title:
Vibrational and stochastic resonances in two coupled overdamped
  anharmonic oscillators

abstract:
  We study the overdamped version of two coupled anharmonic oscillators under
the influence of both low- and high-frequency forces respectively and a
Gaussian noise term added to one of the two state variables of the system. The
dynamics of the system is first studied in the presence of both forces
separately without noise. In the presence of only one of the forces, no
resonance behaviour is observed, however, hysteresis happens there. Then the
influence of the high-frequency force in the presence of a low-frequency, i.e.
biharmonic forcing, is studied. Vibrational resonance is found to occur when
the amplitude of the high-frequency force is varied. The resonance curve
resembles a stochastic resonance-like curve. It is maximum at the value of $g$
at which the orbit lies in one well during one half of the drive cycle of the
low-frequency force and in the other for the remaining half cycle. Vibrational
resonance is characterized using the response amplitude and mean residence
time. We show the occurrence of stochastic resonance behaviour in the
overdamped system by replacing the high-frequency force by Gaussian noise.
Similarities and differences between both types of resonance are presented.


title: Attractor Bifurcation of Three-Dimensional Double-Diffusive Convection

abstract:
  In this article, we present a bifurcation analysis on the double-diffusive
convection. Two pattern selections, rectangles and squares, are investigated.
It is proved that there are two different types of attractor bifurcations
depending on the thermal and salinity Rayleigh numbers for each pattern. The
analysis is based on a newly developed attractor bifurcation theory, together
with eigen-analysis and the center manifold reductions.


title:
Combination of The Cellular Potts Model and Lattice Gas Cellular
  Automata For Simulating The Avascular Cancer Growth

abstract:
  The advantage of Cellular Potts Model (CPM) is due to its ability for
introducing cell-cell interaction based on the well known statistical model
i.e. the Potts model. On the other hand, Lattice gas Cellular Automata (LGCA)
can simulate movement of cell in a simple and correct physical way. These
characters of CPM and LGCA have been combined in a reaction-diffusion frame to
simulate the dynamic of avascular cancer growth on a more physical basis.The
cellular automaton is evolved on a square lattice on which in the diffusion
step tumor cells (C) and necrotic cells (N) propagate in two dimensions and in
the reaction step every cell can proliferate, be quiescent or die due to the
apoptosis and the necrosis depending on its environment. The transition
probabilities in the reaction step have been calculated by the Glauber
algorithm and depend on the KCC, KNC, and KNN (cancer-cancer, necrotic-cancer,
and necrotic-necrotic couplings respectively). It is shown the main feature of
the cancer growth depends on the choice of magnitude of couplings and the
advantage of this method compared to other methods is due to the fact that it
needs only three parameters KCC, KNC and KNN which are based on the well known
physical ground i.e. the Potts model.


title:
Constructing the Critical Curve for the Two-Layer Potts Model Using
  Cellular Automata

abstract:
  The critical points of the 3-states two-layer Potts model on square lattice
for different interlayer couplings (Kx, Ky,and Kz) are calculated with high
precision using probabilistic cellular automata with Glauber algorithm, where
Kx and Ky are the nearest-neighbor interactions within each layer in the x and
y directions, respectively and Kz is the interlayer coupling. The obtained
results are 0.726, 0.807, 0.928, 0.987 and 1.00 for the cases Kz per Kx = 1,
0.5, 0.1, 0.01 and 0.001, respectively. Then, the critical curve has been
constructed for this model.


title: New approximation for nonlinear evolution in periodic potentials

abstract:
  A new approximation for evolution described by Nonlinear Schrodinger Equation
(NLS) with periodic potential is presented. It relies on restricting dynamics
to one band of the bandgap spectrum, and taking into account only one,
dominating Fourier component in the nonlinear Bloch-wave mixing. The resulting
equation has a simple, discrete form in the basis of linear Wannier functions,
and turns out to be very accurate as long as the modes in other bands are not
excited and the potential is not very deep. Widely used approximations, the
tight-binding approximation and the effective mass approximation, are derived
from the new equation as the limiting cases.


title: Modulational Instability in Nonlinearity-Managed Optical Media

abstract:
  We investigate analytically, numerically, and experimentally the modulational
instability in a layered, cubically-nonlinear (Kerr) optical medium that
consists of alternating layers of glass and air. We model this setting using a
nonlinear Schr\"odinger (NLS) equation with a piecewise constant nonlinearity
coefficient and conduct a theoretical analysis of its linear stability,
obtaining a Kronig-Penney equation whose forbidden bands correspond to the
modulationally unstable regimes. We find very good {\it quantitative} agreement
between the theoretical analysis of the Kronig-Penney equation, numerical
simulations of the NLS equation, and the experimental results for the
modulational instability. Because of the periodicity in the evolution variable
arising from the layered medium, we find multiple instability regions rather
than just the one that would occur in uniform media.


title:
Modulational instability of spatially broadband nonlinear optical pulses
  in four-state atomic systems

abstract:
  The modulational instability of broadband optical pulses in a four-state
atomic system is investigated. In particular, starting from a recently derived
generalized nonlinear Schr\"odinger equation, a wave-kinetic equation is
derived. A comparison between coherent and random phase wave states is made. It
is found that the spatial spectral broadening can contribute to the nonlinear
stability of ultra-short optical pulses. In practical terms, this could be
achieved by using random phase plate techniques.


title:
One-dimensional solitary waves in singular deformations of SO(2)
  invariant two-component scalar field theory models

abstract:
  In this paper we study the structure of the manifold of solitary waves in
some deformations of SO(2) symmetric two-component scalar field theoretical
models in two-dimensional Minkowski space. The deformation is chosen in order
to make the analogous mechanical system Hamilton-Jacobi separable in polar
coordinates and displays a singularity at the origin of the internal plane. The
existence of the singularity confers interesting and intriguing properties to
the solitary waves or kink solutions.


title:
New aspects of the continuous phase transition in the scalar noise model
  (SNM) of collective motion

abstract:
  In this paper we present our detailed investigations on the nature of the
phase transition in the scalar noise model (SNM) of collective motion. Our
results confirm the original findings of Vicsek et al. [Phys. Rev. Lett. 75
(1995) 1226] that the disorder-order transition in the SNM is a continuous,
second order phase transition for small particle velocities ($v\leq 0.1$).
However, for large velocities ($v\geq 0.3$) we find a strong anisotropy in the
particle diffusion in contrast with the isotropic diffusion for small
velocities. The interplay between the anisotropic diffusion and the periodic
boundary conditions leads to an artificial symmetry breaking of the solutions
(directionally quantized density waves) and a consequent first order transition
like behavior. Thus, it is not possible to draw any conclusion about the
physical behavior in the large particle velocity regime of the SNM.


title: Adaptive Dynamics for Interacting Markovian Processes

abstract:
  Dynamics of information flow in adaptively interacting stochastic processes
is studied. We give an extended form of game dynamics for Markovian processes
and study its behavior to observe information flow through the system. Examples
of the adaptive dynamics for two stochastic processes interacting through
matching pennies game interaction are exhibited along with underlying causal
structure.


title: Effects of non-denumerable fixed points in finite dynamical systems

abstract:
  The motion of a spinning football brings forth the possible existence of a
whole class of finite dynamical systems where there may be non-denumerably
infinite number of fixed points. They defy the very traditional meaning of the
fixed point that a point on the fixed point in the phase space should remain
there forever, for, a fixed point can evolve as well! Under such considerations
one can argue that a free-kicked football should be non-chaotic.


title: Magnetic field gradients in solar wind plasma and geophysics periods

abstract:
  Using recent data obtained by Advanced Composition Explorer (ACE) the pumping
scale of the magnetic field gradients of the solar wind plasma has been
calculated. This pumping scale is found to be equal to 24h $\pm$ 2h. The ACE
spacecraft orbits at the L1 libration point which is a point of Earth-Sun
gravitational equilibrium about 1.5 million km from Earth. Since the Earth's
magnetosphere extends into the vacuum of space from approximately 80 to 60,000
kilometers on the side toward the Sun the pumping scale cannot be a consequence
of the 24h-period of the Earth's rotation. Vise versa, a speculation is
suggested that for the very long time of the coexistence of Earth and of the
solar wind the weak interaction between the solar wind and Earth could lead to
stochastic synchronization between the Earth's rotation and the pumping scale
of the solar wind magnetic field gradients. This synchronization could
transform an original period of the Earth's rotation to the period close to the
pumping scale of the solar wind magnetic field gradients.


title: Is it possible to control the spread of a globalized culture?

abstract:
  We study a model describing the spread of a globalized culture in a
population of individuals localized at the nodes of a social network. The
influence of this globalized culture, assumed to be foreign to the local
culture, is measured by a probability to convince each individual to adopt its
cultural traits. This probability depends upon the degree s--a real between 0
and 1--of ``wise'' skepticism characterizing the personality of each individual
and a parameter a representing the resistance of the society as a whole to the
spread of the foreign cultural traits. A greater a indicates a stronger
resistance of the local culture to globalization. On the other hand, each
individual interacts with a random number of other individuals--her cultural
neighborhood--uniformly distributed between 1 and a maximum value. The
probability distribution of an individual to belong to the cultural
neighborhood of another individual has a power-law behavior. A small fraction r
of the total population belonging to the tail of this probability distribution
have an s-value equal to 1. They represent the most conservative individuals
firmly attached to their local culture.


title:
Exploring and Simulating Chaotic Advection:A Difference Equations
  Approach

abstract:
  This paper explores the chaotic properties of an advection system expressed
in difference equations form. In the beginning the Aref's blinking vortex
system is examined. Then several new lines are explored related to the sink
problem (one central sink, two symmetric sinks, eccentric sink and others).
Chaotic forms with or without space contraction are presented, analyzed and
simulated. Several chaotic objects are formulated especially when special
rotation angles or a complex sinus rotation angle are introduced in the
rotation-translation difference equations. Very interesting chaotic forms arise
when elliptic rotation-translation equations are applied. The simulated chaotic
images and attractors express several vortex-like forms resulting in various
situations and especially in fluid dynamics.


title: A symmetry test for quasilinear coupled systems

abstract:
  All quasilinear NLS-type systems with fourth order symmetry are listed.
Vector generalizations of some of them are constructed. Local master symmetries
for several systems from the list are found.


