wave
number. We explore BEC band structure in detail using Hamiltonian perturbation
theory and supporting numerical simulations.


title: Burgers Equation Revisited

abstract:
  This paper studies the 1D pressureless turbulence (the Burgers equation). It
shows that reliable numerics in this problem is very easy to produce if one
properly discretizes the Burgers equation. The numerics it presents confirms
the 7/2 power law proposed for probability of observing large negative velocity
gradients in this problem. It also suggests that the entire probability
function for the velocity gradients could be universal, perhaps in some
approximate sense. In particular, the probability that the velocity gradient is
negative appears to be $p \approx 0.21 \pm 0.01$ irrespective of the details of
the random force. Finally, I speculate that the theory initially proposed by
Polyakov, with a particular value of the "anomaly" parameter, may indeed be
exact, at least as far as velocity gradients are concerned.


title: Drag Reduction by Polymers in Wall Bounded Turbulence

abstract:
  We address the mechanism of drag reduction by polymers in turbulent wall
bounded flows. On the basis of the equations of fluid mechanics we present a
quantitative derivation of the "maximum drag reduction (MDR) asymptote" which
is the maximum drag reduction attained by polymers. Based on Newtonian
information only we prove the existence of drag reduction, and with one
experimental parameter we reach a quantitative agreement with the experimental
measurements.


title: Poincar\'e Husimi representation of eigenstates in quantum billiards

abstract:
  For the representation of eigenstates on a Poincar\'e section at the boundary
of a billiard different variants have been proposed. We compare these
Poincar\'e Husimi functions, discuss their properties and based on this select
one particularly suited definition. For the mean behaviour of these Poincar\'e
Husimi functions an asymptotic expression is derived, including a uniform
approximation. We establish the relation between the Poincar\'e Husimi
functions and the Husimi function in phase space from which a direct physical
interpretation follows. Using this, a quantum ergodicity theorem for the
Poincar\'e Husimi functions in the case of ergodic systems is shown.


title: Microscopic Space Dimensions and the Discreteness of Time

abstract:
  We present a simple dynamical systems model for the effect of invisible space
dimensions on the visible ones. There are three premises. A: Orbits consist of
flows of probabilities [P].which is the case in the setting of quantum
mechanics. B: The orbits of probabilities are induced by (continuous time)
differential or partial differential equations. C: Observable orbit are flow of
marginal probabilities where the invisible space variables are averaged out. A
theorem is presented which proves that under certain general conditions the
transfer of marginal probabilities cannot be achieved by continuous time
dynamical systems acting on the space of observable variables but that it can
be achieved by discrete time dynamical systems.


title: Synchronized clusters in coupled map networks: Stability analysis

abstract:
  We study self-organized (s-) and driven (d-) synchronization in coupled map
networks for some simple networks, namely two and three node networks and their
natural generalization to globally coupled and complete bipartite networks. We
use both linear stability analysis and Lyapunov function approach for this
study and determine stability conditions for synchronization. We see that most
of the features of coupled dynamics of small networks with two or three nodes,
are carried over to the larger networks of the same type. The phase diagrams
for the networks studied here have features very similar to the different kinds
of networks studied in Ref. \cite{sarika-REA2}. The analysis of the dynamics of
the difference variable corresponding to any two nodes shows that when the two
nodes are in driven synchronization, all the coupling terms cancel out whereas
when they are in self-organized synchronization, the direct coupling term
between the two nodes adds an extra decay term while the other couplings cancel
out.


title:
A realistic example of chaotic tunneling: The hydrogen atom in parallel
  static electric and magnetic fields

abstract:
  Statistics of tunneling rates in the presence of chaotic classical dynamics
is discussed on a realistic example: a hydrogen atom placed in parallel uniform
static electric and magnetic fields, where tunneling is followed by ionization
along the fields direction. Depending on the magnetic quantum number, one may
observe either a standard Porter-Thomas distribution of tunneling rates or, for
strong scarring by a periodic orbit parallel to the external fields, strong
deviations from it. For the latter case, a simple model based on random matrix
theory gives the correct distribution.


title:
Stability of negative ionization fronts: regularization by electric
  screening?

abstract:
  We recently have proposed that a reduced interfacial model for streamer
propagation is able to explain spontaneous branching. Such models require
regularization. In the present paper we investigate how transversal Fourier
modes of a planar ionization front are regularized by the electric screening
length. For a fixed value of the electric field ahead of the front we calculate
the dispersion relation numerically. These results guide the derivation of
analytical asymptotes for arbitrary fields: for small wave-vector k, the growth
rate s(k) grows linearly with k, for large k, it saturates at some positive
plateau value. We give a physical interpretation of these results.


