title:
Level statistics and eigenfunctions of pseudointegrable systems:
  dependence on energy and genus number

abstract:
  We study the level statistics (second half moment $I_0$ and rigidity
$\Delta_3$) and the eigenfunctions of pseudointegrable systems with rough
boundaries of different genus numbers $g$. We find that the levels form energy
intervals with a characteristic behavior of the level statistics and the
eigenfunctions in each interval. At low enough energies, the boundary roughness
is not resolved and accordingly, the eigenfunctions are quite regular functions
and the level statistics shows Poisson-like behavior. At higher energies, the
level statistics of most systems moves from Poisson-like towards Wigner-like
behavior with increasing $g$. Investigating the wavefunctions, we find many
chaotic functions that can be described as a random superposition of regular
wavefunctions. The amplitude distribution $P(\psi)$ of these chaotic functions
was found to be Gaussian with the typical value of the localization volume
$V_{\rm{loc}}\approx 0.33$. For systems with periodic boundaries we find
several additional energy regimes, where $I_0$ is relatively close to the
Poisson-limit. In these regimes, the eigenfunctions are either regular or
localized functions, where $P(\psi)$ is close to the distribution of a sine or
cosine function in the first case and strongly peaked in the second case. Also
an interesting intermediate case between chaotic and localized eigenfunctions
appears.


title: Soliton stripes in two-dimensional nonlinear photonic lattices

abstract:
  We study experimentally the interaction of a soliton with a nonlinear
lattice. We observe the formation of a novel type of composite soliton created
by strong coupling of mutually incoherent periodic and localized beam
components. By imposing an initial transverse momentum on the soliton stripe,
we observe the effect of lattice compression and deformation.


title: Faraday Wave Pattern Selection Via Multi-Frequency Forcing

abstract:
  We use symmetry considerations to investigate how damped modes affect pattern
selection in multi-frequency forced Faraday waves. We classify and tabulate the
most important damped modes and determine how the corresponding resonant triad
interactions depend on the forcing parameters. The relative phase of the
forcing terms may be used to enhance or suppress the nonlinear interactions. We
compare our predictions with numerical results and discuss their implications
for recent experiments. Our results suggest how to design multi-frequency
forcing functions that favor chosen patterns in the lab.


title:
Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger
  trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type
  of self-trapping transition

abstract:
  Oscillatory instabilities in Hamiltonian anharmonic lattices are known to
appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions
of multibreather type. Here, we analyze the basic mechanisms for this scenario
by considering the simplest possible model system of this kind where they
appear: the three-site discrete nonlinear Schr\"odinger model with periodic
boundary conditions. The stationary solution having equal amplitude and
opposite phases on two sites and zero amplitude on the third is known to be
unstable for an interval of intermediate amplitudes. We numerically analyze the
nature of the two bifurcations leading to this instability and find them to be
of two different types. Close to the lower-amplitude threshold stable
two-frequency quasiperiodic solutions exist surrounding the unstable stationary
solution, and the dynamics remains trapped around the latter so that in
particular the amplitude of the originally unexcited site remains small. By
contrast, close to the higher-amplitude threshold all two-frequency
quasiperiodic solutions are detached from the unstable stationary solution, and
the resulting dynamics is of 'population-inversion' type involving also the
originally unexcited site.


title:
On attractors for a sharp interface model of exothermic phase
  transitions

abstract:
  We study a free interface problem related to combustion of condensed matter
and some non-equilibrium exothermal phase transitions. In spite of a variety of
non-trivial dynamical scenarios exhibited by the model the solutions are
uniformly bounded and the interface velocity is a smooth function. The main
result of the paper establishes existence of a compact connected attractor for
the classical solutions of the problem. Numerical evidence leads to the
conjecture that the fractal dimension of the attractor is finite.


title:
Simulating the time series of a selected gene expression profile in an
  agent-based tumor model

abstract:
  To elucidate the role of environmental conditions in molecular-level dynamics
and to study their impact on macroscopic brain tumor growth patterns, the
expression of the genes Tenascin C and PCNA in a 2D agent-based model for the
migratory trait is calibrated using experimental data from the literature,
while the expression of these genes for the proliferative trait is obtained as
the model output. Numerical results confirm that the gene expression of
Tenascin C is consistently higher in the migratory glioma cell phenotype and
show that the expression of PCNA is consistently higher among proliferating
tumor cells. Furthermore, detrended fluctuation analysis (DFA) suggests that
for prediction purposes, the simulated gene expression profiles of Tenascin C
and PCNA that were determined separately for the migrating and proliferating
phenotypes exhibit lesser predictability than those of the phenotypic mixture
combining all viable tumor cells typically found in clinical biopsies.