title: Classification of integrable polynomial vector evolution equations

abstract:
  Several classes of systems of evolution equations with one or two vector
unknowns are considered. We investigate also systems with one vector and one
scalar unknown. For these classes all equations having the simplest higher
symmetry are listed.


title:
Modulational Instability of Nonlinearly Interacting Incoherent Sea
  States

abstract:
  The modulational instability of nonlinearly interacting spatially incoherent
Stokes waves is analyzed. Starting from a pair of nonlinear Schroedinger
equations, we derive a coupled set of wave-kinetic equations by using the
Wigner transform technique. It is shown that the partial coherence of the
interacting waves induces novel effects on the dynamics of crossing sea states.


title:
On features of scaling in Duffing oscillator under action of impulses
  with random modulation of parameters

abstract:
  In the work a nonlinear Duffing oscillator is considered under impulse
excitation with two ways of introduction of the random additive term simulating
noise, - with help of amplitude modulation and modulation of period of impulses
sequence. The scaling properties both in the Feigenbaum scenario and in the
tricritical case are shown.


title: Population Growth and Periodic Instability of the International System

abstract:
  From the perspective developed in this paper, it can be argued that
exponential population growth resulted in the exponential decrease of the
life-span of consecutive stable periods during the life-span of the European
international system (1480-1945). However, it becomes evident as well that
population growth as such is not a sufficient condition to generate a
punctuated equilibrium dynamic in the war dynamics of the international system:
other conditions and factors - and their interplay - contribute to this typical
dynamic as well. From 1945 until the collapse of the Soviet Union (1991), the
conditions of international system differed fundamentally from the conditions
of the European international system during the period 1480-1945. It can be
argued, that sooner or later a punctuated equilibrium war dynamic will resume.


title:
Interaction and Localization of One-electron Orbitals in an Organic
  Molecule: Fictitious Parameter Analysis for Multi-physics Simulations

abstract:
  We present a new methodology to analyze complicated multi-physics simulations
by introducing a fictitious parameter. Using the method, we study quantum
mechanical aspects of an organic molecule in water. The simulation is
variationally constructed from the ab initio molecular orbital method and the
classical statistical mechanics with the fictitious parameter representing the
coupling strength between solute and solvent. We obtain a number of
one-electron orbital energies of the solute molecule derived from the
Hartree-Fock approximation, and eigenvalue-statistical analysis developed in
the study of nonintegrable systems is applied to them. Based on the results, we
analyze localization properties of the electronic wavefunctions under the
influence of the solvent.


title:
On the general solution for the modified Emden type equation
  $\ddot{x}+\alpha x\dot{x}+\beta x^3=0$

abstract:
  In this paper, we demonstrate that the modified Emden type equation (MEE),
$\ddot{x}+\alpha x\dot{x}+\beta x^3=0$, is integrable either explicitly or by
quadrature for any value of $\alpha$ and $\beta$. We also prove that the MEE
possesses appropriate time-independent Hamiltonian function for the full range
of parameters $\alpha$ and $\beta$. In addition, we show that the MEE is
intimately connected with two well known nonlinear models, namely the
force-free Duffing type oscillator equation and the two dimensional
Lotka-Volterra (LV) equation and thus the complete integrability of the latter
two models can also be understood in terms of the MEE.


title:
Why the Maxwellian Distribution is the Attractive Fixed Point of the
  Boltzmann Equation

abstract:
  The origin of the Boltzmann factor is revisited. An alternative derivation
from the microcanonical picture is given. The Maxwellian distribution in a
mono-dimensional ideal gas is obtained by following this insight. Other
possible applications, as for instance the obtaining of the wealth distribution
in the human society, are suggested in the remarks.

categories: nlin.CD cond-mat.stat-mech cs.MA math.ST physics.class-ph stat.TH


title: Lyapunov Modes for a Nonequilibrium System with a Heat Flux

abstract:
  We present the first numerical observation of Lyapunov modes (mode structure
of Lyapunov vectors) in a system maintained in a nonequilibrium steady state.
The modes show some similarities and some differences when compared with the
results for equilibrium systems. The breaking of energy conservation removes a
zero exponent and introduces a new mode. The transverse modes are only weakly
altered but there are systematic changes to the longitudinal and momentum
dependent modes.


title:
Combinatorial Bethe ansatz and ultradiscrete Riemann theta function with
  rational characteristics

abstract:
  The U_q(\hat{sl}_2) vertex model at q=0 with periodic boundary condition is
an integrable cellular automaton in one-dimension. By the combinatorial Bethe
ansatz, the initial value problem is solved for arbitrary states in terms of an
ultradiscrete analogue of the Riemann theta function with rational
characteristics.


title:
A hierarchy of integrable PDEs in 2+1 dimensions associated with 2 -
  dimensional vector fields

abstract:
  We introduce a hierarchy of integrable PDEs in 2+1 dimensions arising from
the commutation of 2-dimensional vector fields. We also solve the associated
Cauchy problems, using the recently developed Inverse Scattering Transform for
1-parameter families of multidimensional vector fields.


title:
On the Lagrangian and Hamiltonian description of the damped linear
  harmonic oscillator

abstract:
  Using the modified Prelle- Singer approach, we point out that explicit time
independent first integrals can be identified for the damped linear harmonic
oscillator in different parameter regimes. Using these constants of motion, an
appropriate Lagrangian and Hamiltonian formalism is developed and the resultant
canonical equations are shown to lead to the standard dynamical description.
Suitable canonical transformations to standard Hamiltonian forms are also
obtained. It is also shown that a possible quantum mechanical description can
be developed either in the coordinate or momentum representations using the
Hamiltonian forms.


title: Local cause of coherence in Boolean networks

abstract:
  We performed a numerical study on random Boolean networks with power-law rank
outdegree distributions to find local structural cause for the emergence of
high or low degree of coherence in binary state variables. The degree of
randomness and coherence of the binary sequence are measured by entropy and
mutual information, depending on local structure that consists of a node with a
highly connected, called hub and its upstream nodes, and types of Boolean
functions for the nodes. With a large number of output connections from a hub,
the effects of Boolean function on the hub are more prominent. The local
structures that give larger entropy tend to give rise to larger mutual
information. Based on the numerical results and structural conditions we
derived a time-independent transmission characteristic function of state
variables for the local structures. We obtained good relationships between the
numerical and analytical results, which indicate that dynamical properties from
the whole networks can be inferred from the differences in the local
structures.


title:
Exactly-solvable Ising-Heisenberg model for the coupled barotropic fluid
  - rotating solid sphere system - condensation of super and sub-rotating
  barotropic flow states

abstract:
  Exact solutions of a family of Heisenberg-Ising spin-lattice models for a
coupled barotropic flow - massive rotating sphere system under microcanonical
constraint on relative enstrophy is obtained by the method of spherical
constraint. Phase transitions representative of Bose-Einstein condensation in
which highly ordered super and sub-rotating states self-organize from random
initial vorticity states are calculated exactly and related to three key
parameters - spin of sphere, kinetic energy of the barotropic flow which is
specified by the inverse temperature and amount of relative enstrophy which is
held fixed. Angular momentum of the barotropic fluid relative to the rotating
frame of the infinitely massive sphere is the main order parameter in this
statistical mechanics problem $-$ it is not constrained either canonically nor
microcanonically as coupling between the fluid and the rotating sphere by a
complex torque is responsible for its change. This coupling and exchange of
angular momentum is a necessary condition for condensation in this spin-lattice
system. There is no low temperature defects in this model - the partition
function is calculated in closed form for all positive and negative
temperatures. Also note-worthy is the fact that this statistical equilibrium
model is not a mean field model and can be extended to treat fluctuations if
required in more complex coupled flows.


title:
Lie symmetries and solitons in nonlinear systems with spatially
  inhomogeneous nonlinearities

abstract:
  Using Lie group theory and canonical transformations we construct explicit
solutions of nonlinear Schrodinger equations with spatially inhomogeneous
nonlinearities. We present the general theory, use it to show that localized
nonlinearities can support bound states with an arbitrary number solitons and
discuss other applications of interest to the field of nonlinear matter waves.


title: Symmetry-assisted vorticity control in Bose-Einstein condensates

abstract:
  Using group-theoretical methods and numerical simulations we show how to act
on the topological charge of individual vortices in Bose-Einstein condensates
by using control potentials with appropriate discrete symmetries. As examples
of our methodology we study charge inversion and vortex erasing by acting on a
set of control laser gaussian beams generating optical dipole traps.


title:
On the BKP hierarchy: additional symmetries, Fay identity and
  Adler-Shiota-van Moerbeke formula

abstract:
  We give an alternative proof of the Adler-Shiota-van Moerbeke formula for the
BKP hierarchy. The proof is based on a simple expression for the generator of
additional symmetries and the Fay identity of the BKP hierarchy.


title: A Model of a Trust-based Recommendation System on a Social Network

abstract:
  In this paper, we present a model of a trust-based recommendation system on a
social network. The idea of the model is that agents use their social network
to reach information and their trust relationships to filter it. We investigate
how the dynamics of trust among agents affect the performance of the system by
comparing it to a frequency-based recommendation system. Furthermore, we
identify the impact of network density, preference heterogeneity among agents,
and knowledge sparseness to be crucial factors for the performance of the
system. The system self-organises in a state with performance near to the
optimum; the performance on the global level is an emergent property of the
system, achieved without explicit coordination from the local interactions of
agents.


title: Algebro-Geometric Finite-Gap Solutions of the Ablowitz-Ladik Hierarchy

abstract:
  We provide a detailed derivation of all complex-valued algebro-geometric
finite-band solutions of the Ablowitz-Ladik hierarchy. In addition, we survey a
recursive construction of the Ablowitz-Ladik hierarchy and its zero-curvature
and Lax formalism.


