title: Computed Chaos or Numerical Errors

abstract:
  Discrete numerical methods with finite time-steps represent a practical
technique to solve initial-value problems involving nonlinear differential
equations. These methods seem particularly useful to the study of chaos since
no analytical chaotic solution is currently available. Using the well-known
Lorenz equations as an example, it is demonstrated that numerically computed
results and their associated statistical properties are time-step dependent.
There are two reasons for this behavior. First, chaotic differential equations
are unstable so that any small error is amplified exponentially near an
unstable manifold. The more serious and lesser-known reason is that stable and
unstable manifolds of singular points associated with differential equations
can form virtual separatrices. The existence of a virtual separatrix presents
the possibility of a computed trajectory actually jumping through it due to the
finite time-steps of discrete numerical methods. Such behavior violates the
uniqueness theory of differential equations and amplifies the numerical errors
explosively. These reasons imply that, even if computed results are bounded,
their independence on time-step should be established before accepting them as
useful numerical approximations to the true solution of the differential
equations. However, due to these exponential and explosive amplifications of
numerical errors, no computed chaotic solutions of differential equations
independent of integration-time step have been found. Thus, reports of computed
non-periodic solutions of chaotic differential equations are simply
consequences of unstably amplified truncation errors, and are not approximate
solutions of the associated differential equations.

categories: nlin.CD math.CA math.DS math.NA physics.comp-ph physics.flu-dyn


title: Nonlocal competition and logistic growth: patterns, defects and fronts

abstract:
  Logistic growth of diffusing reactants on spatial domains with long range
competition is studied. The bifurcations cascade involved in the transition
from the homogenous state to a spatially modulated stable solution is
presented, and a distinction is made between a modulated phase, dominated by
single or few wavenumbers, and the spiky phase, where localized colonies are
separated by depleted region. The characteristic defects in the periodic
structure are presented for each phase, together with the invasion dynamics in
case of local initiation. It is shown that the basic length scale that controls
the bifurcation is the width of the Fisher front, and that the total population
grows as this width decreases. A mix of analytic results and extensive
numerical simulations yields a comprehensive examination of the possible phases
for logistic growth in the presence of nonlocal competition.


title:
Dynamics of Small Perturbations of Orbits on a Torus in a
  Quasiperiodically Forced 2D Dissipative Map

abstract:
  We consider the dynamics of small perturbations of stable two-frequency
quasiperiodic orbits on an attracting torus in the quasiperiodically forced
Henon map. Such dynamics consists in an exponential decay of the radial
component and in a complex behaviour of the angle component. This behaviour may
be two- or three-frequency quasiperiodicity, or it may be irregular. In the
latter case a graphic image of the dynamics of the perturbation angle is a
fractal object, namely a strange nonchaotic attractor, which appears in
auxiliary map for the angle component. Therefore, we claim that stable
trajectories may approach the attracting torus either in a regular or in an
irregular way. We show that the transition from quasiperiodic dynamics to chaos
in the model system is preceded by the appearance of an irregular behaviour in
the approach of the perturbed quasiperiodic trajectories to the smooth
attracting torus. We also demonstrate a link between the evolution operator of
the perturbation angle and a quasiperiodically forced circle mapping of a
special form and with a Harper equation with quasiperiodic potential.


title: Local to Global Normalization Dynamic by Nonlinear Local Interactions

abstract:
  Here, I present a novel method for normalizing a finite set of numbers, which
is studied by the domain of biological vision. Normalizing in this context
means searching the maximum and minimum number in a set and then rescaling all
numbers such that they fit into a numerical interval. My method computes the
minimum and maximum number by two pseudo-diffusion processes in separate
diffusion layers. Activity of these layers feed into a third layer for
performing the rescaling operation. The dynamic of the network is richer than
merely performing a rescaling of its input, and reveals phenomena like contrast
detection, contrast enhancement, and a transient compression of the numerical
range of the input. Apart from presenting computer simulations, some properties
of the diffusion operators and the network are analyzed mathematically.
Furthermore, a method is proposed for to freeze the model's state when
adaptation is observed.


title: Enstrophy dissipation in freely evolving two-dimensional turbulence

abstract:
  Freely decaying two-dimensional Navier--Stokes turbulence is studied. The
conservation of vorticity by advective nonlinearities renders a class of
Casimirs that decays under viscous effects. A rigorous constraint on the
palinstrophy production by nonlinear transfer is derived, and an upper bound
for the enstrophy dissipation is obtained. This bound depends only on the
decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from
above in terms of initial data of the flows. An upper bound for the enstrophy
dissipation wavenumber is derived and the new result is compared with the
classical dissipation wavenumber.


title: Intermittency in the homopolar disk-dynamo

abstract:
  We study a modified Bullard dynamo and show that this system is equivalent to
a nonlinear oscillator subject to a multiplicative noise. The stability
analysis of this oscillator is performed. Two bifurcations are identified,
first towards an `` intermittent\rq\rq state where the absorbing (non-dynamo)
state is no more stable but the most probable value of the amplitude of the
oscillator is still zero and secondly towards a `` turbulent\rq\rq (dynamo)
state where it is possible to define unambiguously a (non-zero) most probable
value around which the amplitude of the oscillator fluctuates. The bifurcation
diagram of this system exhibits three regions which are analytically
characterized.


title: Quantum Graphology

abstract:
  We review quantum chaos on graphs. We construct a unitary operator which
represents the quantum evolution on the graph and study its spectral and
wavefunction statistics. This operator is the analogue of the classical
evolution operator on the graph. It allow us to establish a connection between
the corresponding periodic orbits and the statistical properties of eigenvalues
and eigenfunctions. Specifically, for the energy-averaged spectral form factor
we derived an exact combinatorial expression which illustrate the role of
correlations between families of isometric orbits. We also show that enhanced
wave function localization due to the presence of short unstable periodic
orbits and strong scarring can rely on completely different mechanisms.


title:
Casorati Determinant Form of Dark Soliton Solutions of the Discrete
  Nonlinear Schr\"odinger Equation

abstract:
  It is shown that the $N$-dark soliton solutions of the integrable discrete
nonlinear Schr\"odinger (IDNLS) equation are given in terms of the Casorati
determinant. The conditions for reduction, complex conjugacy and regularity for
the Casorati determinant solution are also given explicitly. The relationship
between the IDNLS and the relativistic Toda lattice is discussed.


abstract:
  The aim of this paper is to rewrite the Fokker - Planck equation according to
transformation of space coordinates. This is nontrivial problem, because
transformation of space coordinates induces transformation of velocities. We
can use covariant, contravariant or physical velocity components as independent
variables in curvilinear coordinate system. These 3 possibilities are
considered in this paper and 3 kinds of Fokker - Planck equation in curvilinear
coordinates are formulated.


title:
Counting discrete emission steps from intrinsic localized modes in a
  quasi-1D antiferromagnetic lattice

abstract:
  Intrinsic localized modes (ILMs) in a quasi-1D antiferromagnetic material
(C2H5NH3)2CuCl4 are counted by using a novel nonlinear energy magnetometer. The
ILMs are produced by driving the uniform spin wave mode unstable with an
intense microwave pulse. Subsequently a subset of these ILMs become captured by
and locked to a cw driver so that their properties can be examined at a later
time with a tunable cw low power probe source. Four wave mixing is used to
enhance the emission signal from the few large amplitude ILMs over that
associated with the many small amplitude plane wave modes. A discrete step
structure observed in the emission signal is identified with individual ILMs
becoming unlocked from the driver. At most driver power and frequency settings
the resulting emission step structure appears uniformly distributed; however,
sometimes, nearby in parameter space, families of emission steps are evident as
the driver frequency or power is varied. Two different experimental methods
give consistent results for counting individual ILMs. Because of the
discreteness in the emission both the size of an ILM and its energy can be
estimated from these experiments. For the uniformly distributed case each ILM
extends over ~42 antiferromagnetic unit cells and has an energy value of
1.3x10^-12 [J] while for the case with families the ILM length becomes ~54
antiferromagnetic unit cells with an energy of 1.5x10^-12 [J]. An unresolved
puzzle is that the emission step height does not depend on experimental
parameters the way classical numerical simulations suggest.


title: Statistical and Dynamical Measures of Simple Irreversible Processes

abstract:
  A simple model of an irreversible process is introduced. The equation of
iterations in the model includes a noise generation term. We study the
properties of the system when the noise generation term is a stochastic process
(e.g. a random number generator) or a deterministic process (e.g. a chaotic
map). We compare the time series obtained from the above implementations of the
model by use of statistical methods (such as Detrended Fluctuation Analysis).
The conclusion is that using statistical methods the two versions of the model
are indistinguishable. The advantage of this observation is that we may
calculate the Lyapunov exponent for the model. As a result we obtain an
equation relating the DFA exponents (a statistical measure) with the Lyapunov
exponent for such models. On the other hand, typical statistical properties can
also be calculated, as for example the diffusion coefficient for a particle,
which movement is defined by the above model.

categories: nlin.CD cond-mat.stat-mech nlin.CG physics.data-an


title:
Security analysis of communication system based on the synchronization
  of different order chaotic systems

abstract:
  This work analyzes the security weakness of a recently proposed communication
method based on chaotic modulation and masking using synchronization of two
chaotic systems with different orders. It is shown that its application to
secure communication is unsafe, because it can be broken in two different ways,
by high-pass filtering and by reduced order system synchronization, without
knowing neither the system parameter values nor the system key.


title:
Symbolic Computation of Conservation Laws of Nonlinear Partial
  Differential Equations in Multi-dimensions

abstract:
  A direct method for the computation of polynomial conservation laws of
polynomial systems of nonlinear partial differential equations (PDEs) in
multi-dimensions is presented. The method avoids advanced
differential-geometric tools. Instead, it is solely based on calculus,
variational calculus, and linear algebra.
  Densities are constructed as linear combinations of scaling homogeneous terms
with undetermined coefficients. The variational derivative (Euler operator) is
used to compute the undetermined coefficients. The homotopy operator is used to
compute the fluxes.
  The method is illustrated with nonlinear PDEs describing wave phenomena in
fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters,
the method determines the conditions on the parameters so that a sequence of
conserved densities might exist. The existence of a large number of
conservation laws is a predictor for complete integrability. The method is
algorithmic, applicable to a variety of PDEs, and can be implemented in
computer algebra systems such as Mathematica, Maple, and REDUCE.