title: Baxters's Q-operators for the simplest q-deformed model

abstract:
  In the present paper we describe the procedure of the Q-operators
construction for the q-deformed model, described by the Lax operator, which is
important to formulate the Bethe ansatz for the Sin-Gordon model. This Lax
operator can also be considered as some massless limit of the Lax operator of
SG model. We constructed two R-operators which are the universal intertwiners
for the Lax operators. The traces of its monodromies over the auxiliary space
are Baxter operators i.e. the operator solutions of T-Q equation. We also found
the intertwining relations which imply the mutual commutativity of the
corresponding Q-operators.


title: Distribution of injected power fluctuations in electroconvection

abstract:
  We report on the distribution spectra of the fluctations in the amount of
power injected into a liquid crystal undergoing electroconvective flow. The
probability distribution functions (PDFs) of the fluctuations as well as the
magnitude of the fluctuations have been determined in a wide range of imposed
stress both for `unconfined' and `confined' flow geometries. These spectra are
compared to those found in other systems held far from equilibrium, and find
that in certain conditions we obtain the ''universal'' PDF form reported in
[Phys. Rev. Lett. {\bf 84}, 3744 (2000)]. Moreover, the PDF approaches this
universal form via an interesting mechanism whereby the distribution's negative
tail evolves towards form in a different manner than the positive tail.


title:
Continuous symmetries of Lagrangians and exact solutions of discrete
  equations

abstract:
  One of the difficulties encountered when studying physical theories in
discrete space-time is that of describing the underlying continuous symmetries
(like Lorentz, or Galilei invariance). One of the ways of addressing this
difficulty is to consider point transformations acting simultaneously on
difference equations and lattices. In a previous article we have classified
ordinary difference schemes invariant under Lie groups of point
transformations. The present article is devoted to an invariant Lagrangian
formalism for scalar single-variable difference schemes. The formalism is used
to obtain first integrals and explicit exact solutions of the schemes.
Equations invariant under two- and three- dimensional groups of Lagrangian
symmetries are considered.


title: Deformed Harry Dym and Hunter-Zheng Equations

abstract:
  We study the deformed Harry Dym and Hunter-Zheng equations with two arbitrary
deformation parameters. These reduce to various other known models in
appropriate limits. We show that both these systems are bi-Hamiltonian with the
same Hamiltonian structures. They are integrable and belong to the same
hierarchy corresponding to positive and negative flows. We present the Lax pair
description for both the systems and construct the conserved charges of
negative order from the Lax operator. For the deformed Harry Dym equation, we
construct the non-standard Lax representation for two special classes of values
of the deformation parameters. In general, we argue that a non-standard
description will involve a pseudo-differential operator of infinite order.


title: Large $N$ limit of integrable models

abstract:
  We consider a large $N$ limit of the Hitchin type integrable systems.
  The first system is the elliptic rotator on $GL_N$ that corresponds to the
Higgs bundle of degree one over an elliptic curve with a marked point. This
system is gauge equivalent to the $N$-body elliptic Calogero-Moser system, that
is derived from the Higgs bundle of degree zero over the same curve. The large
$N$ limit of the former system is the integrable rotator on the group of the
non-commutative torus. Its classical limit leads to the integrable modification
of 2d hydrodynamics on the two-dimensional torus. We also consider the elliptic
Calogero-Moser system on the group of the non-commutative torus and consider
the systems that arise after the reduction to the loop group.


title:
Darboux transformations for 5-point and 7-point self-adjoint schemes and
  an integrable discretization of the 2D Schrodinger operator

abstract:
  With this paper we begin an investigation of difference schemes that possess
Darboux transformations and can be regarded as natural discretizations of
elliptic partial differential equations. We construct, in particular, the
Darboux transformations for the general self adjoint schemes with five and
seven neighbouring points. We also introduce a distinguished discretization of
the two-dimensional stationary Schrodinger equation, described by a 5-point
difference scheme involving two potentials, which admits a Darboux
transformation.


title: $C_{n}^{(1)}$, $D_{n}^{(1)}$ and $A_{2n-1}^{(2)}$ reflection K-matrices

abstract:
  We investigate the possible regular solutions of the boundary Yang-Baxter
equation for the vertex models associated with the $C_{n}^{(1)}$, $D_{n}^{(1)}$
and $A_{2n-1}^{(2)}$ affine Lie algebras. We find three types of solutions with
$n$, $n-1$ and 1 free parameters,respectively. Special cases and all diagonal
solutions are presented separately.


title:
Discrete solitons and soliton-induced dislocations in partially-coherent
  photonic lattices

abstract:
  We investigate the interaction between a light beam and a two-dimensional
photonic lattice that is photo-induced in a photorefractive crystal using
partially coherent light. We demonstrate that this interaction process is
associated with a host of new phenomena including lattice dislocation, lattice
deformation, and creation of structures akin to optical polarons. In addition,
two-dimensional discrete solitons are realized in such partially coherent
photonic lattices.