title: Charge transport in poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers

abstract:
  We investigate the charge transport in synthetic DNA polymers built up from
single types of base pairs. In the context of a polaron-like model, for which
an electronic tight-binding system and bond vibrations of the double helix are
coupled, we present estimates for the electron-vibration coupling strengths
utilizing a quantum-chemical procedure. Subsequent studies concerning the
mobility of polaron solutions, representing the state of a localized charge in
unison with its associated helix deformation, show that the system for
poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers, respectively possess
quantitatively distinct transport properties. While the former supports
unidirectionally moving electron breathers attributed to highly efficient
long-range conductivity the breather mobility in the latter case is
comparatively restrained inhibiting charge transport. Our results are in
agreement with recent experimental results demonstrating that poly(dG)-poly(dC)
DNA molecules acts as a semiconducting nanowire and exhibits better conductance
than poly(dA)-poly(dT) ones.


title:
Temporal patterns of gene expression via nonmetric multidimensional
  scaling analysis

abstract:
  Motivation: Microarray experiments result in large scale data sets that
require extensive mining and refining to extract useful information. We have
been developing an efficient novel algorithm for nonmetric multidimensional
scaling (nMDS) analysis for very large data sets as a maximally unsupervised
data mining device. We wish to demonstrate its usefulness in the context of
bioinformatics. In our motivation is also an aim to demonstrate that
intrinsically nonlinear methods are generally advantageous in data mining.
Results: The Pearson correlation distance measure is used to indicate the
dissimilarity of the gene activities in transcriptional response of cell
cycle-synchronized human fibroblasts to serum [Iyer et al., Science vol. 283,
p83 (1999)]. These dissimilarity data have been analyzed with our nMDS
algorithm to produce an almost circular arrangement of the genes. The temporal
expression patterns of the genes rotate along this circular arrangement. If an
appropriate preparation procedure may be applied to the original data set,
linear methods such as the principal component analysis (PCA) could achieve
reasonable results, but without data preprocessing linear methods such as PCA
cannot achieve a useful picture. Furthermore, even with an appropriate data
preprocessing, the outcomes of linear procedures are not as clearcut as those
by nMDS without preprocessing.


title: Energy transfer and locality in magnetohydrodynamic turbulence

abstract:
  The shell-to-shell energy transfer rates for magnetohydrodynamic (MHD)
turbulence are computed analytically, which shows local energy transfer rates
from velocity to velocity, velocity to magnetic, magnetic to velocity, and
magnetic to magnetic fields for nonhelical MHD in the inertial range. It is
also found that for kinetic-energy dominated MHD fluid, there is a preferential
shell-to-shell energy transfer from kinetic to magnetic energy; the transfer is
reversed for magnetic-energy dominated MHD fluid. This property is the reason
for the asymptotic value of Alfven ratio to be close to 0.5. The analytical
results are in close agreement with recent numerical results. When magnetic and
kinetic helicities are turned on, the helical contributions are opposite to the
corresponding nonhelical contributions. The helical energy transfers have
significant nonlocal components.


title: Turbulent fields and their recurrences

abstract:
  We introduce a new variational method for finding periodic orbits of flows
and spatio-temporally periodic solutions of classical field theories, a
generalization of the Newton method to a flow in the space of loops. The
feasibility of the method is demonstrated by its application to several
dynamical systems, including the Kuramoto-Sivashinsky system.


title:
Period doubling in glow discharges: local versus global differential
  conductivity

abstract:
  Short planar glow discharges coupled to a resistive layer exhibit a wealth of
spontaneous spatio-temporal patterns. Several authors have suggested effective
reaction-diffusion-models to explore similarities with other pattern forming
systems. To test these effective models, we here investigate the temporal
oscillations of a glow discharge layer coupled to a linear resistor. We find an
unexpected cascade of period doubling events. This shows that the inner
structure of the discharge is more complex than can be described by a
reaction-diffusion-model with negative differential conductivity.