title: Phenomenology of Wall Bounded Newtonian Turbulence

abstract:
  We construct a simple analytic model for wall-bounded turbulence, containing
only four adjustable parameters. Two of these parameters characterize the
viscous dissipation of the components of the Reynolds stress-tensor and other
two parameters characterize their nonlinear relaxation. The model offers an
analytic description of the profiles of the mean velocity and the correlation
functions of velocity fluctuations in the entire boundary region, from the
viscous sub-layer, through the buffer layer and further into the log-layer. As
a first approximation, we employ the traditional return-to-isotropy hypothesis,
which yields a very simple distribution of the turbulent kinetic energy between
the velocity components in the log-layer: the streamwise component contains a
half of the total energy whereas the wall-normal and the cross-stream
components contain a quarter each. In addition, the model predicts a very
simple relation between the von-K\'arm\'an slope $\kappa $ and the turbulent
velocity in the log-law region $v^+$ (in wall units): $v^+=6 \kappa$. These
predictions are in excellent agreement with DNS data and with recent laboratory
experiments.


title: An Overview of Complex Adaptive Systems

abstract:
  Almost every biological, economic and social system is a complex adaptive
system (CAS). Mathematical and computer models are relevant to CAS. Some
approaches to modeling CAS are given. Applications in vaccination and the
immune system are studied. Mathematical topics motivated by CAS are discussed.


title:
Hopping dynamics for localized Lyapunov vectors in many-hard-disk
  systems

abstract:
  The dynamics of the localized region of the Lyapunov vector for the largest
Lyapunov exponent is discussed in quasi-one-dimensional hard-disk systems at
low density. We introduce a hopping rate to quantitatively describe the
movement of the localized region of this Lyapunov vector, and show that it is a
decreasing function of hopping distance, implying spatial correlation of the
localized regions. This behavior is explained quantitatively by a brick
accumulation model derived from hard-disk dynamics in the low density limit, in
which hopping of the localized Lyapunov vector is represented as the movement
of the highest brick position. We also give an analytical expression for the
hopping rate, which is obtained us a sum of probability distributions for brick
height configurations between two separated highest brick sites. The results of
these simple models are in good agreement with the simulation results for
hard-disk systems.


title: Transmitting a signal by amplitude modulation in a chaotic network

abstract:
  We discuss the ability of a network with non linear relays and chaotic
dynamics to transmit signals, on the basis of a linear response theory
developed by Ruelle \cite{Ruelle} for dissipative systems. We show in
particular how the dynamics interfere with the graph topology to produce an
effective transmission network, whose topology depends on the signal, and
cannot be directly read on the ``wired'' network. This leads one to reconsider
notions such as ``hubs''. Then, we show examples where, with a suitable choice
of the carrier frequency (resonance), one can transmit a signal from a node to
another one by amplitude modulation, \textit{in spite of chaos}. Also, we give
an example where a signal, transmitted to any node via different paths, can
only be recovered by a couple of \textit{specific} nodes. This opens the
possibility for encoding data in a way such that the recovery of the signal
requires the knowledge of the carrier frequency \textit{and} can be performed
only at some specific node.


title:
Education, neighbourhood effects and growth: an agent based model
  approach

abstract:
  Endogenous, ideas-led, growth theory and agent based modelling with
neighbourhood effects literature are crossed. In an economic overlapping
generations framework, it is shown how social interactions and neighbourhood
effects are of vital importance in the endogenous determination of the long run
number of skilled workers and therefore of the growth prospects of an economy.
Neighbourhood effects interact with the initial distribution of educated agents
across space and play a key role in the long run stabilisation of the number of
educated individuals. Our model implies a tendency towards segregation, with a
possibly positive influence on growth, if team effects operate. The long run
growth rate is also shown to depend on the rate of time preference. Initial
circumstances are of vital importance for long run outcomes. A poor initial
education endowment will imply a long run reduced number of skilled workers and
a mediocre growth rate, so there no economic convergence tendency. On the
contrary, poor societies will grow less, or will even fall into a poverty trap,
and will diverge continuously from richer ones.


title: Algebraic Closed Geodesics on a Triaxial Ellipsoid

abstract:
  We propose a simple method of explicit description of families of closed
geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in
${\mathbb R}^3$. Such geodesics are either connected components of spatial
elliptic curves or rational curves.
  Our approach is based on elements of the Weierstrass--Poncar\'e reduction
theory for hyperelliptic tangential covers of elliptic curves and the addition
law for elliptic functions.
  For the case of 3-fold and 4-fold coverings, explicit formulas for the
cutting algebraic surfaces are provided and some properties of the
corresponding geodesics are discussed.


title:
On the elimination of the sweeping interactions from theories of
  hydrodynamic turbulence

abstract:
  In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian
same time correlation tensors are equal. This statement allows us to transform
the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic
turbulence back to the Eulerian representation. We define a hierarchy of
homogeneity symmetries between incremental homogeneity and global homogeneity.
It is shown that both the elimination of the sweeping interactions and the
derivation of the 4/5-law require a homogeneity assumption stronger than
incremental homogeneity but weaker than global homogeneity. The
quasi-Lagrangian transformation, on the other hand, requires an even stronger
homogeneity assumption which is many-time rather than one-time but still weaker
than many-time global homogeneity. We argue that it is possible to relax this
stronger assumption and still preserve the conclusions derived from theoretical
work based on the quasi-Lagrangian transformation.


title: Periodicity and Growth in a Lattice Gas with Dynamical Geometry

abstract:
  We study a one-dimensional lattice gas "dynamical geometry model" in which
local reversible interactions of counter-rotating groups of particles on a ring
can create or destroy lattice sites. We exhibit many periodic orbits and and
show that all other solutions have asymptotically growing lattice length in
both directions of time. We explain why the length grows as $\sqrt{t}$ in all
cases examined. We completely solve the dynamics for small numbers of particles
with arbitrary initial conditions.


title:
A probability based approach on ananlyzing dynamics of oscillators on a
  bidirectional ring with propagation delay

abstract:
  In this paper, we presented a model of pulse-coupled oscillators distributed
on a bidirectional ring with propagation delay. In numerical simulations based
on this model, we observed phenomena of asynchrony in a certain range of delay
factor $\alpha$. To find the cause of these phenomena, we used a new
probability based approach of analyzing. In this approach, the mathematical
expectation of influence on one oscillator's phase change caused by its
neighbor, which is regarded as a random factor, is calculated. By adding this
expectation of influence into the firing map $h(\phi)$ introduced by Mirolla
and Strogatz, a probability firing map $H$ is invented. By observing the
behavior of $H$ 's iteration from $H$ 's graph, we successfully constructed a
connection between the asynchrony phenomena and $H$ 's graph.


title: Trigonometric Lax matrix for the Kowalevski gyrostat on so(4)

abstract:
  We present trigonometric Lax matrix and classical $r$-matrix for the
Kowalevski gyrostat on $so(4)$ algebra by using auxiliary matrix algebras
$so(3,2)$ or $sp(4)$.


title: Turbulence and coarsening in active and passive binary mixtures

abstract:
  Phase separation between two fluids in two-dimensions is investigated by
means of Direct Numerical Simulations of coupled Navier-Stokes and
Cahn-Hilliard equations. We study the phase ordering process in the presence of
an external stirring acting on the velocity field. For both active and passive
mixtures we find that, for a sufficiently strong stirring, coarsening is
arrested in a stationary dynamical state characterized by a continuous rupture
and formation of finite domains. Coarsening arrest is shown to be independent
of the chaotic or regular nature of the flow.


title:
Integrable Discrete Equations Derived by Similarity Reduction of the
  Extended Discrete KP Hierarchy

abstract:
  We consider the extended discrete KP hierarchy and show that similarity
reduction of its subhierarchies lead to purely discrete equations with
dependence on some number of parameters together with equations governing
deformations with respect to these parameters. It is written down discrete
equations which naturally generalize the first discrete Painlev\'e equation
$\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations
admit the limit to the first Painlev\'e equation. It is shown that each of
these equations describes B\"acklund transformations of Veselov-Shabat periodic
dressing lattices with odd period known also as Noumi-Yamada systems of type
$A_{2(n-1)}^{(1)}$.


title:
Conformal Properties and Baecklund Transform for the Associated
  Camassa-Holm Equation

abstract:
  Integrable equations exhibit interesting conformal properties and can be
written in terms of the so-called conformal invariants. The most basic and
important example is the KdV equation and the corresponding Schwarz-KdV
equation. Other examples, including the Camassa-Holm equation and the
associated Camassa-Holm equation are investigated in this paper. It is shown
that the Baecklund transform is related to the conformal properties of these
equations. Some particular solutions of the Associated Camassa-Holm Equation
are discussed also.


title:
Detecting Generalized Synchronization Between Chaotic Signals: A
  Kernel-based Approach

abstract:
  A unified framework for analyzing generalized synchronization in coupled
chaotic systems from data is proposed. The key of the proposed approach is the
use of the kernel methods recently developed in the field of machine learning.
Several successful applications are presented, which show the capability of the
kernel-based approach for detecting generalized synchronization. It is also
shown that the dynamical change of the coupling coefficient between two chaotic
systems can be captured by the proposed approach.


title:
Stable three-dimensional solitons in attractive Bose-Einstein
  condensates loaded in an optical lattice

abstract:
  The existence and stability of solitons in Bose-Einstein condensates with
attractive inter-atomic interactions, described by the Gross-Pitaevskii
equation with a three-dimensional (3D) periodic potential, are investigated in
a systematic form. We find a one-parameter family of stable 3D solitons in a
certain interval of values of their norm, provided that the strength of the
potential exceeds a threshold value. The minimum number of $^{7}$Li atoms in
the stable solitons is 60, and the energy of the soliton at the stability
threshold is $\approx 6$ recoil energies in the lattice. The respective
energy-vs.-norm diagram features two cuspidal points, resulting in a typical
\textit{swallowtail pattern}, which is a generic feature of 3D solitons
supported by low- (2D) or fully-dimensional lattice potentials.