title:
Combustion Process in a Spark Ignition Engine: Analysis of Cyclic
  Maximum Pressure and Peak Pressure Angle

abstract:
  In this paper we analyze the cycle-to-cycle variations of maximum pressure
$p_{max}$ and peak pressure angle $\alpha_{pmax}$ in a four-cylinder spark
ignition engine. We examine the experimental time series of $p_{max}$ and
$\alpha_{pmax}$ for three different spark advance angles. Using standard
statistical techniques such as return maps and histograms we show that
depending on the spark advance angle, there are significant differences in the
fluctuations of $p_{max}$ and
  $\alpha_{pmax}$. We also calculate the multiscale entropy of the various time
series to estimate the effect of randomness in these fluctuations. Finally, we
explain how the information on both $p_{max}$ and $\alpha_{pmax}$ can be used
to develop optimal strategies for controlling the combustion process and
improving engine performance.


title:
Semi-classical Expansion of the Smooth Part of the Density of States,
  with Application to the Hydrogen Atom in Uniform Magnetic Field

abstract:
  We give a method to compute the smooth part of the density of states in a
semi-classical expansion, when the Hamiltonian contains a Coulomb potential and
non-cartesian coordinates are appropriate. We apply this method to the case of
the hydrogen atom in a magnetic field with fixed $z$-component of the angular
momentum. This is then compared with numerical results obtained by a high
precision finite element approach. The agreement is excellent especially in the
\emph{chaotic} region of the spectrum. The need to go beyond the Thomas-Fermi
model is clearly established.


title:
A "No-Go" Theorem for the Existence of an Action Principle for Discrete
  Invertible Dynamical Systems

abstract:
  In this paper we study the problem of the existence of a least-action
principle for invertible, second-order dynamical systems, discrete in time and
space. We show that, when the configuration space is finite, a least-action
principle does not exist for such systems. We dichotomize discrete dynamical
systems with infinite configuration spaces into those of finite type for which
this theorem continues to hold, and those not of finite type for which it is
possible to construct a least-action principle. We also show how to recover an
action by restriction of the phase space of certain second-order discrete
dynamical systems. We provide numerous examples to illustrate each of these
results.


title: Quantum mushroom billiards

abstract:
  We report the first large-scale statistical study of very high-lying
eigenmodes (quantum states) of the mushroom billiard proposed by L. Bunimovich
in this journal, vol. 11, 802 (2001). The phase space of this mixed system is
unusual in that it has a single regular region and a single chaotic region, and
no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We
propose a model for dynamical tunneling and show that it predicts well the
chaotic components of predominantly-regular modes. Our model explains our
observed density of such superpositions dying as E^{-1/3} (E is the
eigenvalue). We compare eigenvalue spacing distributions against Random Matrix
Theory expectations, using 16000 odd modes (an order of magnitude more than any
existing study). We outline new variants of mesh-free boundary collocation
methods which enable us to achieve high accuracy and such high mode numbers
orders of magnitude faster than with competing methods.


title:
Influence of weak anisotropy on scaling regimes in a model of advected
  vector field

abstract:
  Influence of weak uniaxial small-scale anisotropy on the stability of
inertial-range scaling regimes in a model of a passive transverse vector field
advected by an incompressible turbulent flow is investigated by means of the
field theoretic renormalization group. Weak anisotropy means that parameters
which describe anisotropy are chosen to be close to zero, therefore in all
expressions it is enough to leave only linear terms in anisotropy parameters.
Turbulent fluctuations of the velocity field are taken to have the Gaussian
statistics with zero mean and defined noise with finite correlations in time.
It is shown that stability of the inertial-range scaling regimes in the
three-dimensional case is not destroyed by anisotropy but the corresponding
stability of the two-dimensional system can be destroyed even by the presence
of weak anisotropy. A borderline dimension $d_c$ below which the stability of
the scaling regime is not present is calculated as a function of anisotropy
parameters.


title:
Influence of anisotropy and compressibility on anomalous scaling of a
  passive scalar field

abstract:
  Influence of uniaxial small-scale anisotropy and compressibility on the
stability of scaling regime and on the anomalous scaling of structure functions
of a scalar field is investigated in the model of a passive scalar field
advected by the compressible Gaussian strongly anisotropic velocity field with
the covariance $\propto \delta(t-t^{\prime})|{\bf x}-{\bf
x^{\prime}}|^{2\epsilon}$ by using the field theoretic renormalization group
and the operator product expansion. The inertial-range stability of the
corresponding scaling regime is established. The anomalous scaling of the
single-time structure functions is studied and the corresponding anomalous
exponents are calculated. Their dependence on the compressibility parameter and
anisotropy parameters is analyzed. It is shown that the presence of
compressibility leads to the decreasing of the critical dimensions of the
important composite operators, i.e., the anomalous scaling is more pronounced
in the compressible systems. This result is demonstrated for the structure
function of the third order. All calculations are done to the first order in
$\epsilon$.


title: The International System: "At the Edge of Chaos"

abstract:
  The assumption that complex systems function optimally at the edge of chaos
seems applicable to the international system as well. In this paper I argue
that the normal chaotic war dynamic of the European international system
(1495-1945) was temporarily (1657-1763) interrupted by a more simplified
dynamic, resulting in more intense Great Power wars and in a delay of the
reorganization of the international system in the 18th century.


title: Normal form for odd periodic FPU chains

abstract:
  In this paper we prove that near the equilibirum position any periodic FPU
chain with an odd number of particles admits a Birkhoff normal form up to order
4, and we obtain an explicit formula of the Hessian of its Hamiltonian at the
fixed point.


title: Clustering of matter in waves and currents

abstract:
  The growth rate of small-scale density inhomogeneities (the entropy
production rate) is given by the sum of the Lyapunov exponents in a random
flow. We derive an analytic formula for the rate in a flow of weakly
interacting waves and show that in most cases it is zero up to the fourth order
in the wave amplitude. We then derive an analytic formula for the rate in a
flow of potential waves and solenoidal currents. Estimates of the rate and the
fractal dimension of the density distribution show that the interplay between
waves and currents is a realistic mechanism for providing patchiness of
pollutant distribution on the ocean surface.

categories: nlin.CD cond-mat.stat-mech physics.flu-dyn physics.geo-ph


title: Bi-Hamiltonian Aspects of a Matrix Harry Dym Hierarchy

abstract:
  We study the Harry Dym hierarchy of nonlinear evolution equations from the
bi-Hamiltonian view point. This is done by using the concept of an S-hierarchy.
It allows us to define a matrix Harry Dym hierarchy of commuting Hamiltonian
flows in two fields that projects onto the scalar Harry Dym hierarchy. We also
show that the conserved densities of the matrix Harry Dym equation can be found
by means of a Riccati-type equation.


title:
R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant
  Spin Chain

abstract:
  The problem of constructing the $SL(N,\mathbb{C})$ invariant solutions to the
Yang-Baxter equation is considered. The solutions ($\mathcal{R}$-operators) for
arbitrarily principal series representations of $SL(N,\mathbb{C})$ are obtained
in an explicit form. We construct the commutative family of the operators
$\mathcal{Q}_k(u)$ which can be identified with the Baxter operators for the
noncompact $SL(N,\mathbb{C})$ spin magnet.


title: Emergence of Chaotic Itinerancy in Simple Ecological Systems

abstract:
  Chaotic itinerancy is a universal dynamical concept that describes itinerant
motion among many different ordered states through chaotic transition in
dynamical systems. Unlike the expectation of the prevalence of chaotic
itinerancy in high-dimensional systems, we identify chaotic itinerant behavior
from a relatively simple ecological system, which consists only of two coupled
consumer-resource pairs. The system exhibits chaotic bursting activity, in
which the explosion and the shrinkage of the population alternate indefinitely,
while the explosion of one pair co-occurs with the shrinkage of the other pair.
We analyze successfully the bursting activity in the framework of chaotic
itinerancy, and find that large duration times of bursts tend to cluster in
time, allowing the effective burst prognosis. We also investigate the control
schemes on the bursting activity, and demonstrate that invoking the competitive
rise of the consumer in one pair can even elongate the burst of the other pair
rather than shorten it.


title: Physical dynamics of quasi-particles in nonlinear wave equations

abstract:
  By treating the centers of solitons as point particles and studying their
discrete dynamics, we demonstrate a new approach to the quantization of the
soliton solutions of the sine-Gordon equation, one of the first model nonlinear
field equations. In particular, we show that a linear superposition of the
non-interacting shapes of two solitons offers a qualitative (and to a good
approximation quantitative) description of the true two-soliton solution,
provided that the trajectories of the centers of the superimposed solitons are
considered unknown. Via variational calculus, we establish that the dynamics of
the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass
terms. The successful identification of the governing equations of the
(discrete) quasi-particles from the (continuous) field equation shows that the
proposed approach provides a basis for the passage from the continuous to a
discrete description of the field.


title:
On some properties of nearly conservative dynamics of Ikeda map and its
  relation with the conservative case

abstract:
  The behavior of the well-known Ikeda map with very weak dissipation (so
called nearly conservative case) is investigated. The changes in the
bifurcation structure of the parameter plane while decreasing the dissipation
are revealed. It is shown that when the dissipation is very weak the system
demonstrates an "intermediate" type of dynamics combining the peculiarities of
conservative and dissipative dynamics. The correspondence between the
trajectories in the phase space in conservative case and the transformations of
the set of initial conditions in the nearly conservative case is revealed. The
dramatic increase of number of coexisting low-period attractors and the
extraordinary growth of the transient time while the dissipation decreases have
been revealed. The method of plotting a bifurcation trees for the set of
initial conditions has been used to classify existing attractors by it's
structure. Also it was shown that most of coexisting attractors are destroyed
by rather small external noise, and the transient time in noisy driven systems
increases still more. The new method of two-parameter analysis of conservative
systems was proposed.


title: Conjugation of cascades

abstract:
  Presented in this work are some results relative to sequences found in the
logistic equation bifurcation diagram, which is the unimodal quadratic map
prototype. All of the different saddle-node bifurcation cascades, associated to
every last appearance p-periodic orbit (p=3,4,5,...), can also be generated
from the very Feigenbaum cascade. In this way it is evidenced the relationship
between both cascades. The orbits of every saddle-node bifurcation cascade,
mentioned above, are located in different chaotic bands, and this determines a
sequence of orbits converging to every band-merging Misiurewicz point. In turn,
these accumulation points form a sequence whose accumulation point is the
Myrberg-Feigenbaum point. It is also proven that the first appearance orbits in
the n-chaotic band converge to the same point as the last appearance orbits of
the (n+1)-chaotic band. The symbolic sequences of band-merging Misiurewicz
points are computed for any window.