title: Variational method for finding periodic orbits in a general flow

abstract:
  A variational principle for determining unstable periodic orbits of flows as
well as unstable spatio-temporally periodic solutions of extended systems is
proposed and implemented. An initial loop approximating a periodic solution is
evolved in the space of loops toward a true periodic solution by a minimization
of local errors along the loop. The ``\descent'' partial differential equation
that governs this evolution is an infinitesimal step version of the damped
Newton-Raphson iteration. The feasibility of the method is demonstrated by its
application to the H\'enon-Heiles system, the circular restricted three-body
problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.


title: Stochastic Resonance in Two Dimensional Landau Ginzburg Equation

abstract:
  We study the mechanism of stochastic resonance in a two dimensional Landau
Ginzburg equation perturbed by a white noise. We shortly review how to
renormalize the equation in order to avoid ultraviolet divergences. Next we
show that the renormalization amplifies the effect of the small periodic
perturbation in the system. We finally argue that stochastic resonance can be
used to highlight the effect of renormalization in spatially extended system
with a bistable equilibria.


title:
Non-Vanishing Profiles for the Kuramoto-Sivashinsky Equation on the
  Infinite Line

abstract:
  We study the Kuramoto-Sivashinsky equation on the infinite line with initial
conditions having arbitrarily large limits $\pm Y$ at $x=\pm\infty$. We show
that the solutions have the same limits for all positive times. This implies
that an attractor for this equation cannot be defined in $L^\infty$. To prove
this, we consider profiles with limits at $x=\pm\infty$, and show that initial
conditions $L^2$-close to such profiles lead to solutions which remain
$L^2$-close to the profile for all times. Furthermore, the difference between
these solutions and the initial profile tends to 0 as $x\to\pm\infty$, for any
fixed time $t>0$. Analogous results hold for $L^2$-neighborhoods of periodic
stationary solutions. This implies that profiles and periodic stationary
solutions partition the phase space into mutually unattainable regions.


title: Integrable SU(N) vertex models with general toroidal boundary conditions

abstract:
  We formulate the algebraic Bethe ansatz solution of the SU(N) vertex models
with rather general non-diagonal toroidal boundary conditions. The reference
states needed in the Bethe ansatz construction are found by performing gauge
transformations on the Boltzmann weights in the manner of Baxter [1]. The
structure of the transfer matrix eigenvectors consists of multi-particle states
over such pseudovacuums and the corresponding eigenvalues depend crucially on
the boundary matrix eigenvalues. We also discuss for N=2 the peculiar case of
twisted boundaries associated to singular matrices.


title:
Exact solitary wave solutions of the nonlinear Schr\"odinger equation
  with a source

abstract:
  We use a fractional transformation to connect the traveling wave solutions of
the nonlinear Schr\"odinger equation (NLSE), phase-locked with a source, to the
elliptic functions satisfying, $f^{\prime\prime}\pm af\pm \lambda f^{3}=0$. The
solutions are {\it necessarily} of the rational form, containing both
trigonometric and hyperbolic types as special cases. Bright, and dark solitons,
respectively, for attractive and repuslive type of nonlinearities, as also
singular solitons, are obtained in suitable range of parameter values.


title: Emergence of scale-free properties in Hebbian networks

abstract:
  The fundamental `plasticity' of the nervous system (i.e high adaptability at
different structural levels) is primarily based on Hebbian learning mechanisms
that modify the synaptic connections. The modifications rely on neural activity
and assign a special dynamic behavior to the neural networks. Another striking
feature of the nervous system is that spike based information transmission,
which is supposed to be robust against noise, is noisy in itself: the variance
of the spiking of the individual neurons is surprisingly large which may
deteriorate the adequate functioning of the Hebbian mechanisms. In this paper
we focus on networks in which Hebbian-like adaptation is induced only by
external random noise and study spike-timing dependent synaptic plasticity. We
show that such `HebbNets' are able to develop a broad range of network
structures, including scale-free small-world networks. The development of such
network structures may provide an explanation of the role of noise and its
interplay with Hebbian plasticity. We also argue that this model can be seen as
a unification of the famous Watts-Strogatz and preferential attachment models
of small-world nets.


title: Analytical results for coupled map lattices with long-range interactions

abstract:
  We obtain exact analytical results for lattices of maps with couplings that
decay with distance as $r^{-\alpha}$. We analyze the effect of the coupling
range on the system dynamics through the Lyapunov spectrum. For lattices whose
elements are piecewise linear maps, we get an algebraic expression for the
Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the
Lyapunov spectrum for a completely synchronized state is analytically obtained.
The critical lines characterizing the synchronization transition are determined
from the expression for the largest transversal Lyapunov exponent. In
particular, it is shown that in the thermodynamical limit, such transition is
only possible for sufficiently long-range interactions, namely, for $\alpha\le
alpha_c<d$, where $d$ is the lattice dimension.