title:
Group-invariant Solutions of the 2+1-dimensional Cubic Schr\"odinger
  Equation

abstract:
  Lie point symmetries of the 2+1-dimensional cubic Schr\"odinger equation to
obtain new analytic solutions in a systematic manner. We present an analysis of
the reduced ODEs, and in particular show that although the original equation is
not integrable they typically can belong to the class of Painlev\'e type
equations.


title:
Reaction-diffusion model for pattern formation in E. coli swarming
  colonies with slime

abstract:
  A new experimental colonial pattern and pattern transition observed in E.
coli MG1655 swarming cells grown on semi-solid agar are described. We present a
reaction-diffusion model that, taking into account the slime generated by these
cells and its influence on the bacterial differentiation and motion, reproduces
the pattern and successfully predicts the observed changes when the colonial
collective motility is limited. In spite of having small non-hyperflagellated
swarming cells, under these experimental conditions E. coli MG1655 can very
rapidly colonize a surface, with a low branching rate, thanks to a strong fluid
production and a local incremented density of motile, lubricating cells.


title: Noise-induced clustering in Hamiltonian systems

abstract:
  The motion of oscillatory-like nonlinear Hamiltonian systems, driven by a
weak noise, is considered. A general method to find regions of stability in the
phase space of a randomly-driven system, based on a specific Poincar\'e map, is
proposed and justified. Physical manifestations of these regions of stability,
the so-called coherent clusters, are demonstrated with two models in ocean
physics. We find bunches of sound rays propagating coherently in an underwater
waveguide through a randomly fluctuating ocean at long distances. We find
clusters of passive particles to be advected coherently by a random
two-dimensional flow modelling mixing around a topographic eddy in the ocean.


title: Completeness of the cubic and quartic H\'enon-Heiles Hamiltonians

abstract:
  The quartic H\'enon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(\Omega_1
Q_1^2+\Omega_2 Q_2^2)/2
  +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4
  +(1/2)(\alpha/Q_1^2+\beta/Q_2^2) - \gamma Q_1$ passes the Painlev\'e test for
only four sets of values of the constants. Only one of these, identical to the
traveling wave reduction of the Manakov system, has been explicitly integrated
(Wojciechowski, 1985), while the three others are not yet integrated in the
generic case $(\alpha,\beta,\gamma)\not=(0,0,0)$. We integrate them by building
a birational transformation to two fourth order first degree equations in the
classification (Cosgrove, 2000) of such polynomial equations which possess the
Painlev\'e property. This transformation involves the stationary reduction of
various partial differential equations (PDEs). The result is the same as for
the three cubic H\'enon-Heiles Hamiltonians, namely, in all four quartic cases,
a general solution which is meromorphic and hyperelliptic with genus two. As a
consequence, no additional autonomous term can be added to either the cubic or
the quartic Hamiltonians without destroying the Painlev\'e integrability
(completeness property).


title:
Hamiltonians with two degrees of freedom admitting a singlevalued
  general solution

abstract:
  Following the basic principles stated by Painlev\'e, we first revisit the
process of selecting the admissible time-independent Hamiltonians
$H=(p_1^2+p_2^2)/2+V(q_1,q_2)$ whose some integer power $q_j^{n_j}(t)$ of the
general solution is a singlevalued function of the complex time $t$. In
addition to the well known rational potentials $V$ of H\'enon-Heiles, this
selects possible cases with a trigonometric dependence of $V$ on $q_j$. Then,
by establishing the relevant confluences, we restrict the question of the
explicit integration of the seven (three ``cubic'' plus four ``quartic'')
rational H\'enon-Heiles cases to the quartic cases. Finally, we perform the
explicit integration of the quartic cases, thus proving that the seven rational
cases have a meromorphic general solution explicitly given by a genus two
hyperelliptic function.


title: Darboux Transformation for the Non-stationary Shr\"odinger Equation

abstract:
  The Lax representation for the nonstationary Schr\"odinger equation with
rather arbitrary potential is proposed. Some examples of the construction of
exact solutions are given by means of Darboux Transformation method.


title:
Chaotic dynamics of the elliptical stadium billiard in the full
  parameter space

abstract:
  Dynamical properties of the elliptical stadium billiard, which is a
generalization of the stadium billiard and a special case of the recently
introduced mushroom billiards, are investigated analytically and numerically.
In dependence on two shape parameters delta and gamma, this system reveals a
rich interplay of integrable, mixed and fully chaotic behavior. Poincare
sections, the box counting method and the stability analysis determine the
structure of the parameter space and the borders between regions with different
behavior. Results confirm the existence of a large fully chaotic region
surrounding the straight line delta=1-gamma corresponding to the Bunimovich
circular stadium billiard. Bifurcations due to the hour-glass and multidiamond
orbits are described. For the quantal elliptical stadium billiard, statistical
properties of the level spacing fluctuations are examined and compared with
classical results.


title: Vortex interaction, chaos and quantum probabilities

abstract:
  The motion of a single vortex is able to originate chaos in the quantum
trajectories defined in Bohm's interpretation of quantum mechanics. In this
Letter, we show that this is also the case in the general situation, in which
many interacting vortices exist. This result gives support to recent attempts
in which Born's probability rule is derived in terms of an irreversible time
evolution to equilibrium, rather than being postulated.


title: Kinetic equation for a dense soliton gas

abstract:
  We propose a general method to derive kinetic equations for dense soliton
gases in physical systems described by integrable nonlinear wave equations. The
kinetic equation describes evolution of the spectral distribution function of
solitons due to soliton-soliton collisions. Owing to complete integrability of
the soliton equations, only pairwise soliton interactions contribute to the
solution and the evolution reduces to a transport of the eigenvalues of the
associated spectral problem with the corresponding soliton velocities modified
by the collisions. The proposed general procedure of the derivation of the
kinetic equation is illustrated by the examples of the Korteweg -- de Vries
(KdV) and nonlinear Schr\"odinger (NLS) equations. As a simple physical example
we construct an explicit solution for the case of interaction of two cold NLS
soliton gases.


title:
Systematic Density Expansion of the Lyapunov Exponents for a
  Two-dimensional Random Lorentz Gas

abstract:
  We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at
low density. The positive Lyapunov exponent may be obtained either by a direct
analysis of the dynamics, or by the use of kinetic theory methods. To leading
orders in the density of scatterers it is of the form
$A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}$, where $A_{0}$ and $B_{0}$ are
known constants and $\tilde{n}$ is the number density of scatterers expressed
in dimensionless units. In this paper, we find that through order
$(\tilde{n}^{2})$, the positive Lyapunov exponent is of the form
$A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}+A_{1}\tilde{n}^{2}\ln\tilde{n}
+B_{1}\tilde{n}^{2}$. Explicit numerical values of the new constants $A_{1}$
and $B_{1}$ are obtained by means of a systematic analysis. This takes into
account, up to $O(\tilde{n}^{2})$, the effects of {\it all\/} possible
trajectories in two versions of the model; in one version overlapping scatterer
configurations are allowed and in the other they are not.


title:
Properties of Fourier spectrum of the signal, generated at the
  accumulation point of period-tripling bifurcations

abstract:
  Universal regularities of the Fourier spectrum of signal, generated by
complex analytic map at the period-tripling bifurcations accumulation point are
considered. The difference between intensities of the subharmonics at the
values of frequency corresponding to the neighbor hierarchical levels of the
spectrum is characterized by a constant $\gamma=21.9$ dB?, which is an analogue
of the known value $\gamma_F=13.4$ dB, intrinsic to the Feigenbaum critical
point. Data of the physical experiment, directed to the observation of the
spectrum at period-tripling accumulation point, are represented.


title:
Regularization of moving boundaries in a Laplacian field by a mixed
  Dirichlet-Neumann boundary condition: exact results

abstract:
  The dynamics of ionization fronts that generate a conducting body, are in
simplest approximation equivalent to viscous fingering without regularization.
Going beyond this approximation, we suggest that ionization fronts can be
modeled by a mixed Dirichlet-Neumann boundary condition. We derive exact
uniformly propagating solutions of this problem in 2D and construct a single
partial differential equation governing small perturbations of these solutions.
For some parameter value, this equation can be solved analytically which shows
that the uniformly propagating solution is linearly convectively stable.


title:
Nonlinear Hamiltonian dynamics of Lagrangian transport and mixing in the
  ocean

abstract:
  Methods of dynamical system's theory are used for numerical study of
transport and mixing of passive particles (water masses, temperature, salinity,
pollutants, etc.) in simple kinematic ocean models composed with the main
Eulerian coherent structures in a randomly fluctuating ocean -- a jet-like
current and an eddy. Advection of passive tracers in a periodically-driven flow
consisting of a background stream and an eddy (the model inspired by the
phenomenon of topographic eddies over mountains in the ocean and atmosphere) is
analyzed as an example of chaotic particle's scattering and transport. A
numerical analysis reveals a nonattracting chaotic invariant set $\Lambda$ that
determines scattering and trapping of particles from the incoming flow. It is
shown that both the trapping time for particles in the mixing region and the
number of times their trajectories wind around the vortex have hierarchical
fractal structure as functions of the initial particle's coordinates.
Scattering functions are singular on a Cantor set of initial conditions, and
this property should manifest itself by strong fluctuations of quantities
measured in experiments. The Lagrangian structures in our numerical experiments
are shown to be similar to those found in a recent laboratory dye experiment at
Woods Hole. Transport and mixing of passive particles is studied in the
kinematic model inspired by the interaction of a jet current (like the Gulf
Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a
non-trivial phenomenon of noise-induced clustering of passive particles and
propose a method to find such clusters in numerical experiments. These clusters
are patches of advected particles which can move together in a random velocity
field for comparatively long time.


title: Measuring Generalized Preferential Attachment in Dynamic Social Networks

abstract:
  The mechanism of preferential attachment underpins most recent social network
formation models. Yet few authors attempt to check or quantify assumptions on
this mechanism. We call generalized preferential attachment any kind of
preference to interact with other agents with respect to any node property. We
then introduce tools for measuring empirically and characterizing
comprehensively such phenomena, and apply these tools to a socio-semantic
network of scientific collaborations, investigating in particular homophilic
behavior. This opens the way to a whole class of realistic and credible social
network morphogenesis models.