title: Colliding Particles in Highly Turbulent Flows

abstract:
  We discuss relative velocities and the collision rate of small particles
suspended in a highly turbulent fluid. In the limit where the viscous damping
is very weak, we estimate the relative velocities using the Kolmogorov cascade
principle.


title:
The macro-behavior of agents' opinion under the influence of an external
  field

abstract:
  In this paper, a model about the evolution of opinion on small world networks
is proposed. We studied the macro-behavior of the agents' opinion and the
relative change rate as time elapses. The external field was found to play an
important role in making the opinion $s(t)$ balance or increase, and without
the influence of the external field, the relative change rate $\gamma(t)$ shows
a nonlinear increasing behavior as time runs. What's more, this nonlinear
increasing behavior is independent of the initial condition, the strength of
the external field and the time that we cancel the external field. Maybe the
results can reflect some phenomenon in our society, such as the function of the
macro-control in China or the Mass Media in our society.


title:
Random global coupling induces synchronization and nontrivial collective
  behavior in networks of chaotic maps

abstract:
  The phenomena of synchronization and nontrivial collective behavior are
studied in a model of coupled chaotic maps with random global coupling. The
mean field of the system is coupled to a fraction of elements randomly chosen
at any given time. It is shown that the reinjection of the mean field to a
fraction of randomly selected elements can induce synchronization and
nontrivial collective behavior in the system. The regions where these
collective states emerge on the space of parameters of the system are
calculated.


title:
Expanded boundary integral method and chaotic time-reversal doublets in
  quantum billiards

abstract:
  We present the expanded boundary integral method for solving the planar
Helmholtz problem, which combines the ideas of the boundary integral method and
the scaling method and is applicable to arbitrary shapes. We apply the method
to a chaotic billiard with unidirectional transport, where we demonstrate
existence of doublets of chaotic eigenstates, which are quasi-degenerate due to
time-reversal symmetry, and a very particular level spacing distribution that
attains a chaotic Shnirelman peak at short energy ranges and exhibits GUE-like
statistics for large energy ranges. We show that, as a consequence of such
particular level statistics or algebraic tunneling between disjoint chaotic
components connected by time-reversal operation, the system exhibits quantum
current reversals.


title:
Exact Moving and Stationary Solutions of a Generalized Discrete
  Nonlinear Schrodinger Equation

abstract:
  We obtain exact moving and stationary, spatially periodic and localized
solutions of a generalized discrete nonlinear Schr\"odinger equation. More
specifically, we find two different moving periodic wave solutions and a
localized moving pulse solution. We also address the problem of finding exact
stationary solutions and, for a particular case of the model when stationary
solutions can be expressed through the Jacobi elliptic functions, we present a
two-point map from which all possible stationary solutions can be found.
Numerically we demonstrate the generic stability of the stationary pulse
solutions and also the robustness of moving pulses in long-term dynamics.


title:
State Differentiation by Transient Truncation in Coupled Threshold
  Dynamics

abstract:
  Dynamics with a threshold input--output relation commonly exist in gene,
signal-transduction, and neural networks. Coupled dynamical systems of such
threshold elements are investigated, in an effort to find differentiation of
elements induced by the interaction. Through global diffusive coupling, novel
states are found to be generated that are not the original attractor of
single-element threshold dynamics, but are sustained through the interaction
with the elements located at the original attractor. This stabilization of the
novel state(s) is not related to symmetry breaking, but is explained as the
truncation of transient trajectories to the original attractor due to the
coupling. Single-element dynamics with winding transient trajectories located
at a low-dimensional manifold and having turning points are shown to be
essential to the generation of such novel state(s) in a coupled system.
Universality of this mechanism for the novel state generation and its relevance
to biological cell differentiation are briefly discussed.


title: Multi-scale model of gradient evolution in turbulent flows

abstract:
  A multi-scale model for the evolution of the velocity gradient tensor in
fully developed turbulence is proposed. The model is based on a coupling
between a ``Restricted Euler'' dynamics [{\it P. Vieillefosse, Physica A, {\bf
14}, 150 (1984)}] which describes gradient self-stretching, and a deterministic
cascade model which allows for energy exchange between different scales. We
show that inclusion of the cascade process is sufficient to regularize the
well-known finite time singularity of the Restricted Euler dynamics. At the
same time, the model retains topological and geometrical features of real
turbulent flows: these include the alignment between vorticity and the
intermediate eigenvector of the strain-rate tensor and the typical teardrop
shape of the joint probability density between the two invariants, $R-Q$, of
the gradient tensor. The model also possesses skewed, non-Gaussian longitudinal
gradient fluctuations and the correct scaling of energy dissipation as a
function of Reynolds number. Derivative flatness coefficients are in good
agreement with experimental data.


title:
Features of pulsed synchronization of a systems with a tree-dimensional
  phase space

abstract:
  Features of synchronization picture in the system with the limit cycle
embedded in a three-dimensional phase space are considered. By the example of
Ressler system and Dmitriev - Kislov generator under the action of a periodic
sequence of delta - function it is shown, that synchronization picture
significantly depends on the direction of pulse action. Features of
synchronization tons appeared in these models are observed.


title:
Supplement to the article "Approximating multi-dimensional Hamiltonian
  flows by billiards": Proof of C$^{0}$ and C$^{r}$ - closeness Theorems

abstract:
  This supplement to the article "Approximating multi-dimensional Hamiltonian
flows by billiards" contains the proofs of C$^{0}$ and C$^{r}$ - closeness
theorems stated in the article.


title:
Discrete breathers in a two-dimensional hexagonal Fermi-Pasta-Ulam
  lattice

abstract:
  We consider a two-dimensional Fermi-Pasta-Ulam (FPU) lattice with hexagonal
symmetry. Using asymptotic methods based on small amplitude ansatz, at third
order we obtain a reduction to a cubic nonlinear Schrodinger equation (NLS) for
the breather envelope. However, this does not support stable soliton solutions,
so we pursue a higher-order analysis yielding a generalised NLS, which includes
known stabilising terms. We present numerical results which suggest that
long-lived stationary and moving breathers are supported by the lattice. We
find breather solutions which move in an arbitrary direction, an ellipticity
criterion for the wavenumbers of the carrier wave, asymptotic estimates for the
breather energy, and a minimum threshold energy below which breathers cannot be
found. This energy threshold is maximised for stationary breathers, and becomes
vanishingly small near the boundary of the elliptic domain where breathers
attain a maximum speed. Several of the results obtained are similar to those
obtained for the square FPU lattice (Butt & Wattis, J Phys A, 39, 4955,
(2006)), though we find that the square and hexagonal lattices exhibit
different properties in regard to the generation of harmonics, and the isotropy
of the generalised NLS equation.


title: Bottleneck crossover between classical and quantum superfluid turbulence

abstract:
  We consider superfluid turbulence near absolute zero of temperature generated
by classical means, e.g. towed grid or rotation but not by counterflow. We
argue that such turbulence consists of a {\em polarized} tangle of mutually
interacting vortex filaments with quantized vorticity. For this system we
predict and describe a bottleneck accumulation of the energy spectrum at the
classical-quantum crossover scale $\ell$. Demanding the same energy flux
through scales, the value of the energy at the crossover scale should exceed
the Kolmogorov-41 spectrum by a large factor $\ln^{10/3} (\ell/a_0)$ ($\ell$ is
the mean intervortex distance and $a_0$ is the vortex core radius) for the
classical and quantum spectra to be matched in value. One of the important
consequences of the bottleneck is that it causes the mean vortex line density
to be considerably higher that based on K41 alone, and this should be taken
into account in (re)interpretation of new (and old) experiments as well as in
further theoretical studies.


title: Solvable Nonlinear Evolution PDEs in Multidimensional Space

abstract:
  A class of solvable (systems of) nonlinear evolution PDEs in multidimensional
space is discussed. We focus on a rotation-invariant system of PDEs of
Schr\"odinger type and on a relativistically-invariant system of PDEs of
Klein-Gordon type. Isochronous variants of these evolution PDEs are also
considered.


title:
A study on bifurcation diagrams in relation to synchronisation in
  chaotic systems

abstract:
  We numerically study some of the 3-D dynamical systems which exhibit complete
synchronisation as well as generalised synchronisation (GS) to show that these
systems can be conveniently partitioned into equivalent classes facilitating
the study of bifurcation diagrams (BDs) within each class. We demonstrate how
BDs may be helpful in assessing the robustness of GS and in predicting the
nature of the driven system by knowing the BD of driving system and vice versa.
We extend the study to include the possible GS between elements of two
different equivalent classes by taking the example of the
Rossler-driven-Lorenz-system.


title:
First Experimental Observation of Superscars in a Pseudointegrable
  Barrier Billiard

abstract:
  With a perturbation body technique intensity distributions of the electric
field strength in a flat microwave billiard with a barrier inside up to mode
numbers as large as about 700 were measured. A method for the reconstruction of
the amplitudes and phases of the electric field strength from those intensity
distributions has been developed. Recently predicted superscars have been
identified experimentally and - using the well known analogy between the
electric field strength and the quantum mechanical wave function in a
two-dimensional microwave billiard - their properties determined.


title: Differential Geometry of Hydrodynamic Vlasov Equations

abstract:
  We consider hydrodynamic chains in $(1+1)$ dimensions which are Hamiltonian
with respect to the Kupershmidt-Manin Poisson bracket. These systems can be
derived from single $(2+1)$ equations, here called hydrodynamic Vlasov
equations, under the map $A^n =\int_{-\infty}^\infty p^n f dp.$ For these
equations an analogue of the Dubrovin-Novikov Hamiltonian structure is
constructed. The Vlasov formalism allows us to describe objects like the
Haantjes tensor for such a chain in a much more compact and computable way. We
prove that the necessary conditions found by Ferapontov and Marshall in
(arXiv:nlin.SI/0505013) for the integrability of these hydrodynamic chains are
also sufficient.