title:
A Lax Pair for the Dynamics of DNA Modeled as a Shearable and Extensible
  Elastic Rod: III. Discretizatioon of Arc Length and Time

abstract:
  We find a Lax pair for the geometrically exact discrete Hamiltonian equations
for the discrete elastic rod. This is paper III of a series.


title:
Soliton transverse instabilities in anisotropic nonlocal self-focusing
  media

abstract:
  We study, both theoretically and experimentally, the transverse modulational
instability of spatial stripe solitons in anisotropic nonlocal photorefractive
media. We demonstrate that the instability scenarios depend strongly on the
stripe orientation, but the anisotropy-induced features are largely suppressed
for spatial solitons created by self-trapping of partially incoherent light.


title: Dynamic Domains in Strongly Driven Ferromagnetic Films

abstract:
  The spatiotemporal structure formation problem is investigated in the region
far above the transverse ferromagnetic resonance instability. The
investigations are based on the dissipative Landau-Lifshitz equation and have
been performed on a model which takes external fields, isotropic exchange
fields, anisotropy fields and the demagnetizing part of the dipolar field into
consideration. The numerical simulations for these models exhibit stationary
domain structure in the rotating frame. Employing analytical methods and
simplifying the model, certain features, such as the magnetization within the
domains and the proportion of the system in each domain, are described
analytically.


title: Integrable mappings derived from the $\Delta \Delta RsG$ equation

abstract:
  In this paper we consider the integrability of the mappings derived from the
double discrete related sine-Gordon $(\Delta \Delta RsG)$ equation, which we
recently introduced in the paper Higher dimensional integrable mappings, under
an appropriate periodicity condition.


title: Dual Multifractal Spectra

abstract:
  The multifractal formalism characterizes the scaling properties of a physical
density rho as a function of the distance L. To each singularity alpha of the
field is attributed a fractal dimension for its support f(alpha). An
alternative representation has been proposed by Jensen considering the
distribution of distances associated to a fixed mass. Computing these spectra
for a multifractal Cantor set, it is shown that these two approaches are dual
to each other, and that both spectra as well as the moment scaling exponents
are simply related. We apply the same inversion formalism to exponents obtained
for turbulent statistics in the GOY shell model and observe that the same
duality relation holds here.


title: Selection of the ground state for nonlinear Schroedinger equations

abstract:
  We prove for a class of nonlinear Schr\"odinger systems (NLS) having two
nonlinear bound states that the (generic) large time behavior is characterized
by decay of the excited state, asymptotic approach to the nonlinear ground
state and dispersive radiation. Our analysis elucidates the mechanism through
which initial conditions which are very near the excited state branch evolve
into a (nonlinear) ground state, a phenomenon known as {\it ground state
selection}.
  Key steps in the analysis are the introduction of a particular linearization
and the derivation of a normal form which reflects the dynamics on all time
scales and yields, in particular, nonlinear Master equations.
  Then, a novel multiple time scale dynamic stability theory is developed.
Consequently, we give a detailed description of the asymptotic behavior of the
two bound state NLS for all small initial data. The methods are general and can
be extended to treat NLS with more than two bound states and more general
nonlinearities including those of Hartree-Fock type.


title:
Feedback Control of Traveling Wave Solutions of the Complex Ginzburg
  Landau Equation

abstract:
  Through a linear stability analysis, we investigate the effectiveness of a
noninvasive feedback control scheme aimed at stabilizing traveling wave
solutions of the one-dimensional complex Ginzburg Landau equation (CGLE) in the
Benjamin-Feir unstable regime. The feedback control is a generalization of the
time-delay method of Pyragas, which was proposed by Lu, Yu and Harrison in the
setting of nonlinear optics. It involves both spatial shifts, by the wavelength
of the targeted traveling wave, and a time delay that coincides with the
temporal period of the traveling wave. We derive a single necessary and
sufficient stability criterion which determines whether a traveling wave is
stable to all perturbation wavenumbers. This criterion has the benefit that it
determines an optimal value for the time-delay feedback parameter. For various
coefficients in the CGLE we use this algebraic stability criterion to
numerically determine stable regions in the (K,rho) parameter plane, where rho
is the feedback parameter associated with the spatial translation and K is the
wavenumber of the traveling wave. We find that the combination of the two
feedbacks greatly enlarges the parameter regime where stabilization is
possible, and that the stability regions take the form of stability tongues in
the (K,rho)--plane. We discuss possible resonance mechanisms that could account
for the spacing with K of the stability tongues.