categories: nlin.AO cond-mat.dis-nn cs.DM cs.NI physics.soc-ph


title:
Optimized suppression of coherent noise from seismic data using the
  Karhunen-Loeve transform

abstract:
  Signals obtained in land seismic surveys are usually contaminated with
coherent noise, among which the ground roll (Rayleigh surface waves) is of
major concern for it can severely degrade the quality of the information
obtained from the seismic record. Properly suppressing the ground roll from
seismic data is not only of great practical importance but also remains a
scientific challenge. Here we propose an optimized filter based on the
Karhunen--Lo\'eve transform for processing seismic data contaminated with
ground roll. In our method, the contaminated region of the seismic record, to
be processed by the filter, is selected in such way so as to correspond to the
maximum of a properly defined coherence index. The main advantages of the
method are that the ground roll is suppressed with negligible distortion of the
remanent reflection signals and that the filtering can be performed on the
computer in a largely unsupervised manner. The method has been devised to
filter seismic data, however it could also be relevant for other applications
where localized coherent structures, embedded in a complex spatiotemporal
dynamics, need to be identified in a more refined way.


title: Topological Chaos in Spatially Periodic Mixers

abstract:
  Topologically chaotic fluid advection is examined in two-dimensional flows
with either or both directions spatially periodic. Topological chaos is created
by driving flow with moving stirrers whose trajectories are chosen to form
various braids. For spatially periodic flows, in addition to the usual
stirrer-exchange braiding motions, there are additional
topologically-nontrivial motions corresponding to stirrers traversing the
periodic directions. This leads to a study of the braid group on the cylinder
and the torus. Methods for finding topological entropy lower bounds for such
flows are examined. These bounds are then compared to numerical stirring
simulations of Stokes flow to evaluate their sharpness. The sine flow is also
examined from a topological perspective.


title:
The Transition from Regular to Irregular Motions, Explained as Travel on
  Riemann Surfaces

abstract:
  We introduce and discuss a simple Hamiltonian dynamical system, interpretable
as a 3-body problem in the complex plane and providing the prototype of a
mechanism explaining the transition from regular to irregular motions as travel
on Riemann surfaces. The interest of this phenomenology -- illustrating the
onset in a deterministic context of irregular motions -- is underlined by its
generality, suggesting its eventual relevance to understand natural phenomena
and experimental investigations. Here only some of our main findings are
reported, without detailing their proofs: a more complete presentation will be
published elsewhere.


title:
Feshbach Resonance Management of Bose-Einstein Condensates in Optical
  Lattices

abstract:
  We analyze gap solitons in trapped Bose-Einstein condensates (BECs) in
optical lattice potentials under Feshbach resonance management. Starting with
an averaged Gross-Pitaevsky (GP) equation with a periodic potential, we employ
an envelope wave approximation to derive coupled-mode equations describing the
slow BEC dynamics in the first spectral gap of the optical lattice. We
construct exact analytical formulas describing gap soliton solutions and
examine their spectral stability using the Chebyshev interpolation method. We
show that these gap solitons are unstable far from the threshold of local
bifurcation and that the instability results in the distortion of their shape.
We also predict the threshold of the power of gap solitons near the local
bifurcation limit.

categories: nlin.PS cond-mat.other math.DS physics.atom-ph


title:
Power expansions for solution of the fourth-order analog to the first
  Painlev\'{e} equation

abstract:
  One of the fourth-order analog to the first Painlev\'{e} equation is studied.
All power expansions for solutions of this equation near points $z=0$ and
$z=\infty$ are found by means of the power geometry method. The exponential
additions to the expansion of solution near $z=\infty$ are computed. The
obtained results confirm the hypothesis that the fourth-order analog of the
first Painlev\'{e} equation determines new transcendental functions.


title:
Enhanced soliton transport in quasi-periodic lattices with short-range
  aperiodicity

abstract:
  We study linear transmission and nonlinear soliton transport through
quasi-periodic structures, which profiles are described by multiple modulation
frequencies. We show that resonant scattering at mixed-frequency resonances
limits transmission efficiency of localized wave packets, leading to radiation
and possible trapping of solitons. We obtain an explicit analytical expression
for optimal quasi-periodic lattice profiles, where additional aperiodic
modulations suppress mixed-frequency resonances, resulting in dramatic
enhancement of soliton mobility. Our results can be applied to the design of
photonic waveguide structures, and arrays of magnetic micro-traps for atomic
Bose-Einstein condensates.


title: Angularly localized Skyrmions

abstract:
  Quantized Skyrmions with baryon numbers $B=1,2$ and 4 are considered and
angularly localized wavefunctions for them are found. By combining a few low
angular momentum states, one can construct a quantum state whose spatial
density is close to that of the classical Skyrmion, and has the same
symmetries. For the B=1 case we find the best localized wavefunction among
linear combinations of $j=1/2$ and $j=3/2$ angular momentum states. For B=2, we
find that the $j=1$ ground state has toroidal symmetry and a somewhat reduced
localization compared to the classical solution. For B=4, where the classical
Skyrmion has cubic symmetry, we construct cubically symmetric quantum states by
combining the $j=0$ ground state with the lowest rotationally excited $j=4$
state. We use the rational map approximation to compare the classical and
quantum baryon densities in the B=2 and B=4 cases.


title: Unsteady undular bores in fully nonlinear shallow-water theory

abstract:
  We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type which contain the familiar fully nonlinear dissipationless
shallow-water dynamics and the leading-order fully nonlinear dispersive terms.
This system contains one horizontal space dimension and time and can be
systematically derived from the full Euler equations for irrotational flows
with a free surface using a standard long-wave asymptotic expansion.
  In this context the system was first derived by Su and Gardner. It coincides
with the one-dimensional flat-bottom reduction of the Green-Naghdi system and,
additionally, has recently found a number of fluid dynamics applications other
than the present context of shallow-water gravity waves. We then use the
Whitham modulation theory for a one-phase periodic travelling wave to obtain an
asymptotic analytical description of an undular bore in the Su-Gardner system
for a full range of "depth" ratios across the bore. The positions of the
leading and trailing edges of the undular bore and the amplitude of the leading
solitary wave of the bore are found as functions of this "depth ratio". The
formation of a partial undular bore with a rapidly-varying finite-amplitude
trailing wave front is predicted for ``depth ratios'' across the bore exceeding
1.43. The analytical results from the modulation theory are shown to be in
excellent agreement with full numerical solutions for the development of an
undular bore in the Su-Gardner system.


title: Synchronization of coupled chaotic oscillators as a phase transition

abstract:
  We characterize the synchronization of an array of coupled chaotic elements
as a phase transition where order parameters related to the joint probability
at two sites obey power laws versus the mutual coupling strength; the phase
transition corresponds to a change in the exponent of the power law. Since
these studies are motivated by the behaviour of the cortical neurons in
cognitive tasks, we account for the short time available to any brain decision
by studying how the mutual coupling affects the transient behaviour of a
synchronization transition over a fixed time interval. We present a novel
feature, namely, the absence of decay of the initial defect density for small
coupling.


title: Soliton topology versus discrete symmetry in optical lattices

abstract:
  We address the existence of vortex solitons supported by azimuthally
modulated lattices and reveal how the global lattice discrete symmetry has
fundamental implications on the possible topological charges of solitons. We
set a general ``charge rule'' using group-theory techniques, which holds for
all lattices belonging to a given symmetry group. Focusing in the case of
Bessel lattices allows us to derive also a overall stability rule for the
allowed vortex solitons.


title: Aerodynamics at the Particle Level

abstract:
  This paper is intended to clarify some of the rather well-known aerodynamic
phenomena. It is also intended to pique the interest of the layman as well as
the professional. All aerodynamic forces on a surface are caused by collisions
of fluid particles with the surface. While the standard approach to fluid
dynamics, which is founded on the fluid approximation, is effective in
providing a means of calculating various behavior and properties, it begs the
question of causality. The determination of the causes of many of the most
important aerodynamic effects requires a microscopic examination of the fluid
and of the surface with which it interacts. The Kutta-Joukowski theorem is
investigated from first physical principles. It is noted that the circulation
does not arise alone as a physical phenomenon, e.g. air doesn't flow forward
under a wing, but must be added to the translation flow. The circulation term
is necessary to take into account the vertical deflection of the air flow by a
wing. Various aerodynamic devices are discussed, e.g. rocket engine exhaust
diffuser and the perfume atomizer. The section on slurries discusses
pyroclastic flow as occurs in violent volcanic eruptions.


title: Finite-size effects on open chaotic advection

abstract:
  We study the effects of finite-sizeness on small, neutrally buoyant,
spherical particles advected by open chaotic flows. We show that, when
projected onto configuration space, the advected finite-size particles disperse
about the unstable manifold of the chaotic saddle that governs the passive
advection. Using a discrete-time system for the dynamics, we obtain an
expression predicting the dispersion of the finite-size particles in terms of
their Stokes parameter at the onset of the finite-sizeness induced dispersion.
We test our theory in a system derived from a flow and find remarkable
agreement between our expression and the numerically measured dispersion.


title:
On the Two-point Correlation of Potential Vorticity in Rotating and
  Stratified Turbulence

abstract:
  A framework is developed to describe the two-point statistics of potential
vorticity in rotating and stratified turbulence as described by the Boussinesq
equations. The Karman-Howarth equation for the dynamics of the two-point
correlation function of potential vorticity reveals the possibility of
inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and
Reynolds number parameters. For the case of large Rossby and Froude numbers,
and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3
prefactor is derived for the third-order mixed correlation between potential
vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law
for the third-order velocity structure function in turbulence theory.


title:
Generalized synchronization in coupled Ginzburg-Landau equations and
  mechanisms of its arising

abstract:
  Generalized chaotic synchronization regime is observed in the
unidirectionally coupled one-dimensional Ginzburg-Landau equations. The
mechanism resulting in the generalized synchronization regime arising in the
coupled spatially extended chaotic systems demonstrating spatiotemporal chaotic
oscillations has been described. The cause of the generalized synchronization
occurrence is studied with the help of the modified Ginzburg-Landau equation
with additional dissipation.