title:
High Lewis number combustion wavefronts: a perturbative Melnikov
  analysis

abstract:
  The wavefronts associated with a one-dimensional combustion model with
Arrhenius kinetics and no heat loss are analyzed within the high Lewis number
perturbative limit. This situation, in which fuel diffusivity is small in
comparison to that of heat, is appropriate for highly dense fluids. A formula
for the wavespeed is established by a non-standard application of Melnikov's
method and slow manifold theory from dynamical systems, and compared to
numerical results. A simple characterization of the wavespeed correction is
obtained: it is proportional to the ratio between the exothermicity parameter
and the Lewis number. The perturbation method developed herein is also
applicable to more general coupled reaction-diffusion equations with strongly
differing diffusivities. The stability of the wavefronts is also tested using a
numerical Evans function method.


title:
Life cycle of a minimal protocell - a dissipative particle dynamics
  (DPD) study

abstract:
  Cross-reactions and other systematic issues generated by the coupling of
functional chemical subsystems pose the largest challenge for assembling a
viable protocell in the laboratory. Our current work seeks to identify and
clarify such key issues as we represent and analyze in simulation a full
implementation of a minimal protocell. Using a 3D dissipative particle dynamics
(DPD) simulation method we are able to address the coupled diffusion,
self-assembly, and chemical reaction processes, required to model a full life
cycle of the protocell, the protocell being composed of coupled genetic,
metabolic, and container subsystems. Utilizing this minimal structural and
functional representation of the constituent molecules, their interactions, and
their reactions, we identify and explore the nature of the many linked
processes for the full protocellular system.


title: A family of the Poisson brackets compatible with the Sklyanin bracket

abstract:
  We introduce a family of compatible Poisson brackets on the space of $2\times
2$ polynomial matrices, which contains the Sklyanin bracket, and use it to
derive a multi-Hamiltonian structure for a set of integrable systems that
includes $XXX$ Heisenberg magnet, the open and periodic Toda lattices, the
discrete self-trapping model and the Goryachev-Chaplygin gyrostat.


title: Linear response, susceptibility and resonances in chaotic toy models

abstract:
  We consider simple examples illustrating some new features of the linear
response theory developed by Ruelle for dissipative and chaotic systems [{\em
J. of Stat. Phys.} {\bf 95} (1999) 393]. In this theory the concepts of linear
response, susceptibility and resonance, which are familiar to physicists, have
been revisited due to the dynamical contraction of the whole phase space onto
attractors. In particular the standard framework of the
"fluctuation-dissipation" theorem breaks down and new resonances can show up
oustside the powerspectrum. In previous papers we proposed and used new
numerical methods to demonstrate the presence of the new resonances predicted
by Ruelle in a model of chaotic neural network. In this article we deal with
simpler models which can be worked out analytically in order to gain more
insights into the genesis of the ``stable'' resonances and their consequences
on the linear response of the system. We consider a class of 2-dimensional
time-discrete maps describing simple rotator models with a contracting radial
dynamics onto the unit circle and a chaotic angular dynamics $\theta_{t+1} = 2
\theta_t (\mod 2\pi)$. A generalisation of this system to a network of
interconnected rotators is also analysed and related with our previous studies
\cite{CS1,CS2}. These models permit us to classify the different types of
resonances in the susceptibility and to discuss in particular the relation
between the relaxation time of the system to equilibrium with the {\em mixing}
time given by the decay of the correlation functions. Also it enables one to
propose some general mechanisms responsible for the creation of stable
resonances with arbitrary frequencies, widths, and dependency on the pair of
perturbed/observed variables.


abstract:
  The method of solving of nonlinear Schr\"odinger equation is considered. Some
examples of its applications are demonstrated.


title:
Nonlinear Supression of Tunneling and Strong Localization in
  Bose-Einstein Condensates with Spatially Inhomogeneous Interactions

abstract:
  We study the properties of the ground state of Bose-Einstein condensates with
spatially inhomogeneous interactions and show that the atom density experiences
a strong localization at the spatial region where the scattering length is
close to zero while tunneling to regions with positive values of the scattering
length is strongly supressed by the nonlinear interactions.


title: A Kinetic Theory of Coupled Oscillators

abstract:
  We present an approach for the description of fluctuations that are due to
finite system size induced correlations in the Kuramoto model of coupled
oscillators. We construct a hierarchy for the moments of the density of
oscillators that is analogous to the BBGKY hierarchy in the kinetic theory of
plasmas and gases. To calculate the lowest order system size effect, we
truncate this hierarchy at second order and solve the resulting closed
equations for the two-oscillator correlation function around the incoherent
state. We use this correlation function to compute the fluctuations of the
order parameter, including the effect of transients, and compare this
computation with numerical simulations.


title: Intracultural diversity in a model of social dynamics

abstract:
  We study the consequences of introducing individual nonconformity in social
interactions, based on Axelrod's model for the dissemination of culture. A
constraint on the number of situations in which interaction may take place is
introduced in order to lift the unavoidable ho mogeneity present in the final
configurations arising in Axelrod's related models. The inclusion of this
constraint leads to the occurrence of complex patterns of intracultural
diversity whose statistical properties and spatial distribution are
characterized by means of the concepts of cultural affinity and cultural cli
ne. It is found that the relevant quantity that determines the properties of
intracultural diversity is given by the fraction of cultural features that
characterizes the cultural nonconformity of individuals.


title:
Bright N-solitons for the intermediate nonlinear Schr\"{o}dinger
  equation

abstract:
  A previously unknown bright N-soliton solution for an intermediate nonlinear
Schr\"{o}dinger equation of focusing type is presented. This equation is
constructed as a reduction of an integrable system related to a Sato equation
of a 2-component KP hierarchy for certain differential--difference dispersion
relations. Bright soliton solutions are obtained in the form of double
Wronskian determinants.


title:
A new measure of phase synchronization for a pair of time series and
  seizure focus localization

abstract:
  Defining and measuring phase synchronization in a pair of nonlinear time
series are highly nontrivial. This can be done with the help of Fourier
transform, when it exists, for a pair of stored (hence stationary) signals. In
a time series instantaneous phase is often defined with the help of Hilbert
transform. In this paper phase of a time series has been defined with the help
of Fourier transform. This gives rise to a deterministic method to detect phase
synchronization in its most general form between a pair of time series. Since
this is a stricter method than the statistical methods based on instantaneous
phase, this can be used for lateralization and source localization of epileptic
seizures with greater accuracy. Based on this method a novel measure of phase
synchronization, called syn function, has been defined, which is capable of
quantifying neural phase synchronization and asynchronization as important
parameters of epileptic seizure dynamics. It has been shown that such a strict
measure of phase synchronization has potential application in seizure focus
localization from scalp electroencephalogram (EEG) data, without any knowledge
of electrical conductivity of the head.


title:
Statistical Properties of Nonlinear Shell Models of Turbulence from
  Linear Advection Models: Rigorous Results

abstract:
  In a recent paper it was proposed that for some nonlinear shell models of
turbulence one can construct a linear advection model for an auxiliary field
such that the scaling exponents of all the structure functions of the linear
and nonlinear fields coincide.
  The argument depended on an assumption of continuity of the solutions as a
function of a parameter. The aim of this paper is to provide a rigorous proof
for the validity of the assumption. In addition we clarify here when the swap
of a nonlinear model by a linear one will not work.


title:
New approach to the treatment of separatrix chaos and its application to
  the global chaos onset between adjacent separatrices

abstract:
  We have developed the {\it general method} for the description of {\it
separatrix chaos}, basing on the analysis of the separatrix map dynamics.
Matching it with the resonant Hamiltonian analysis, we show that, for a given
amplitude of perturbation, the maximum width of the chaotic layer in energy may
be much larger than it was assumed before. We apply the above theory to explain
the drastic facilitation of global chaos onset in time-periodically perturbed
Hamiltonian systems possessing two or more separatrices, previously discovered
(PRL 90, 174101 (2003)). The theory well agrees with simulations. We also
discuss generalizations and applications. Examples of applications of the
facilitation include: the increase of the DC conductivity in spatially periodic
structures, the reduction of activation barriers for noise-induced transitions
and the related acceleration of spatial diffusion, the facilitation of the
stochastic web formation in a wave-driven or kicked oscillator.


title: Energy and enstrophy fluxes in the double cascade of 2d turbulence

abstract:
  High resolution direct numerical simulations of two-dimensional turbulence in
stationary conditions are presented. The development of an energy-enstrophy
double cascade is studied and found to be compatible with the classical
Kraichnan theory in the limit of extended inertial ranges. The analysis of the
joint distribution of energy and enstrophy fluxes in physical space reveals a
small value of cross correlation. This result supports many experimental and
numerical studies where only one cascade is generated.


title:
The general solution of the matrix equation
  $w_t+\sum\limits_{k=1}^nw_{x_k}\rho^{(k)}(w)=\rho(w)+[w,T\tilde\rho(w)]$

abstract:
  We construct the general solution of the equation
$w_t+\sum\limits_{k=1}^nw_{x_k}\rho^{(k)}(w)=\rho(w)+[w,T\tilde\rho(w)]$, for
the $N\times N$ matrix $w$, where $T$ is any constant diagonal matrix, $n, N
\in \NN_+$ and $\rho^{(k)}, \rho, \tilde\rho: \RR \to \RR$ are arbitrary
analytic functions. Such a solution is based on the observation that, as $w$
evolves according to the above equation, the evolution of its spectrum
decouples, and it is ruled by the scalar analogue of the above equation.
Therefore the eigenvalues of $w$ and suitably normalized eigenvectors are the
$N^2$ Riemann invariants. We also obtain, in the case $\rho=\tilde\rho=0$, a
system of $N^2$ non-differential equations characterizing such a general
solution. We finally discuss reductions of the above matrix equation to systems
of $N$ equations admitting, as Riemann invariants, the eigenvalues of $w$. The
simplest example of such reductions is a particular case of the gas dynamics
equations


title: A Discretization of the Nonholonomic Chaplygin Sphere Problem

abstract:
  The celebrated problem of a non-homogeneous sphere rolling over a horizontal
plane was proved to be integrable and was reduced to quadratures by Chaplygin.
Applying the formalism of variational integrators (discrete Lagrangian systems)
with nonholonomic constraints and introducing suitable discrete constraints, we
construct a discretization of the n-dimensional generalization of the Chaplygin
sphere problem, which preserves the same first integrals as the continuous
model, except the energy. We then study the discretization of the classical
3-dimensional problem for a class of special initial conditions, when an analog
of the energy integral does exist and the corresponding map is given by an
addition law on elliptic curves. The existence of the invariant measure in this
case is also discussed.