title: Breather Statics and Dynamics in Klein--Gordon Chains with a Bend

abstract:
  In this communication, we examine a nonlinear model with an impurity
emulating a bend. We justify the geometric interpretation of the model and
connect it with earlier work on models including geometric effects. We focus on
both the bifurcation and stability analysis of the modes that emerge as a
function of the strength of the bend angle, but we also examine dynamical
effects including the scattering of mobile localized modes (discrete breathers)
off of such a geometric structure. The potential outcomes of such numerical
experiments (including transmission, trapping within the bend as well as
reflection) are highlighted and qualitatively explained. Such models are of
interest both theoretically in understanding the interplay of breathers with
curvature, but also practically in simple models of photonic crystals or of
bent chains of DNA.


title: Introduction to Chaos in Deterministic Systems

abstract:
  The scope of this teaching package is to make a brief introduction to some
notions and properties of chaotic systems. We first make a brief introduction
to chaos in general and then we show some important properties of chaotic
systems using the logistic map and its bifurcation diagram. We also show the
universality found in "the route to chaos". The user is only required to have
notions of algebra, so it is quite accessible. The formal basis of chaos theory
are not covered in this introduction, but are pointed out for the reader
interested in them. Therefore, this package is also useful for people who are
interested in going deep into the mathematical theories, because it is a simple
introduction of the terminology, and because it points out which are the
original sources of information (so there is no danger in falling in the trap
of "Learn Chaos in 48 hours" or "Bifurcation Diagrams for Dummies"). The
included exercises are suggested for consolidating the covered topics. The
on-line resources are highly recommended for extending this brief induction.


title:
A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein
  Condensates

abstract:
  We apply Lindstedt's method and multiple scale perturbation theory to analyze
spatio-temporal structures in nonlinear Schr\"odinger equations and thereby
study the dynamics of quasi-one-dimensional Bose-Einstein condensates with
mean-field interactions. We determine the dependence of the intensity of
modulated amplitude waves on their wave number. We also explore the band
structure of Bose-Einstein condensates in detail using Hamiltonian perturbation
theory and supporting numerical simulations.


title: Energy dissipation in body-forced plane shear flow

abstract:
  We study the problem of body-force driven shear flows in a plane channel of
width l with free-slip boundaries. A mini-max variational problem for upper
bounds on the bulk time averaged energy dissipation rate epsilon is derived
from the incompressible Navier-Stokes equations with no secondary assumptions.
This produces rigorous limits on the power consumption that are valid for
laminar or turbulent solutions. The mini-max problem is solved exactly at high
Reynolds numbers Re = U*l/nu, where U is the rms velocity and nu is the
kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic
dissipation factor beta=epsilon*l/U^3 that depends only on the ``shape'' of the
shearing body force. For a simple half-cosine force profile, for example, the
high Reynolds number bound is beta <= pi^2/sqrt{216} = .6715... . We also
report extensive direct numerical simulations for this particular force shape
up to Re approximately 400; the observed dissipation rates are about a factor
of three below the rigorous high-Re bound. Interestingly, the high-Re optimal
solution of the variational problem bears some qualitative resemblence to the
observed mean flow profiles in the simulations. These results extend and refine
the recent analysis for body-forced turbulence in J. Fluid Mech. 467, 289-306
(2002).


title: Uniform approximations for pitchfork bifurcation sequences

abstract:
  In non-integrable Hamiltonian systems with mixed phase space and discrete
symmetries, sequences of pitchfork bifurcations of periodic orbits pave the way
from integrability to chaos. In extending the semiclassical trace formula for
the spectral density, we develop a uniform approximation for the combined
contribution of pitchfork bifurcation pairs. For a two-dimensional double-well
potential and the familiar H\'enon-Heiles potential, we obtain very good
agreement with exact quantum-mechanical calculations. We also consider the
integrable limit of the scenario which corresponds to the bifurcation of a
torus from an isolated periodic orbit. For the separable version of the
H\'enon-Heiles system we give an analytical uniform trace formula, which also
yields the correct harmonic-oscillator SU(2) limit at low energies, and obtain
excellent agreement with the slightly coarse-grained quantum-mechanical density
of states.