title: Ergodicity Breaking in a Deterministic Dynamical System

abstract:
  The concept of weak ergodicity breaking is defined and studied in the context
of deterministic dynamics. We show that weak ergodicity breaking describes a
weakly chaotic dynamical system: a nonlinear map which generates subdiffusion
deterministically. In the non-ergodic phase non-trivial distribution of the
fraction of occupation times is obtained. The visitation fraction remains
uniform even in the non-ergodic phase. In this sense the non-ergodicity is
quantified, leading to a statistical mechanical description of the system even
though it is not ergodic.


title:
Forecasting non-stationary financial time series through genetic
  algorithm

abstract:
  We utilize a recently developed genetic algorithm, in conjunction with
discrete wavelets, for carrying out successful forecasts of the trend in
financial time series, that includes the NASDAQ composite index. Discrete
wavelets isolate the local, small scale variations in these non-stationary time
series, after which the genetic algorithm's predictions are found to be quite
accurate. The power law behavior in Fourier domain reveals an underlying
self-affine dynamical behavior, well captured by the algorithm, in the form of
an analytic equation. Remarkably, the same equation captures the trend of the
Bombay stock exchange composite index quite well.


title:
Boltzmann entropy and chaos in a large assembly of weakly interacting
  systems

abstract:
  We introduce a high dimensional symplectic map, modeling a large system
consisting of weakly interacting chaotic subsystems, as a toy model to analyze
the interplay between single-particle chaotic dynamics and particles
interactions in thermodynamic systems. We study the growth with time of the
Boltzmann entropy, S_B, in this system as a function of the coarse graining
resolution. We show that a characteristic scale emerges, and that the behavior
of S_B vs t, at variance with the Gibbs entropy, does not depend on the coarse
graining resolution, as far as it is finer than this scale. The interaction
among particles is crucial to achieve this result, while the rate of entropy
growth depends essentially on the single-particle chaotic dynamics (for t not
too small). It is possible to interpret the basic features of the dynamics in
terms of a suitable Markov approximation.


title: BLP dissipative structures in plane

abstract:
  We study the Darboux and Laplace transformations for the
Boiti-Leon-Pempinelli equations (BLP). These equations are the (1+2)
generalization of the sinh-Gordon equation. In addition, the BLP equations
reduced to the Burgers (and anti-Burgers) equation in a one-dimensional limit.
Localized nonsingular solutions in both spatial dimensions and (anti) "blow-up"
solutions are constructed. The Burgers equation's "dressing" procedure is
suggested. This procedure allows us to construct such solutions of the BLP
equations which are reduced to the solutions of the dissipative Burgers
equations when $t\to \infty$. These solutions we call the BLP dissipative
structures.


title: Hopf bifurcations in time-delay systems with band-limited feedback

abstract:
  We investigate the steady-state solution and its bifurcations in time-delay
systems with band-limited feedback. This is a first step in a rigorous study
concerning the effects of AC-coupled components in nonlinear devices with
time-delayed feedback. We show that the steady state is globally stable for
small feedback gain and that local stability is lost, generically, through a
Hopf bifurcation for larger feedback gain. We provide simple criteria that
determine whether the Hopf bifurcation is supercritical or subcritical based on
the knowledge of the first three terms in the Taylor-expansion of the
nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the
steady state is shown, which indicates possible quasiperiodic and chaotic
dynamics in these systems. As a result of this investigation, we find that
AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda
et al., Physica D 29 (1987) 223-235] already at the level of steady-state
bifurcations, e.g. bifurcations exist in which limit cycles are created with
periods other than the fundamental ``period-2'' mode found in Ikeda-type
systems.


title:
Detecting Generalized Synchronization of Chaotic Dynamical Systems: A
  Kernel-based Method and Choice of Its Parameter

abstract:
  An approach based on the kernel methods for capturing the nonlinear
interdependence between two signals is introduced. It is demonstrated that the
proposed approach is useful for characterizing generalized synchronization with
a successful simple example. An attempt to choose an optimal kernel parameter
based on cross validation is also discussed.


title:
Is the subdominant part of the energy spectrum due to downscale energy
  cascade hidden in quasi-geostrophic turbulence?

abstract:
  In systems governing two-dimensional turbulence, surface quasi-geostrophic
turbulence, (more generally $\alpha$-turbulence), two-layer quasi-geostrophic
turbulence, etc., there often exist two conservative quadratic quantities, one
``energy''-like and one ``enstrophy''-like. In a finite inertial range there
are in general two spectral fluxes, one associated with each conserved
quantity. We derive here an inequality comparing the relative magnitudes of the
``energy'' and ``enstrophy'' fluxes for finite or infinitesimal dissipations,
and for hyper or hypo viscosities. When this inequality is satisfied, as is the
case of 2D turbulence,where the energy flux contribution to the energy spectrum
is small, the subdominant part will be effectively hidden. In sQG turbulence,
it is shown that the opposite is true: the downscale energy flux becomes the
dominant contribution to the energy spectrum. A combination of these two
behaviors appears to be the case in 2-layer QG turbulence, depending on the
baroclinicity of the system.


title:
Chaotic Dynamics of a Free Particle Interacting Linearly with a Harmonic
  Oscillator

abstract:
  We study the closed Hamiltonian dynamics of a free particle moving on a ring,
over one section of which it interacts linearly with a single harmonic
oscillator. On the basis of numerical and analytical evidence, we conjecture
that at small positive energies the phase space of our model is completely
chaotic except for a single region of complete integrability with a smooth
sharp boundary showing no KAM-type structures of any kind. This results in the
cleanest mixed phase space structure possible, in which motions in the
integrable region and in the chaotic region are clearly separated and
independent of one another. For certain system parameters, this mixed phase
space structure can be tuned to make either of the two components disappear,
leaving a completely integrable or completely chaotic phase space. For other
values of the system parameters, additional structures appear, such as KAM-like
elliptic islands, and one parameter families of parabolic periodic orbits
embedded in the chaotic sea. The latter are analogous to bouncing ball orbits
seen in the stadium billiard. The analytical part of our study proceeds from a
geometric description of the dynamics, and shows it to be equivalent to a
linked twist map on the union of two intersecting disks.


title:
Global existence in infinite lattices of nonlinear oscillators: The
  Discrete Klein-Gordon equation

abstract:
  Pointing out the difference between the Discrete Nonlinear Schr\"odinger
equation with the classical power law nonlinearity-for which solutions exist
globally, independently of the sign and the degree of the nonlinearity, the
size of the initial data and the dimension of the lattice-we prove either
global existence or nonexistence in time, for the Discrete Klein-Gordon
equation with the same type of nonlinearity (but of ``blow-up'' sign), under
suitable conditions on the initial data, and some times on the dimension of the
lattice. The results, consider both the conservative and the linearly damped
lattice. Similarities and differences with the continuous counterparts, are
remarked. We also make a short comment, on the existence of excitation
thresholds, for forced solutions of damped and parametrically driven,
Klein-Gordon lattices.


title: An ant-based algorithm for annular sorting

abstract:
  In this paper we describe a minimal model for annular sorting by Leptothorax
ants. Simulation results are consistent with the structures observed in actual
ant colonies.


title:
Brownian Motion Model of Quantization Ambiguity and Universality in
  Chaotic Systems

abstract:
  We examine spectral equilibration of quantum chaotic spectra to universal
statistics, in the context of the Brownian motion model. Two competing time
scales, proportional and inversely proportional to the classical relaxation
time, jointly govern the equilibration process. Multiplicity of quantum systems
having the same semiclassical limit is not sufficient to obtain equilibration
of any spectral modes in two-dimensional systems, while in three-dimensional
systems equilibration for some spectral modes is possible if the classical
relaxation rate is slow. Connections are made with upper bounds on
semiclassical accuracy and with fidelity decay in the presence of a weak
perturbation.


title: Propagation of Discrete Solitons in Inhomogeneous Networks

abstract:
  In many physical applications solitons propagate on supports whose
topological properties may induce new and interesting effects. In this paper,
we investigate the propagation of solitons on chains with a topological
inhomogeneity generated by the insertion of a finite discrete network on the
chain. For networks connected by a link to a single site of the chain, we
derive a general criterion yielding the momenta for perfect reflection and
transmission of traveling solitons and we discuss solitonic motion on chains
with topological inhomogeneities.


title: Discrete Solitons and Vortices on Anisotropic Lattices

abstract:
  We consider effects of anisotropy on solitons of various types in
two-dimensional nonlinear lattices, using the discrete nonlinear
Schr{\"{o}}dinger equation as a paradigm model. For fundamental solitons, we
develop a variational approximation, which predicts that broad quasi-continuum
solitons are unstable, while their strongly anisotropic counterparts are
stable. By means of numerical methods, it is found that, in the general case,
the fundamental solitons and simplest on-site-centered vortex solitons ("vortex
crosses") feature enhanced or reduced stability areas, depending on the
strength of the anisotropy. More surprising is the effect of anisotropy on the
so-called "super-symmetric" intersite-centered vortices ("vortex squares"),
with the topological charge $S$ equal to the square's size $M$: we predict in
an analytical form by means of the Lyapunov-Schmidt theory, and confirm by
numerical results, that arbitrarily weak anisotropy results in dramatic changes
in the stability and dynamics in comparison with the \emph{degenerate}, in this
case, isotropic limit.


title: Exact nonparametric inference for detection of nonlinear determinism

abstract:
  We propose an exact nonparametric inference scheme for the detection of
nonlinear determinism. The essential fact utilized in our scheme is that, for a
linear stochastic process with jointly symmetric innovations, its ordinary
least square (OLS) linear prediction error is symmetric about zero. Based on
this viewpoint, a class of linear signed rank statistics, e.g. the Wilcoxon
signed rank statistic, can be derived with the known null distributions from
the prediction error. Thus one of the advantages of our scheme is that, it can
provide exact confidence levels for our null hypothesis tests. Furthermore, the
exactness is applicable for finite samples with arbitrary length. We
demonstrate the test power of this statistic through several examples.


title: Soliton dynamics in deformable nonlinear lattices

abstract:
  We describe wave propagation and soliton localization in photonic lattices
which are induced in a nonlinear medium by an optical interference pattern,
taking into account the inherent lattice deformations at the soliton location.
We obtain exact analytical solutions and identify the key factors defining
soliton mobility, including the effects of gap merging and lattice imbalance,
underlying the differences with discrete and gap solitons in conventional
photonic structures.