title:
Controlling pulse propagation in optical fibers through nonlinearity and
  dispersion management

abstract:
  In case of the nonlinear Schr\"odinger equation with designed group velocity
dispersion, variable nonlinearity and gain/loss; we analytically demonstrate
the phenomenon of chirp reversal crucial for pulse reproduction. Two different
scenarios are exhibited, where the pulses experience identical dispersion
profiles, but show entirely different propagation behavior. Exact expressions
for dynamical quasi-solitons and soliton bound-states relevant for fiber
communication are also exhibited.


title: Transport control in deterministic ratchet system

abstract:
  We study the control of transport properties in a deterministic inertia
ratchet system via the extended delay feedback method. A chaotic current of a
deterministic inertia ratchet system is controlled to a regular current by
stabilizing unstable periodic orbits embedded in a chaotic attractor of the
unperturbed system. By selecting an unstable periodic orbit, which has a
desired transport property, and stabilizing it via the extended delay feedback
method, we can control transport properties of the deterministic inertia
ratchet system. Also, we show that the extended delay feedback method can be
utilized for separation of particles in the deterministic inertia ratchet
system as a particle's initial condition varies.


title: Geometric characterization of nodal domains: the area-to-perimeter ratio

abstract:
  In an attempt to characterize the distribution of forms and shapes of nodal
domains in wave functions, we define a geometric parameter - the ratio $\rho$
between the area of a domain and its perimeter, measured in units of the
wavelength $1/\sqrt{E}$. We show that the distribution function $P(\rho)$ can
distinguish between domains in which the classical dynamics is regular or
chaotic. For separable surfaces, we compute the limiting distribution, and show
that it is supported by an interval, which is independent of the properties of
the surface. In systems which are chaotic, or in random-waves, the
area-to-perimeter distribution has substantially different features which we
study numerically. We compare the features of the distribution for chaotic wave
functions with the predictions of the percolation model to find agreement, but
only for nodal domains which are big with respect to the wavelength scale. This
work is also closely related to, and provides a new point of view on
isoperimetric inequalities.


title:
Application of Dressing Method for Long Wave-Short Wave Resonance
  Interaction Equation

abstract:
  In this paper we investigate the application of Zakharov - Shabat dressing
method to (2+1) - dimensional long wave - short wave resonance interaction
equation (LSRI). Using this method we can construct the exact N - soliton
solution of this equation depending on arbitrary constants. It contains both
solutions which don't decay along N different directions in space infinity, and
"dromion" ones, or localized solutions that decay exponentially in all
directions.


title:
Eigenvalues of Killing Tensors and Separable Webs on Riemannian and
  Pseudo-Riemannian Manifolds

abstract:
  Given a $n$-dimensional Riemannian manifold of arbitrary signature, we
illustrate an algebraic method for constructing the coordinate webs separating
the geodesic Hamilton-Jacobi equation by means of the eigenvalues of $m \leq n$
Killing two-tensors. Moreover, from the analysis of the eigenvalues,
information about the possible symmetries of the web foliations arises. Three
cases are examined: the orthogonal separation, the general separation,
including non-orthogonal and isotropic coordinates, and the conformal
separation, where Killing tensors are replaced by conformal Killing tensors.
The method is illustrated by several examples and an application to the
L-systems is provided.


title: Predictability of El Ni\~no as a Nonlinear Stochastic Limit Cycle

abstract:
  The El Ni\~no phenomenon, synonymously El Ni\~no-Southern Oscillation (ENSO),
is an anomalous climatic oscillation in the Equatorial Pacific that occurs once
every 3-8 years. It affects the earth's climate on a global scale. Whether it
is a cyclic or a sporadic event, or whether its apparently random behaviour can
be explained by stochastic dynamics have remained matters of debate. Herein
ENSO is viewed, unconventionally, as a two-dimensional dynamical system on a
desktop. The main features of ENSO: irregularity, interannual variability, and
the asymmetry between El Ni\~no and La Ni\~na are captured, simply,
comprehensibly, quickly and cheaply, in a nonlinear stochastic limit cycle
paradigm. Its predictability for ENSO compares remarkably well with that of the
best state-of-the-art complex models from the European and American
meteorological centres. Additionally, for the first time, by analyzing
subsurface Equatorial Pacific data since 1960, this model finds that long-term
variations are not caused by ENSO itself, but by external sources.


title:
Study of transients in the propagation of nonlinear waves in some
  reaction diffusion systems

abstract:
  We study the transient dynamics of single species reaction diffusion systems
whose reaction terms $f(u)$ vary nonlinearly near $u\approx 0$, specifically as
$f(u)\approx u^{2}$ and $f(u)\approx u^{3}$. We consider three cases, calculate
their traveling wave fronts and speeds \emph{analytically} and solve the
equations numerically with different initial conditions to study the approach
to the asymptotic front shape and speed. Observed time evolution is found to be
quite sensitive to initial conditions and to display in some cases nonmonotonic
behavior. Our analysis is centered on cases with $f'(0)=0$, and uncovers
findings qualitatively as well quantitatively different from the more familiar
reaction diffusion equations with $f'(0)>0$. These differences are ascribable
to the disparity in time scales between the evolution of the front interior and
the front tail.


title: Numerical simulations of flow reversal in Rayleigh--B\'enard convection

abstract:
  We investigate numerically the statistical properties of the large scale flow
in Rayleigh--B\'enard convection. By using an external random perturbation on
the temperature field, we were able to decrease the effective Prandtl number of
the flow while keeping the Rayleigh number relatively small; this increases the
Reynolds number thus making possible the numerical investigation of the
long--term flow statistics. We also propose a simple and quantitative
explanation for the experimental findings on the statistical distribution of
flow reversals and reorientations.


title: On the dressing method for Dunajski anti-self-duality equation

abstract:
  A dressing scheme applicable to Dunajski equation is developed. Simple
example of constructing solutions in terms of implicit functions is considered.
Dunajski equation hierarchy is described, its Lax-Sato form is presented.
Dunajsky equation hierarchy is characterised by conservation of
three-dimensional volume form, in which a spectral variable is taken into
account. Some reductions of Dunajsky equation hierarchy, including
waterbag-type reduction, are studied.


title: Boolean Delay Equations: A simple way of looking at complex systems

abstract:
  Boolean Delay Equations (BDEs) are semi-discrete dynamical models with
Boolean-valued variables that evolve in continuous time. Systems of BDEs can be
classified into conservative or dissipative, in a manner that parallels the
classification of ordinary or partial differential equations. Solutions to
certain conservative BDEs exhibit growth of complexity in time. They represent
therewith metaphors for biological evolution or human history. Dissipative BDEs
are structurally stable and exhibit multiple equilibria and limit cycles, as
well as more complex, fractal solution sets, such as Devil's staircases and
``fractal sunbursts``. All known solutions of dissipative BDEs have stationary
variance. BDE systems of this type, both free and forced, have been used as
highly idealized models of climate change on interannual, interdecadal and
paleoclimatic time scales. BDEs are also being used as flexible, highly
efficient models of colliding cascades in earthquake modeling and prediction,
as well as in genetics. In this paper we review the theory of systems of BDEs
and illustrate their applications to climatic and solid earth problems. The
former have used small systems of BDEs, while the latter have used large
networks of BDEs. We moreover introduce BDEs with an infinite number of
variables distributed in space (``partial BDEs``) and discuss connections with
other types of dynamical systems, including cellular automata and Boolean
networks. This research-and-review paper concludes with a set of open
questions.


title:
New multidimensional partially integrable generalization of S-integrable
  N-wave equation

abstract:
  This paper develops a modification of the dressing method based on the
inhomogeneous linear integral equation with integral operator having nonempty
kernel. Method allows one to construct the systems of multidimensional Partial
Differential Equations (PDEs) having the differential polynomial forms in any
dimension n. Associated solution space is not full, although it is parametrized
by a certain number of arbitrary functions of (n-1)-variables. We consider
4-dimensional generalization of the classical (2+1)-dimensional S-integrable
N-wave equation as an example.


title:
Ring Intermittency in Coupled Chaotic Oscillators at the Boundary of
  Phase Synchronization

abstract:
  A new type of intermittent behavior is described to occur near the boundary
of phase synchronization regime of coupled chaotic oscillators. This mechanism,
called ring intermittency, arises for sufficiently high initial mismatches in
the frequencies of the two coupled systems. The laws for both the distribution
and the mean length of the laminar phases versus the coupling strength are
analytically deduced. A very good agreement between the theoretical results and
the numerically calculated data is shown. We discuss how this mechanism is
expected to take place in other relevant physical circumstances.


title:
On-Off Intermittency in Time Series of Spontaneous Paroxysmal Activity
  in Rats with Genetic Absence Epilepsy

abstract:
  Dynamic behavior of complex neuronal ensembles is a topic comprising a
streamline of current researches worldwide. In this article we study the
behavior manifested by epileptic brain, in the case of spontaneous
non-convulsive paroxysmal activity. For this purpose we analyzed archived
long-term recording of paroxysmal activity in animals genetically susceptible
to absence epilepsy, namely WAG/Rij rats. We first report that the brain
activity alternated between normal states and epilepsy paroxysms is the on-off
intermittency phenomenon which has been observed and studied earlier in the
different nonlinear systems.

categories: nlin.CD nlin.AO physics.bio-ph physics.med-ph


title:
Complex variables for separation of Hamilton-Jacobi equation on
  three-dimensional Minkowski space

abstract:
  The real coordinates separating geodesic Hamilton-Jacobi equation on
three-dimensional Minkowski space in several cases cannot be defined in the
whole space. We show through an example how to naturally extend them to complex
variables defined everywhere (excluding the singular surfaces of each
coordinate system only) and still separating the same equation.


title: Growing condensate in two-dimensional turbulence

abstract:
  We report a numerical study, supplemented by phenomenological explanations,
of ``energy condensation'' in forced 2D turbulence in a biperiodic box.
Condensation is a finite size effect which occurs after the standard inverse
cascade reaches the size of the system. It leads to emergence of a coherent
vortex dipole. We show that the time growth of the dipole is self-similar, and
it contains most of the injected energy, thus resulting in an energy spectrum
which is markedly steeper than the standard $k^{-5/3}$ one. Once the coherent
component is subtracted, however, the remaining fluctuations have a spectrum
close to $k^{-1}$. The fluctuations decay slowly as the coherent part grows.