title:
Whirling Hexagons and Defect Chaos in Hexagonal Non-Boussinesq
  Convection

abstract:
  We study hexagon patterns in non-Boussinesq convection of a thin rotating
layer of water. For realistic parameters and boundary conditions we identify
various linear instabilities of the pattern. We focus on the dynamics arising
from an oscillatory side-band instability that leads to a spatially disordered
chaotic state characterized by oscillating (whirling) hexagons. Using
triangulation we obtain the distribution functions for the number of pentagonal
and heptagonal convection cells. In contrast to the results found for defect
chaos in the complex Ginzburg-Landau equation and in inclined-layer convection,
the distribution functions can show deviations from a squared Poisson
distribution that suggest non-trivial correlations between the defects.


title:
Some relations between Lagrangian models and synthetic random velocity
  fields

abstract:
  We propose an alternative interpretation of Markovian transport models based
on the well-mixedness condition, in terms of the properties of a random
velocity field with second order structure functions scaling linearly in the
space time increments. This interpretation allows direct association of the
drift and noise terms entering the model, with the geometry of the turbulent
fluctuations. In particular, the well known non-uniqueness problem in the
well-mixedness approach is solved in terms of the antisymmetric part of the
velocity correlations; its relation with the presence of non-zero mean helicity
and other geometrical properties of the flow is elucidated. The well-mixedness
condition appears to be a special case of the relation between conditional
velocity increments of the random field and the one-point Eulerian velocity
distribution, allowing generalization of the approach to the transport of
non-tracer quantities. Application to solid particle transport leads to a model
satisfying, in the homogeneous isotropic turbulence case, all the conditions on
the behaviour of the correlation times for the fluid velocity sampled by the
particles. In particular, correlation times in the gravity and in the inertia
dominated case, respectively, longer and shorter than in the passive tracer
case; in the gravity dominated case, correlation times longer for velocity
components along gravity, than for the perpendicular ones. The model produces,
in channel flow geometry, particle deposition rates in agreement with
experiments.


title: Is Arnold diffusion relevant to global diffusion?

abstract:
  Global diffusion of Hamiltonian dynamical systems is investigated by using a
coupled standard maps. Arnold web is visualized in the frequency space, using
local rotation numbers, while Arnold diffusion and resonance overlaps are
distinguished by the residence time distributions at resonance layers. Global
diffusion in the phase space is shown to be accelerated by diffusion across
overlapped resonances generated by the coupling term, rather than Arnold
diffusion along the lower-order resonances. The former plays roles of hubs for
transport in the phase space, and accelerate the diffusion.


title: Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems

abstract:
  Discretization of phase space usually nullifies chaos in dynamical systems.
We show that if randomness is associated with discretization dynamical chaos
may survive and be indistinguishable from that of the original chaotic system,
when an entropic, coarse-grained analysis is performed. Relevance of this
phenomenon to the problem of quantum chaos is discussed.


title:
Perturbation of coupling matrices and its effect on the
  synchronizability in arrays of coupled chaotic systems

abstract:
  In a recent paper, wavelet analysis was used to perturb the coupling matrix
in an array of identical chaotic systems in order to improve its
synchronization. As the synchronization criterion is determined by the second
smallest eigenvalue $\lambda_2$ of the coupling matrix, the problem is
equivalent to studying how $\lambda_2$ of the coupling matrix changes with
perturbation. In the aforementioned paper, a small percentage of the wavelet
coefficients are modified. However, this result in a perturbed matrix where
every element is modified and nonzero. The purpose of this paper is to present
some results on the change of $\lambda_2$ due to perturbation. In particular,
we show that as the number of systems $n \to \infty$, perturbations which only
add local coupling will not change $\lambda_2$. On the other hand, we show that
there exists perturbations which affect an arbitrarily small percentage of
matrix elements, each of which is changed by an arbitrarily small amount and
yet can make $\lambda_2$ arbitrarily large. These results give conditions on
what the perturbation should be in order to improve the synchronizability in an
array of coupled chaotic systems. This analysis allows us to prove and explain
some of the synchronization phenomena observed in a recently studied network
where random coupling are added to a locally connected array. Finally we
classify various classes of coupling matrices such as small world networks and
scale free networks according to their synchronizability in the limit.


title: Observation of transverse instabilities in optically-induced lattices

abstract:
  We study experimentally the Bloch-wave instabilities in optically-induced
lattices. We reveal two different instability scenarios associated with either
the transverse modulational instability of a single Bloch wave, or the
nonlinear inter-band coupling between different Bloch waves. These effects are
compared with the transverse instability in homogeneous media, and it is shown
that the periodic modulation of the refractive index greatly enhances the
transverse instability.