title: Wave turbulence and vortices in Bose-Einstein condensation

abstract:
  We report a numerical study of turbulence and Bose-Einstein condensation
within the two-dimmensional Gross-Pitaevski model with repulsive interaction.
In presence of weak forcing localized around some wave number in the Fourier
space, we observe three qualitatively different evolution stages. At the
initial stage a thermodynamic energy equipartition spectrum forms at both
smaller and larger scales with respect to the forcing scale. This agrees with
predictions of the the four-wave kinetic equation of the Wave Turbulence (WT)
theory. At the second stage, WT breaks down at large scales and the
interactions become strongly nonlinear. Here, we observe formation of a gas of
quantum vortices whose number decreases due to an annihilation process helped
by the acoustic component. This process leads to formation of a coherent-phase
Bose-Einstein condensate. After such a coherent-phase condensate forms,
evolution enters a third stage characterised by three-wave interactions of
acoustic waves that can be described again using the WT theory.


title: Thin front propagation in random shear flows

abstract:
  Front propagation in time dependent laminar flows is investigated in the
limit of very fast reaction and very thin fronts, i.e. the so-called
geometrical optics limit. In particular, we consider fronts evolving in time
correlated random shear flows, modeled in terms of Ornstein-Uhlembeck
processes. We show that the ratio between the time correlation of the flow and
an intrinsic time scale of the reaction dynamics (the wrinkling time $t_w$) is
crucial in determining both the front propagation speed and the front spatial
patterns. The relevance of time correlation in realistic flows is briefly
discussed in the light of the bending phenomenon, i.e. the decrease of
propagation speed observed at high flow intensities.


title: Large-scale effects on meso-scale modeling for scalar transport

abstract:
  The transport of scalar quantities passively advected by velocity fields with
a small-scale component can be modeled at meso-scale level by means of an
effective drift and an effective diffusivity, which can be determined by means
of multiple-scale techniques. We show that the presence of a weak large-scale
flow induces interesting effects on the meso-scale scalar transport. In
particular, it gives rise to non-isotropic and non-homogeneous corrections to
the meso-scale drift and diffusivity. We discuss an approximation that allows
us to retain the second-order effects caused by the large-scale flow. This
provides a rather accurate meso-scale modeling for both asymptotic and
pre-asymptotic scalar transport properties. Numerical simulations in model
flows are used to illustrate the importance of such large-scale effects.


title: A Model of Coupled-Maps for Economic Dynamics

abstract:
  An array system of coupled maps is proposed as a model for economy evolution.
The local dynamics of each map or agent is controlled by two parameters. One of
them represents the growth capacity of the agent and the other one is a control
term representing the local environmental pressure which avoids an exponential
growth. The asymptotic state of the system evolution displays a complex
behavior. The distribution of the maps values in this final regime is of power
law type. In the model, inequality emerges as a result of the dynamical
processes taking place in the microscopic scales.

categories: nlin.AO cond-mat.dis-nn physics.data-an q-bio.PE q-fin.GN


title:
A $U_q\bigl(\hat{gl}(2|2)\bigr)_1$-Vertex Model: Creation Algebras and
  Quasi-Particles I

abstract:
  The infinite configuration space of an integrable vertex model based on
$U_q\bigl(\hat{gl}(2|2)\bigr)_1$ is studied at $q=0$. Allowing four particular
boundary conditions, the infinite configurations are mapped onto the
semi-standard supertableaux of pairs of infinite border strips. By means of
this map, a weight-preserving one-to-one correspondence between the infinite
configurations and the normal forms of a pair of creation algebras is
established for one boundary condition. A pair of type-II vertex operators
associated with an infinite-dimensional $U_q\bigl(gl(2|2)\bigr)$-module
$\mathring V$ and its dual $\mathring V^*$ is introduced. Their existence is
conjectured relying on a free boson realization. The realization allows to
determine the commutation relation satisfied by two vertex operators related to
the same $U_q\bigl(gl(2|2)\bigr)$-module. Explicit expressions are provided for
the relevant R-matrix elements. The formal $q\to0$ limit of these commutation
relations leads to the defining relations of the creation algebras. Based on
these findings it is conjectured that the type II vertex operators associated
with $\mathring V$ and $\mathring V^*$ give rise to part of the eigenstates of
the row-to-row transfer matrix of the model. A partial discussion of the
R-matrix elements introduced on $\mathring V\otimes \mathring V^*$ is given.


title: The restricted two-body problem in constant curvature spaces

abstract:
  We perform the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$.
An analogue of the Delaunay variables is introduced. We investigate the motion
of a point mass in the field of the Newtonian center moving along a geodesic on
$S^2$ and $L^2$ (the restricted two-body problem). When the curvature is small,
the pericenter shift is computed using the perturbation theory. We also present
the results of the numerical analysis based on the analogy with the motion of
rigid body.


title: Reduction and chaotic behavior of point vortices on a plane and a sphere

abstract:
  We offer a new method of reduction for a system of point vortices on a plane
and a sphere. This method is similar to the classical node elimination
procedure. However, as applied to the vortex dynamics, it requires substantial
modification. Reduction of four vortices on a sphere is given in more detail.
We also use the Poincare surface-of-section technique to perform the reduction
a four-vortex system on a sphere.


title:
New periodic solutions for three or four identical vortices on a plane
  and a sphere

abstract:
  In this paper we describe new classes of periodic solutions for point
vortices on a plane and a sphere. They correspond to similar solutions
(so-called choreographies) in celestial mechanics.


title:
CGL Defect Chaos and Bursts in Hexagonal Rotating non-Boussinesq
  Convection

abstract:
  We employ numerical computations of the full Navier-Stokes equations to
investigate non-Boussinesq convection in a rotating system using water as a
working fluid. We identify two regimes. For weak non-Boussinesq effects the
Hopf bifurcation from steady to oscillating (whirling) hexagons is
supercritical and typical states exhibit defect chaos that is systematically
described by the cubic complex Ginzburg-Landau equation. For stronger
non-Boussinesq effects the Hopf bifurcation becomes subcritical and the
oscillations exhibit localized chaotic bursting, which can be modeled by a
quintic complex Ginzburg-Landau equation.


title: Motion of vortex sources on a plane and a sphere

abstract:
  The Equations of motion of vortex sources (examined earlier by Fridman and
Polubarinova) are studied, and the problems of their being Hamiltonian and
integrable are discussed. A system of two vortex sources and three
sources-sinks was examined. Their behavior was found to be regular. Qualitative
analysis of this system was made, and the class of Liouville integrable systems
is considered. Particular solutions analogous to the homothetic configurations
in celestial mechanics are given.


title: Integrable Equations on Time Scales

abstract:
  Integrable systems are usually given in terms of functions of continuous
variables (on ${\mathbb R}$), functions of discrete variables (on ${\mathbb
Z}$) and recently in terms of functions of $q$-variables (on ${\mathbb
K}_{q}$). We formulate the Gel'fand-Dikii (GD) formalism on time scales by
using the delta differentiation operator and find more general integrable
nonlinear evolutionary equations. In particular they yield integrable equations
over integers (difference equations) and over $q$-numbers ($q$-difference
equations). We formulate the GD formalism also in terms of shift operators for
all regular-discrete time scales. We give a method to construct the recursion
operators for integrable systems on time scales. Finally, we give a trace
formula on time scales and then construct infinitely many conserved quantities
(Casimirs) of the integrable systems on time scales.


title:
On negative flows of the AKNS hierarchy and a class of deformations of
  bihamiltonian structure of hydrodynamic type

abstract:
  A deformation parameter of a bihamiltonian structure of hydrodynamic type is
shown to parameterize different extensions of the AKNS hierarchy to include
negative flows. This construction establishes a purely algebraic link between,
on the one hand, two realizations of the first negative flow of the AKNS model
and, on the other, two-component generalizations of Camassa-Holm and Dym type
equations.
  The two-component generalizations of Camassa-Holm and Dym type equations can
be obtained from the negative order Hamiltonians constructed from the Lenard
relations recursively applied on the Casimir of the first Poisson bracket of
hydrodynamic type. The positive order Hamiltonians, which follow from Lenard
scheme applied on the Casimir of the second Poisson bracket of hydrodynamic
type, are shown to coincide with the Hamiltonians of the AKNS model. The AKNS
Hamiltonians give rise to charges conserved with respect to equations of motion
of two-component Camassa-Holm and two-component Dym type equations.


title:
Secure Communication Scheme Based on Asymptotic Model of Deterministic
  Randomness

abstract:
  We propose a new cryptosystem by combing the Lissajous map, which is the
asymptotic model of deterministic randomness, with the one-way coupled map
lattice (OCML) system. The key space, the encryption efficiency, and the
security are investigated. We find that the parameter sensitivity can reach the
computational precision when the system size is only three, all the lattice
outputs can be treated as key stream parallelly, and the system is resistible
against various attacks including the differential-like chosen cipher attack.
The findings of this paper are a strong indication of the importance of
deterministic randomness in secure communications.


title: Optical transparency modes in anisotropic media

abstract:
  The modes of nonlinear propagation of the two-component electromagnetic
pulses through optically uniaxial media containing resonant particles are
studied. The features of their manifestation in the "dense" media and in the
media with expressed positive and negative birefringences are discussed. It is
shown that exponentially and rationally decreasing solutions of the system of
material and wave equations allow us also to describe the propagation of the
self-induced transparency pulses in isotropic media in the case, when the
direct electric dipole-dipole interaction between the resonant particles is
taken into account.


title:
Hamiltonian formalism of the DNLS equation with nonvanished boundary
  value

abstract:
  Hamiltonian formalism of the DNLS equation with nonvanishing boundary value
is developed by the standard procedure.


title: Structured Information in Metric Neural Networks

abstract:
  The retrieval abilities of spatially uniform attractor networks can be
measured by the average overlap between patterns and neural states. We found
that metric networks, with local connections, however, can carry information
structured in blocks without any global overlap. and blocks attractors. We
propose a way to measure the block information, related to the fluctuation of
the overlap. The phase-diagram with the transition from local to global
information, shows that the stability of blocks grows with dilution, but
decreases with the storage rate and disappears for random topologies.