title: Synchronized dynamics of cortical neurons with time-delay feedback

abstract:
  The dynamics of three mutually coupled cortical neurons with time delays in
the coupling are explored numerically and analytically. The neurons are coupled
in a line, with the middle neuron sending a somewhat stronger projection to the
outer neurons than the feedback it receives, to model for instance the relay of
a signal from primary to higher cortical areas. For a given coupling
architecture, the delays introduce correlations in the time series at the
time-scale of the delay. It was found that the middle neuron leads the outer
ones by the delay time, while the outer neurons are synchronized with zero lag
times. Synchronization is found to be highly dependent on the synaptic time
constant, with faster synapses increasing both the degree of synchronization
and the firing rate. Analysis shows that presynaptic input during the
interspike interval stabilizes the synchronous state, even for arbitrarily weak
coupling, and independent of the initial phase. The finding may be of
significance to synchronization of large groups of cells in the cortex that are
spatially distanced from each other.


title: Rotating Accelerator-Mode Islands

abstract:
  The existence of rotating accelerator-mode islands (RAIs), performing
quasiregular motion in rotational resonances of order $m>1$ of the standard
map, is firmly established by an accurate numerical analysis of all the known
data. It is found that many accelerator-mode islands for relatively small
nonintegrability parameter $K$ are RAIs visiting resonances of different orders
$m\leq 3$. For sufficiently large $K$, one finds also ``pure'' RAIs visiting
only resonances of the {\em same} order, $m=2$ or $m=3$. RAIs, even quite small
ones, are shown to exhibit sufficient stickiness to produce an anomalous
chaotic transport. The RAIs are basically different in nature from
accelerator-mode islands in resonances of the ``forced'' standard map which was
extensively studied recently in the context of quantum accelerator modes.


title:
Two-dimensional nonlocal vortices, multipole solitons and azimuthons in
  dipolar Bose-Einstein condensates

abstract:
  We have performed numerical analysis of the two-dimensional (2D) soliton
solutions in Bose-Einstein condensates with nonlocal dipole-dipole
interactions. For the modified 2D Gross-Pitaevski equation with nonlocal and
attractive local terms, we have found numerically different types of nonlinear
localized structures such as fundamental solitons, radially symmetric vortices,
nonrotating multisolitons (dipoles and quadrupoles), and rotating multisolitons
(azimuthons). By direct numerical simulations we show that these structures can
be made stable.


title: How periodic orbit bifurcations drive multiphoton ionization

abstract:
  The multiphoton ionization of hydrogen by a strong bichromatic microwave
field is a complex process prototypical for atomic control research. Periodic
orbit analysis captures this complexity: Through the stability of periodic
orbits we can match qualitatively the variation of experimental ionization
rates with a control parameter, the relative phase between the two modes of the
field. Moreover, an empirical formula reproduces quantum simulations to a high
degree of accuracy. This quantitative agreement shows how short periodic orbits
organize the dynamics in multiphoton ionization.


title: Optimization of synchronization in gradient clustered networks

abstract:
  We consider complex clustered networks with a gradient structure, where sizes
of the clusters are distributed unevenly. Such networks describe more closely
actual networks in biophysical systems and in technological applications than
previous models. Theoretical analysis predicts that the network
synchronizability can be optimized by the strength of the gradient field but
only when the gradient field points from large to small clusters. A remarkable
finding is that, if the gradient field is sufficiently strong,
synchronizability of the network is mainly determined by the properties of the
subnetworks in the two largest clusters. These results are verified by
numerical eigenvalue analysis and by direct simulation of synchronization
dynamics on coupled-oscillator networks.


title: Escape time statistics for mushroom billiards

abstract:
  Chaotic orbits of mushroom billiards display intermittent behaviors. We
investigate statistical properties of this system by constructing an infinite
partition on the chaotic part of a Poincar\'e surface which illustrates details
of chaotic dynamics. Each piece of the infinite partition has an unique escape
time from the half disk region, and from this result it is shown that, for
fixed values of the system parameters, the escape time distribution obeys power
law $1/t_{\rm esc}^3$.


title:
Interactions of Parametrically Driven Dark Solitons. I: N\'eel-N\'eel
  and Bloch-Bloch interactions

abstract:
  We study interactions between the dark solitons of the parametrically driven
nonlinear Schr\"odinger equation, Eq.(\ref{NLS}). When the driving strength,
$h$, is below $\sqrt{\gamma^2 +1/9}$, two well-separated N\'eel walls may repel
or attract. They repel if their initial separation $2z(0)$ is larger than the
distance $2z_u$ between the constituents in the unstable stationary complex of
two walls. They attract and annihilate if $2z(0)$ is smaller than $2z_u$. Two
N\'eel walls with $h$ lying between $\sqrt{\gamma^2 + 1/9}$ and a threshold
driving strength $h_{sn}$ attract for $2z(0)<2z_u$ and evolve into a stable
stationary bound state for $2z(0)>2z_u$. Finally, the N\'eel walls with $h$
greater than $h_{sn}$ attract and annihilate -- irrespective of their initial
separation. Two Bloch walls of opposite chiralities attract, while Bloch walls
of like chiralities repel -- except near the critical driving strength, where
the difference between the like-handed and oppositely-handed walls becomes
negligible. In this limit, similarly-handed walls at large separations repel
while those placed at shorter distances may start moving in the same direction
or transmute into an oppositely-handed pair and attract. The collision of two
Bloch walls or two nondissipative N\'eel walls typically produces a quiescent
or moving breather.


title: On the One Class of Hyperbolic Systems

abstract:
  The classification problem is solved for some type of nonlinear lattices.
These lattices are closely related to the lattices of Ruijsenaars-Toda type and
define the Backlund auto-transformations for the class of two-component
hyperbolic systems.


title: Unmixing in Random Flows

abstract:
  We consider particles suspended in a randomly stirred or turbulent fluid.
When effects of the inertia of the particles are significant, an initially
uniform scatter of particles can cluster together. We analyse this 'unmixing'
effect by calculating the Lyapunov exponents for dense particles suspended in
such a random three-dimensional flow, concentrating on the limit where the
viscous damping rate is small compared to the inverse correlation time of the
random flow (that is, the regime of large Stokes number). In this limit
Lyapunov exponents are obtained as a power series in a parameter which is a
dimensionless measure of the inertia. We report results for the first seven
orders. The perturbation series is divergent, but we obtain accurate results
from a Pade-Borel summation. We deduce that particles can cluster onto a
fractal set and show that its dimension is in satisfactory agreement with
previously reported in simulations of turbulent Navier-Stokes flows. We also
investigate the rate of formation of caustics in the particle flow.


title:
Energy diffusion in strongly driven quantum chaotic systems: Role of
  correlations of the matrix elements

abstract:
  The energy evolution of a quantum chaotic system under the perturbation that
harmonically depends on time is studied for the case of large perturbation, in
which the rate of transition calculated from the Fermi golden rule (FGR) is
about or exceeds the frequency of perturbation. For this case the models of
Hamiltonian with random non-correlated matrix elements demonstrate that the
energy evolution retains its diffusive character, but the rate of diffusion
increases slower than the square of the magnitude of perturbation, thus
destroying the quantum-classical correspondence for the energy diffusion and
the energy absorption in the classical limit $\hbar \to 0$. The numerical
calculation carried out for a model built from the first principles (the
quantum analog of the Pullen - Edmonds oscillator) demonstrates that the
evolving energy distribution, apart from the diffusive component, contains a
ballistic one with the energy dispersion that is proportional to the square of
time. This component originates from the chains of matrix elements with
correlated signs and vanishes if the signs of matrix elements are randomized.
The presence of the ballistic component formally extends the applicability of
the FGR to the non-perturbative domain and restores the quantum-classical
correspondence.


title: Bethe Ansatz Solutions of the Bose-Hubbard Dimer

abstract:
  The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for
describing tunneling phenomena of Bose-Einstein condensates. One of the
significant mathematical properties of the model is that it can be exactly
solved by Bethe ansatz methods. Here we review the known exact solutions,
highlighting the contributions of V.B. Kuznetsov to this field. Two of the
exact solutions arise in the context of the Quantum Inverse Scattering Method,
while the third solution uses a differential operator realisation of the su(2)
Lie algebra.


title:
Generalized N=4 supersymmetric Toda lattice hierarchy and N=4
  superintegrable mapping

abstract:
  It is shown that the one-dimensional generalized N=4 supersymmetric Toda
lattice (TL) hierarchy (nlin.Si/0311030) contains the N=4 super-KdV hierarchy
with the first flow time in the role of space coordinate. Two different N=2
superfield forms of the generalized N=4 supersymmetric TL equation, which are
useful when solving the N=4 super-KdV and (1,1)-GNLS hierarchies, are
discussed.


title:
Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic
  Deformations

abstract:
  We study the topology of quasiperiodic solutions of the vortex filament
equation in a neighborhood of multiply covered circles. We construct these
solutions by means of a sequence of isoperiodic deformations, at each step of
which a real double point is "unpinched" to produce a new pair of branch points
and therefore a solution of higher genus. We prove that every step in this
process corresponds to a cabling operation on the previous curve, and we
provide a labelling scheme that matches the deformation data with the knot type
of the resulting filament.


title:
Hypergeometric solutions to the q-Painlev\'e equation of type
  $A_4^{(1)}$

abstract:
  We consider the q-Painlev\'e equation of type $A_4^{(1)}$ (a version of
q-Painlev\'e V equation) and construct a family of solutions expressible in
terms of certain basic hypergeometric series. We also present the determinant
formula for the solutions.