title: Detecting local synchronization in coupled chaotic systems

abstract:
  We introduce a technique to detect and quantify local functional dependencies
between coupled chaotic systems. The method estimates the fraction of locally
syncronized configurations, in a pair of signals with an arbitrary state of
global syncronization. Application to a pair of interacting Rossler oscillators
shows that our method is capable to quantify the number of dynamical
configurations where a local prediction task is possible, also in absence of
global synchronization features.


title:
Current reversal with type-I intermittency in deterministic inertia
  ratchets

abstract:
  The intermittency is investigated when the current reversal occurs in a
deterministic inertia ratchet system. To determine which type the intermittency
belongs to, we obtain the return map of velocities of particle using
stroboscopic recording, and numerically calculate the distribution of average
laminar length ${<}l{>}$. The distribution follows the scaling law of ${<}l{>}
\propto {\epsilon}^{-1/2}$, the characteristic relation of type-I
intermittency.


title: Boundary effects and the onset of Taylor vortices

abstract:
  It is well established that the onset of spatially periodic vortex states in
the Taylor-Couette flow between rotating cylinders occurs at the value of
Reynold's number predicted by local bifurcation theory. However, the symmetry
breaking induced by the top and bottom plates means that the true situation
should be a disconnected pitchfork. Indeed, experiments have shown that the
fold of the disconnected branch can occur at more than double the Reynold's
number of onset. This leads to an apparent contradiction: why should Taylor
vortices set in so sharply at the value Reynold's number predicted by the
symmetric theory, given such large symmetry-breaking effects caused by the
boundary conditions? This paper offers a generic explanation. The details are
worked out using a Swift-Hohenberg pattern formation model that shares the same
qualitative features as the Taylor-Couette flow. Onset occurs via a wall mode
whose exponential tail penetrates further into the bulk of the domain as the
driving parameter increases. In a large domain of length L, we show that the
wall mode creates significant amplitude in the centre at parameter values that
are O(L^{-2}) away from the value of onset in the problem with ideal boundary
conditions. We explain this as being due to a Hamiltonian Hopf bifurcation in
space, which occurs at the same parameter value as the pitchfork bifurcation of
the temporal dynamics. The disconnected anomalous branch remains O(1) away from
the onset parameter since it does not arise as a bifurcation from the wall
mode.


title: Lifetimes of noisy repellors

abstract:
  We study the effects of additive noise on the lifetimes of chaotic repellors.
Using first order perturbation theory we argue that noise will increase the
lifetime if the escape holes lie in regions where the unperturbed density is
higher than in the immediate vicinity and that it decreases if the density is
lower. Numerical experiments support the qualitative conclusions also beyond
perturbation theory.


title: Stability of synchronized oscillations in networks of phase-oscillators

abstract:
  We derive simple conditions for the stability or instability of the
synchronized oscillation of a class of networks of coupled phase-oscillators,
which includes many of the systems used in neural modelling.


title:
Estimating the distribution of dynamic invariants: Illustrated with an
  application to human photo-plethysmographic time series

abstract:
  Dynamic invariants are often estimated from experimental time series with the
aim of differentiating between different physical states in the underlying
system. The most popular schemes for estimating dynamic invariants are capable
of estimating confidence intervals, however such confidence intervals do not
reflect variability in the underlying dynamics. In this communication we
propose a surrogate based method to estimate the expected distribution of
values under the null hypothesis that the underlying deterministic dynamics are
stationary. We demonstrate the application of this method by considering four
recordings of human pulse waveforms in differing physiological states and
provide conclusive evidence that correlation dimension is capable of
differentiating between three (but not all four) of these states.


title: Non-Integrability of a weakly integrable Hamiltonian system

abstract:
  The geometric approach to mechanics based on the Jacobi metric allows to
easily construct natural mechanical systems which are integrable (actually
separable) at a fixed value of the energy. The aim of the present paper is to
investigate the dynamics of a simple prototype system outside the zero-energy
hypersurface. We find that the general situation is that in which integrability
is not preserved at arbitrary values of the energy. The structure of the
Hamiltonian in the separating coordinates at zero energy allows a perturbation
treatment of this system at energies slightly different from zero, by which we
obtain an analytical proof of non-integrability.