title: Quantifying Self-Organization in Cyclic Cellular Automata

abstract:
  Cyclic cellular automata (CCA) are models of excitable media. Started from
random initial conditions, they produce several different kinds of spatial
structure, depending on their control parameters. We introduce new tools from
information theory that let us calculate the dynamical information content of
spatial random processes. This complexity measure allows us to quantitatively
determine the rate of self-organization of these cellular automata, and
establish the relationship between parameter values and self-organization in
CCA. The method is very general and can easily be applied to other cellular
automata or even digitized experimental data.


title:
Automatic Filters for the Detection of Coherent Structure in
  Spatiotemporal Systems

abstract:
  Most current methods for identifying coherent structures in
spatially-extended systems rely on prior information about the form which those
structures take. Here we present two new approaches to automatically filter the
changing configurations of spatial dynamical systems and extract coherent
structures. One, local sensitivity filtering, is a modification of the local
Lyapunov exponent approach suitable to cellular automata and other discrete
spatial systems. The other, local statistical complexity filtering, calculates
the amount of information needed for optimal prediction of the system's
behavior in the vicinity of a given point. By examining the changing
spatiotemporal distributions of these quantities, we can find the coherent
structures in a variety of pattern-forming cellular automata, without needing
to guess or postulate the form of that structure. We apply both filters to
elementary and cyclical cellular automata (ECA and CCA) and find that they
readily identify particles, domains and other more complicated structures. We
compare the results from ECA with earlier ones based upon the theory of formal
languages, and the results from CCA with a more traditional approach based on
an order parameter and free energy. While sensitivity and statistical
complexity are equally adept at uncovering structure, they are based on
different system properties (dynamical and probabilistic, respectively), and
provide complementary information.


title: Synchronization in delayed discrete-time complex networks

abstract:
  In this paper, we study synchronization in the delayed discrete-time complex
networks. Several criterions of synchronization stability for such networks are
established. And illustrative examples are presented. The numerical simulations
coincide with the theoretical analysis.


title: Asymptotic Calculation of Discrete Nonlinear Wave Interactions

abstract:
  We illustrate how to compute asymptotic interactions between discrete
solitary waves of dispersive equations, using the approach proposed by Manton
[Nucl. Phys. B 150, 397 (1979)]. We also discuss the complications arising due
to discreteness and showcase the application of the method in nonlinear
Schrodinger, as well as in Klein-Gordon lattices, finding excellent agreement
with direct numerical computations.


title: Foliations of Isonergy Surfaces and Singularities of Curves

abstract:
  It is well known that changes in the Liouville foliations of the isoenergy
surfaces of an integrable system imply that the bifurcation set has
singularities at the corresponding energy level. We formulate certain
genericity assumptions for two degrees of freedom integrable systems and we
prove the opposite statement: the essential critical points of the bifurcation
set appear only if the Liouville foliations of the isoenergy surfaces change at
the corresponding energy levels. Along the proof, we give full classification
of the structure of the isoenergy surfaces near the critical set under our
genericity assumptions and we give their complete list using Fomenko graphs.
This may be viewed as a step towards completing the Smale program for relating
the energy surfaces foliation structure to singularities of the momentum
mappings for non-degenerate integrable two degrees of freedom systems.


title: Unifying quantization for inhomogeneous integrable models

abstract:
  Integrable inhomogeneous versions of the models like NLS, Toda chain,
Ablowitz-Ladik model etc., though well known at the classical level, have never
been investigated for their possible quantum extensions. We propose a unifying
scheme for constructing and solving such quantum integrable inhomogeneous
models including a novel inhomogeneous sine-Gordon model, which avoids the
difficulty related to the customary non-isospectral flow by introducing the
inhomogeneities through some central elements of the underlying algebra.

categories: nlin.SI cond-mat.stat-mech hep-th math-ph math.MP


title: Metamimetic Games : Modeling Metadynamics in Social Cognition

abstract:
  Imitation is fundamental in the understanding of social system dynamics. But
the diversity of imitation rules employed by modelers proves that the modeling
of mimetic processes cannot avoid the traditional problem of endogenization of
all the choices, including the one of the mimetic rules. Starting from the
remark that human reflexive capacities are the ground for a new class of
mimetic rules, I propose a formal framework, metamimetic games, that enable to
endogenize the distribution of imitation rules while being human specific. The
corresponding concepts of equilibrium - counterfactually stable state - and
attractor are introduced. Finally, I give an interpretation of social
differentiation in terms of cultural co-evolution among a set of possible
motivations, which departs from the traditional view of optimization indexed to
criteria that exist prior to the activity of agents.


title:
The N-soliton of the focusing nonlinear Schroedinger equation for N
  large

abstract:
  We present a detailed analysis of the solution of the focusing nonlinear
Schr\"odinger equation with initial condition $\psi(x,0)=N {\rm sech}(x)$ in
the limit $N\to\infty$. We begin by presenting new and more accurate numerical
reconstructions of the $N$-soliton by inverse scattering (numerical linear
algebra) for N=5, 10, 20, and 40. We then recast the inverse-scattering problem
as a Riemann-Hilbert problem and provide a rigorous asymptotic analysis of this
problem in the large-N limit. For those $(x,t)$ where results have been
obtained by other authors, we improve the error estimates from $O(N^{-1/3})$ to
$O(N^{-1})$. We also analyze the Fourier power spectrum in this regime and
relate the results to the optical phenomenon of supercontinuum generation. We
then study the $N$-soliton for values of $(x,t)$ where analysis has not been
carried out before, and we uncover new phenomena. The main discovery of this
paper is the mathematical mechanism for a secondary caustic (phase transition),
which turns out to differ from the mechanism that generates the primary
caustic. The mechanism for the generation of the secondary caustic depends
essentially on the discrete nature of the spectrum for the $N$-soliton, and
more significantly, cannot be recovered from an analysis of an ostensibly
similar Riemann-Hilbert problem in the conditions of which a certain formal
continuum limit is taken on an {\em ad hoc} basis


title: Scattering fidelity in elastodynamics

abstract:
  The recent introduction of the concept of scattering fidelity, causes us to
revisit the experiment by Lobkis and Weaver [Phys. Rev. Lett. 90, 254302
(2003)]. There, the ``distortion'' of the coda of an acoustic signal is
measured under temperature changes. This quantity is in fact the negative
logarithm of scattering fidelity. We re-analyse their experimental data for two
samples, and we find good agreement with random matrix predictions for the
standard fidelity. Usually, one may expect such an agreement for chaotic
systems only. While the first sample, may indeed be assumed chaotic, for the
second sample, a perfect cuboid, such an agreement is more surprising. For the
first sample, the random matrix analysis yields a perturbation strength
compatible with semiclassical predictions. For the cuboid the measured
perturbation strength is much larger than expected, but with the fitted values
for this strength, the experimental data are well reproduced.


title: Spectral correlations of individual quantum graphs

abstract:
  We investigate the spectral properties of chaotic quantum graphs. We
demonstrate that the `energy'--average over the spectrum of individual graphs
can be traded for the functional average over a supersymmetric non--linear
$\sigma$--model action. This proves that spectral correlations of individual
quantum graphs behave according to the predictions of Wigner--Dyson random
matrix theory. We explore the stability of the universal random matrix behavior
with regard to perturbations, and discuss the crossover between different types
of symmetries.


title: Exact Elliptic Compactons in Generalized Korteweg-DeVries Equations

abstract:
  We derive a general theorem relating the energy, momentum and velocity of any
solitary wave solution of the generalized KdV equation which enables us to
relate the amplitude, width, and momentum to the velocity of these solutions.
We obtain the general condition for linear and Lyapunov stability. We then
obtain a two parameter family of exact solutions to these equations which
include elliptic and hyper-elliptic compacton solutions. For this general
family we explicitly verify both the theorem and the stability criteria.


title: Chaotic and Regular Motion in Dissipative Gravitational Billiards

abstract:
  We consider the motion of a particle subjected to the constant gravitational
field and scattered inelasticaly by hard boundaries which possess the shape of
parabola, wedge, and hyperbola. The billiard itself performs oscillations. The
linear dependence of the restitution coefficient on the particle velocity is
assumed. We demonstrate that this dynamical system can be either regular or
chaotic, which depends on the billiard shape and the oscillation frequency. The
trajectory calculations are compared with the experimental data; a good
agreement has been achieved. Moreover, the properties of the system has been
studied by means of the Lyapunov exponents and the Kaplan-Yorke dimension.
Chaotic and nonuniform patterns visible in the experimental data are
interpreted as a result of large embedding dimension.


title: Acceleration statistics of heavy particles in turbulence

abstract:
  We present the results of direct numerical simulations of heavy particle
transport in homogeneous, isotropic, fully developed turbulence, up to
resolution $512^3$ ($R_\lambda\approx 185$). Following the trajectories of up
to 120 million particles with Stokes numbers, $St$, in the range from 0.16 to
3.5 we are able to characterize in full detail the statistics of particle
acceleration. We show that: ({\it i}) The root-mean-squared acceleration
$a_{\rm rms}$ sharply falls off from the fluid tracer value already at quite
small Stokes numbers; ({\it ii}) At a given $St$ the normalised acceleration
$a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently
with the trend observed for fluid tracers; ({\it iii}) The tails of the
probability density function of the normalised acceleration $a/a_{\rm rms}$
decrease with $St$. Two concurrent mechanisms lead to the above results:
preferential concentration of particles, very effective at small $St$, and
filtering induced by the particle response time, that takes over at larger
$St$.


title:
Discrete optic breathers in zigzag chain: analytical study and computer
  simulation

abstract:
  We present analytical and numerical study of discrete breathers identified as
localized deformations of valence angles accompanied by change of valence bonds
in crystalline polyethylene (PE). It is shown that such breathers can exist
inside the optic wave number band and can propagate along the macromolecule
chain with subsonic velocities. Analytical results are confirmed by numerical
simulation using Molecular Dynamics procedure. We examine also stability of the
breathers relative to thermal excitations and mutual collisions as well as to
collisions with acoustic non-topological and topological solitons. The
conditions of breathers existence depend strongly on their generating
frequency, relationship between stiffnesses of valence bonds and valence angles
as well as on type of nonlinear characteristics of intrachain interactions.