title:
Failure of Parameter Identification Based on Adaptive Synchronization
  Techniques

abstract:
  In the paper, several concrete examples, as well as their numerical
simulations, are given to show that parameter identification based on the
so-called adaptive synchronization techniques might be failed if those
functions with parameters pending for identification in coupled systems are
designed to be mutually linearly dependent or approximately linearly dependent
on the orbit in the synchronization manifold. This failure might be emergent
not only when the synchronized orbit is selected to be some sort of equilibrium
or some sort of periodic oscillation, but also when it is taken as some type of
chaotic attractor produced by driving system. This result implies that chaotic
property of driving signal is not necessary to achievement of parameter
identification. The mechanism inducing such a failure, as well as the bounded
property of all trajectories generated by coupled systems, is theoretically
expatiated. New synchronization techniques are proposed to rigorously realize
the complete synchronization and parameter identification in a class of systems
where the nonlinearity is not globally Lipschitz. In addition, parameter
identification are discussed for systems with time delay.


title: Restricted Flows and the Soliton Equation with Self-Consistent Sources

abstract:
  The KdV equation is used as an example to illustrate the relation between the
restricted flows and the soliton equation with self-consistent sources.
Inspired by the results on the Backlund transformation for the restricted flows
(by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations
for the KdV equation with self-consistent sources (KdVES). These Darboux
transformations are used to get some explicit solutions of the KdVES, which
include soliton, rational, positon, and negaton solutions.


title: On the Darboux-Nijenhuis Variables for the Open Toda Lattice

abstract:
  We discuss two known constructions proposed by Moser and by Sklyanin of the
Darboux-Nijenhuis coordinates for the open Toda lattice.


title:
Interactions of Parametrically Driven Dark Solitons. II: N\'eel-Bloch
  interactions

abstract:
  The interaction between a Bloch and a N\'eel wall in the parametrically
driven nonlinear Schr\"odinger equation is studied by following the
dissociation of their unstable bound state. Mathematically, the analysis
focusses on the splitting of a four-fold zero eigenvalue associated with a pair
of infinitely separated Bloch and N\'eel walls. It is shown that a Bloch and a
N\'eel wall interact as two classical particles, one with positive and the
other one with negative mass.


title: Continuous and Discrete (Classical) Heisenberg Spin Chain Revised

abstract:
  Most of the work done in the past on the integrability structure of the
Classical Heisenberg Spin Chain (CHSC) has been devoted to studying the $su(2)$
case, both at the continuous and at the discrete level. In this paper we
address the problem of constructing integrable generalized ''Spin Chains''
models, where the relevant field variable is represented by a $N\times N$
matrix whose eigenvalues are the $N^{th}$ roots of unity. To the best of our
knowledge, such an extension has never been systematically pursued. In this
paper, at first we obtain the continuous $N\times N$ generalization of the CHSC
through the reduction technique for Poisson-Nijenhuis manifolds, and exhibit
some explicit, and hopefully interesting, examples for $3\times 3$ and $4\times
4$ matrices; then, we discuss the much more difficult discrete case, where a
few partial new results are derived and a conjecture is made for the general
case.


title:
Spatial and temporal feedback control of traveling wave solutions of the
  two-dimensional complex Ginzburg-Landau equation

abstract:
  Previous work has shown that Benjamin-Feir unstable traveling waves of the
complex Ginzburg-Landau equation (CGLE) in two spatial dimensions cannot be
stabilized using a particular time-delayed feedback control mechanism known as
`time-delay autosynchronisation'. In this paper, we show that the addition of
similar spatial feedback terms can be used to stabilize such waves. This type
of feedback is a generalization of the time-delay method of Pyragus (Phys.
Letts. A 170, 1992) and has been previously used to stabilize waves in the
one-dimensional CGLE by Montgomery and Silber (Nonlinearity 17, 2004). We
consider two cases in which the feedback contains either one or two spatial
terms. We focus on how the spatial terms may be chosen to select the direction
of travel of the plane waves. Numerical linear stability calculations
demonstrate the results of our analysis.


title:
Asymptotically pivotal statistic for surrogate testing with extended
  hypothesis

abstract:
  The method of surrogate data provides a framework for testing observed data
against a hierarchy of alternative hypotheses. The aim of applying this method
is to exclude the possibility that the data are consistent with simple linear
explanations before seeking complex nonlinear causes. However, in recent time
the method has attracted considerable criticism, largely as a result of
ambiguity about the formation of the underlying null hypotheses, or about the
power of the chosen statistic. In this communication we show that by employing
a special family of ranks statistics these problems can be avoided and the
method of surrogate data placed of a firm statistical foundation.


title:
Stabilization of high-order solutions of the cubic Nonlinear Schrodinger
  Equation

abstract:
  In this paper we consider the stabilization of non-fundamental unstable
stationary solutions of the cubic nonlinear Schrodinger equation. Specifically
we study the stabilization of radially symmetric solutions with nodes and
asymmetric complex stationary solutions. For the first ones we find partial
stabilization similar to that recently found for vortex solutions while for the
later ones stabilization does not seem possible.


title: Weakly nonassociative algebras, Riccati and KP hierarchies

abstract:
  It has recently been observed that certain nonassociative algebras (called
"weakly nonassociative", WNA) determine, via a universal hierarchy of ordinary
differential equations, solutions of the KP hierarchy with dependent variable
in an associative subalgebra (the middle nucleus). We recall central results
and consider a class of WNA algebras for which the hierarchy of ODEs reduces to
a matrix Riccati hierarchy, which can be easily solved. The resulting solutions
of a matrix KP hierarchy then determine (under a rank 1 condition) solutions of
the scalar KP hierarchy. We extend these results to the discrete KP hierarchy.
Moreover, we build a bridge from the WNA framework to the Gelfand-Dickey
formulation of the KP hierarchy.


title:
Nonstationary excitations in Bose-Einstein condensates under the action
  of periodically varying scattering length with time dependent frequencies

abstract:
  We investigate nonstationary excitations in 3D-Bose-Einstein condensates in a
spherically symmetric trap potential under the modulation of scattering length
with slowly varying frequencies (adiabatic modulation). By numerically solving
the Gross-Pitaevskii equation we observe a step-wise increase in the amplitude
of oscillation due to successive phase locking between driving frequency and
nonlinear frequency. Such a nonstationary excitation has been shown to exist by
an analytic approach using variational procedure and perturbation theory in the
action-angle variables. By using a canonical perturbation theory, we have
identified the successive resonance excitations whenever the driven frequency
matches the nonlinear frequency or its subharmonics.


title: Chaos in Simple Rotation-Translation Models

abstract:
  The chaotic properties of simple two-dimensional rotation-translation models
are explored and simulated. The models are given in difference equation forms,
while the corresponding differential equations systems are studied and the
resulting trajectories in the plane are explored and illustrated in the
computer experiments done. Characteristic patterns, egg-shaped forms and
central chaotic bulges are present when particles are introduced in the
rotating system. The resulting forms and chaotic attractors mainly depend on
the form of the nonlinear function expressing the rotation angle. Several cases
are studied corresponding to a central force rotation system.


title: Special polynomials associated with the K2 hierarchy

abstract:
  New special polynomials associated with the rational solutions of analogue to
the Painleve hierarchies are introduced. The Hirota relations for these special
polynomials are found. Differential - difference hierarchies for finding
special polynomials are presented. These formulae allow us to search the
special polynomials associated with the hierarchy studied. The ordinary
differential hierarchy for the Yablonskii - Vorob'ev polynomials is given.


title:
Nonlinear Dynamical Model of Regime Switching Between Conventions and
  Business Cycles

abstract:
  We introduce and study a non-equilibrium continuous-time dynamical model of
the price of a single asset traded by a population of heterogeneous interacting
agents in the presence of uncertainty and regulatory constraints. The model
takes into account (i) the price formation delay between decision and
investment by the second-order nature of the dynamical equations, (ii) the
linear and nonlinear mean-reversal or their contrarian in the form of
speculative price trading, (iii) market friction, (iv) uncertainty in the
fundamental value which controls the amplitude of mispricing, (v) nonlinear
speculative momentum effects and (vi) market regulations that may limit large
mispricing drifts. We find markets with coexisting equilibrium, conventions and
business cycles, which depend on (a) the relative strength of value-investing
versus momentum-investing, (b) the level of uncertainty on the fundamental
value and (c) the degree of market regulation. The stochastic dynamics is
characterized by nonlinear geometric random walk-like processes with
spontaneous regime shifts between different conventions or business cycles.
This model provides a natural dynamical framework to model regime shifts
between different market phases that may result from the interplay between the
effects (i-vi).


title: Exact solvability of superintegrable Benenti systems

abstract:
  We establish quantum and classical exact solvability for two large classes of
maximally superintegrable Benenti systems in $n$ dimensions with arbitrarily
large $n$. Namely, we solve the Hamilton--Jacobi and Schr\"odinger equations
for the systems in question. The results obtained are illustrated for a model
with the cubic potential.


title: A model of coupled maps with Pareto behavior

abstract:
  A deterministic system of coupled maps is proposed as a model for economic
activity among interacting agents. The values of the maps represent the wealth
of the agents. The dynamics of the system is controlled by two parameters. One
parameter expresses the growth capacity of the agents and the other describes
the local environmental pressure. For some values of the parameters, the system
exhibits nontrivial collective behavior, characterized by macroscopic periodic
oscillations of the average wealth of the system, emerging out of local chaos.
The probability distribution of wealth in the asymptotic regime shows a power
law behavior for some ranges of parameters.


title: On the scaling properties of 2d randomly stirred Navier--Stokes equation

abstract:
  We inquire the scaling properties of the 2d Navier-Stokes equation sustained
by a forcing field with Gaussian statistics, white-noise in time and with
power-law correlation in momentum space of degree $2-2 \eps$. This is at
variance with the setting usually assumed to derive Kraichan's classical
theory. We contrast accurate numerical experiments with the different
predictions provided for the small $\eps$ regime by Kraichnan's double cascade
theory and by renormalization group (RG) analysis. We give clear evidence that
for all $\eps$ Kraichnan's theory is consistent with the observed
phenomenology. Our results call for a revision in the RG analysis of (2d) fully
developed turbulence.


title: Enhancement of particle trapping in the wave-particle interaction

abstract:
  The saturated dynamics of a Single-Pass Free Electron Laser is considered
within a simplified mean-field approach. A method is proposed to increase the
size of the macro-particle, which is responsible for the oscillations of the
intensity of the wave. This approach is based on the reconstruction of
invariant tori of the dynamics of test particles. To this aim a dedicated
control term is derived, the latter acting as a small apt perturbation of the
system dynamics. Implications of these findings are discussed in relation to
the optimization of the laser source.