title: Dynamical failure of Turing patterns

abstract:
  The emergence of stable disordered patterns in reactive system on spatially
homogenous substrate is studied in the context of vegetation patterns in the
semi-arid climatic zone. It is shown that reaction-diffusion systems that allow
for Turing instability may exhibit heterogeneous "glassy" steady state, with no
characteristic wavelength, if the diffusion rate associated with the "slow"
reactant is very small. Upon decreasing the diffusion constant of the slow
reactant three phases are identified: strong diffusion yields a stable
homogenous phase, intermediate diffusion implies Turing (crystal like) patterns
while in the slow diffusion limit the glassy state is the generic stable
solution. In this disordered phase the dynamics is of crucial importance, with
strong differences between local and global initiation.


title: Superposition in nonlinear wave and evolution equations

abstract:
  Real and bounded elliptic solutions suitable for applying the Khare-Sukhatme
superposition procedure are presented and used to generate superposition
solutions of the generalized modified Kadomtsev-Petviashvili equation (gmKPE)
and the nonlinear cubic-quintic Schroedinger equation (NLCQSE).


title:
Fano resonance in two-dimensional optical waveguide arrays with a
  bi-modal defect

abstract:
  We study the two-dimensional extension of the Fano-Anderson model on the
basis of a two-dimensional optical waveguide array with a bi-modal defect. We
demonstrate numerically the persistence of the Fano resonance in wavepacket
scattering process by the defect. An analytical approximation is derived for
the total scattered light power.


title: Fractal Stationary Density in Coupled Maps

abstract:
  We study the invariant measure or the stationary density of a coupled
discrete dynamical system as a function of the coupling parameter \epsilon (0 <
\epsilon < 1/4). The dynamical system considered is chaotic and unsynchronized
for this range of parameter values. We find that the stationary density,
restricted on the synchronization manifold, is a fractal function. We find the
lower bound on the fractal dimension of the graph of this function and show
that it changes continuously with the coupling parameter


title: Chaotic and pseudochaotic attractors of perturbed fractional oscillator

abstract:
  We consider a nonlinear oscillator with fractional derivative of the order
alpha. Perturbed by a periodic force, the system exhibits chaotic motion called
fractional chaotic attractor (FCA). The FCA is compared to the ``regular''
chaotic attractor. The properties of the FCA are discussed and the
``pseudochaotic'' case is demonstrated.


title:
Nodal Domain Distribution for a Nonintegrable Two-Dimensional Anharmonic
  Oscillator

abstract:
  We investigate the transition from integrable to chaotic dynamics in the
quantum mechanical wave functions from the point of view of the nodal structure
by employing a two dimensional quartic oscillator. We find that the number of
nodal domains is drastically reduced as the dynamics of the system changes from
integrable to nonintegrable, and then gradually increases as the system becomes
chaotic. The number of nodal intersections with the classical boundary as a
function of the level number shows a characteristic dependence on the dynamics
of the system, too. We also calculate the area distribution of nodal domains
and study the emergence of the power law behavior with the Fisher exponent in
the chaotic limit.


title: Asymptotic properties of mathematical models of excitability

abstract:
  We analyse small parameters in selected models of biological excitability,
including Hodgkin-Huxley (1952) model of nerve axon, Noble (1962) model of
heart Purkinje fibres, and Courtemanche et al. (1998) model of human atrial
cells. Some of the small parameters are responsible for differences in the
characteristic timescales of dynamic variables, as in the traditional singular
perturbation approaches. Others appear in a way which makes the standard
approaches inapplicable. We apply this analysis to study the behaviour of
fronts of excitation waves in spatially-extended cardiac models. Suppressing
the excitability of the tissue leads to a decrease in the propagation speed,
but only to a certain limit; further suppression blocks active propagation and
leads to a passive diffusive spread of voltage. Such a dissipation may happen
if a front propagates into a tissue recovering after a previous wave, e.g.
re-entry. A dissipated front does not recover even when the excitability
restores. This has no analogy in FitzHugh-Nagumo model and its variants, where
fronts can stop and then start again. In two spatial dimensions, dissipation
accounts for break-ups and self-termination of re-entrant waves in excitable
media with Courtemanche et al. (1998) kinetics.


title: The sixth Painleve equation as similarity reduction of gl_3 hierarchy

abstract:
  Scaling symmetry of gl_n-type Drinfel'd-Sokolov hierarchy is investigated.
Applying similarity reduction to the hierarchy, one can obtain the Schlesinger
equation with (n+1) regular singularities. Especially in the case of n=3, the
hierarchy contains the three-wave resonant system and the similarity reduction
gives the generic case of the Painleve VI equation. We also discuss Weyl group
symmetry of the hierarchy.


title: Discrete embedded solitons

abstract:
  We address the existence and properties of discrete embedded solitons (ESs),
i.e., localized waves existing inside the phonon band in a nonlinear
dynamical-lattice model. The model describes a one-dimensional array of optical
waveguides with both the quadratic (second-harmonic generation) and cubic
nonlinearities. A rich family of ESs was previously known in the continuum
limit of the model. First, a simple motivating problem is considered, in which
the cubic nonlinearity acts in a single waveguide. An explicit solution is
constructed asymptotically in the large-wavenumber limit. The general problem
is then shown to be equivalent to the existence of a homoclinic orbit in a
four-dimensional reversible map. From properties of such maps, it is shown that
(unlike ordinary gap solitons), discrete ESs have the same codimension as their
continuum counterparts. A specific numerical method is developed to compute
homoclinic solutions of the map, that are symmetric under a specific reversing
transformation. Existence is then studied in the full parameter space of the
problem. Numerical results agree with the asymptotic results in the appropriate
limit and suggest that the discrete ESs may be semi-stable as in the continuous
case.


title:
Bivariate $t$-distribution for transition matrix elements in
  Breit-Wigner to Gaussian domains of interacting particle systems

abstract:
  Interacting many-particle systems with a mean-field one body part plus a
chaos generating random two-body interaction having strength $\lambda$, exhibit
Poisson to GOE and Breit-Wigner (BW) to Gaussian transitions in level
fluctuations and strength functions with transition points marked by
$\lambda=\lambda_c$ and $\lambda=\lambda_F$, respectively; $\lambda_F >>
\lambda_c$. For these systems theory for matrix elements of one-body transition
operators is available, as valid in the Gaussian domain, with $\lambda >
\lambda_F$, in terms of orbitals occupation numbers, level densities and an
integral involving a bivariate Gaussian in the initial and final energies. Here
we show that, using bivariate $t$-distribution, the theory extends below from
the Gaussian regime to the BW regime up to $\lambda=\lambda_c$. This is well
tested in numerical calculations for six spinless fermions in twelve single
particle states.


title: Resonance- and Chaos-Assisted Tunneling

abstract:
  We consider dynamical tunneling between two symmetry-related regular islands
that are separated in phase space by a chaotic sea. Such tunneling processes
are dominantly governed by nonlinear resonances, which induce a coupling
mechanism between ``regular'' quantum states within and ``chaotic'' states
outside the islands. By means of a random matrix ansatz for the chaotic part of
the Hamiltonian, one can show that the corresponding coupling matrix element
directly determines the level splitting between the symmetric and the
antisymmetric eigenstates of the pair of islands. We show in detail how this
matrix element can be expressed in terms of elementary classical quantities
that are associated with the resonance. The validity of this theory is
demonstrated with the kicked Harper model.


title:
Foundations of Nonlinear Dynamics or Real Dynamics, Ideal Dynamics,
  Unpredictable Dynamics and the "Schroedinger's Cat Paradox"

abstract:
  The paper discusses the basic paradoxes of thermodynamics and quantum
mechanics. The approaches to solution of these paradoxes are suggested. The
first one relies on the influence of the external observer (environment), which
disrupts the correlations in the system. The second one is based on the limits
of self-knowledge of the system in case of both the external observer and the
environment is included in the considered system. The concepts of Real
Dynamics, Ideal Dynamics, and Unpredictable dynamics are introduced. The
phenomenon of Life is contemplated from the point of view of these Dynamics.


title: Variational method for locating invariant tori

abstract:
  We formulate a variational fictitious-time flow which drives an initial guess
torus to a torus invariant under given dynamics. The method is general and
applies in principle to continuous time flows and discrete time maps in
arbitrary dimension, and to both Hamiltonian and dissipative systems.


title:
Controlled switching of intrinsic localized modes in a 1-D
  antiferromagnet

abstract:
  Nearly steady-state locked intrinsic localized modes (ILMs) in the quasi-1d
antiferromagnet (C2H5NH3)2CuCl4 are detected via four-wave mixing emission or
the uniform mode absorption. Exploiting the long-time stability of these locked
ILMs, repeatable nonlinear switching is observed by varying the sample
temperature, and localized modes with various amplitudes are created by
modulation of the microwave driver power. This steady-state ILM locking
technique could be used to produce energy localization in other atomic
lattices.


title: Normal Forms for Symplectic Maps with Twist Singularities

abstract:
  We derive a normal form for a near-integrable, four-dimensional symplectic
map with a fold or cusp singularity in its frequency mapping. The normal form
is obtained for when the frequency is near a resonance and the mapping is
approximately given by the time-$T$ mapping of a two-degree-of freedom
Hamiltonian flow. Consequently there is an energy-like invariant. The fold
Hamiltonian is similar to the well-studied, one-degree-of freedom case but is
essentially nonintegrable when the direction of the singular curve in action
does not coincide with curves of the resonance module. We show that many
familiar features, such as multiple island chains and reconnecting invariant
manifolds, are retained even in this case. The cusp Hamiltonian has an
essential coupling between its two degrees of freedom even when the singular
set is aligned with the resonance module. Using averaging, we approximately
reduced this case to one degree of freedom as well. The resulting Hamiltonian
and its perturbation with small cusp-angle is analyzed in detail.


title: Coupled KdV equations derived from atmospherical dynamics

abstract:
  Some types of coupled Korteweg de-Vries (KdV) equations are derived from an
atmospheric dynamical system. In the derivation procedure, an unreasonable
$y$-average trick (which is usually adopted in literature) is removed. The
derived models are classified via Painlev\'e test. Three types of
$\tau$-function solutions and multiple soliton solutions of the models are
explicitly given by means of the exact solutions of the usual KdV equation. It
is also interesting that for a non-Painlev\'e integrable coupled KdV system
there may be multiple soliton solutions.