title:
Topological constraints on spiral wave dynamics in spherical geometries
  with inhomogeneous excitability

abstract:
  We analyze the way topological constraints and inhomogeneity in the
excitability influence the dynamics of spiral waves on spheres and punctured
spheres of excitable media. We generalize the definition of an index such that
it characterizes not only each spiral but also each hole in punctured,
oriented, compact, two-dimensional differentiable manifolds and show that the
sum of the indices is conserved and zero. We also show that heterogeneity and
geometry are responsible for the formation of various spiral wave attractors,
in particular, pairs of spirals in which one spiral acts as a source and a
second as a sink -- the latter similar to an antispiral. The results provide a
basis for the analysis of the propagation of waves in heterogeneous excitable
media in physical and biological systems.


title: Kink propagation in a two-dimensional curved Josephson junction

abstract:
  We consider the propagation of sine-Gordon kinks in a planar curved strip as
a model of nonlinear wave propagation in curved wave guides. The homogeneous
Neumann transverse boundary conditions, in the curvilinear coordinates, allow
to assume a homogeneous kink solution. Using a simple collective variable
approach based on the kink coordinate, we show that curved regions act as
potential barriers for the wave and determine the threshold velocity for the
kink to cross. The analysis is confirmed by numerical solution of the 2D
sine-Gordon equation.


title: Multiple Timescale Dynamics in Economic Production Networks

abstract:
  We describe a new complex system model of an evolving production economy.
This model is the simplest we can envisage which incorporates the new
observation that the rate of an economic production process depends only on the
minimum of its supplies of inputs. We describe how this condition gives rise to
a new type of complex multiple timescale dynamical evolution through a novel
type of bifurcation we call a `trapping bifurcation', which is also shown to be
one cause of non-equilibrium economic behaviour. Such dynamics is an example of
meta-level coupling which may also arise in other fields such as the
organization of a living cell as a network of molecular machines.


title:
Breather lattice and its stabilization for the modified Korteweg-de
  Vries equation

abstract:
  We obtain an exact solution for the breather lattice solution of the modified
Korteweg-de Vries (MKdV) equation. Numerical simulation of the breather lattice
demonstrates its instability due to the breather-breather interaction. However,
such multi-breather structures can be stabilized through the concurrent
application of ac driving and viscous damping terms.


title: Semiclassical Construction of Random Wave Functions for Confined Systems

abstract:
  We develop a statistical description of chaotic wavefunctions in closed
systems obeying arbitrary boundary conditions by combining a semiclassical
expression for the spatial two-point correlation function with a treatment of
eigenfunctions as Gaussian random fields. Thereby we generalize Berry's
isotropic random wave model by incorporating confinement effects through
classical paths reflected at the boundaries. Our approach allows to explicitly
calculate highly non-trivial statistics, such as intensity distributions, in
terms of usually few short orbits, depending on the energy window considered.
We compare with numerical quantum results for the Africa billiard and derive
non-isotropic random wave models for other prominent confinement geometries.


title:
Periodicity of chaotic trajectories in realizations of finite computer
  precisions and its implication in chaos communications

abstract:
  Fundamental problems of periodicity and transient process to periodicity of
chaotic trajectories in computer realization with finite computation precision
is investigated by taking single and coupled Logistic maps as examples.
Empirical power law relations of the period and transient iterations with the
computation precisions and the sizes of coupled systems are obtained. For each
computation we always find, by randomly choosing initial conditions, a single
dominant periodic trajectory which is realized with major portion of
probability. These understandings are useful for possible applications of
chaos, e.g., chaotic cryptography in secure communication.


title: Basin structure in the two-dimensional dissipative circle map

abstract:
  Fractal basin structure in the two-dimensional dissipative circle map is
examined in detail. Numerically obtained basin appears to be riddling in the
parameter region where two periodic orbits co-exist near a boundary crisis, but
it is shown to consist of layers of thin bands.


title:
Characterization of QPM gratings in quadratic media via
  second-harmonic-power measurements only

abstract:
  A new scheme for non-destructive characterization of quasi-phase-matching
grating structures and temperature gradients via inverse Fourier theory using
second-harmonic-generation experiments is proposed. We show how it is possible
to retrieve the relevant information via measuring only the power in the
generated second harmonic field, thus avoiding more complicated phase
measurements. The potential of the scheme is emphasized through theoretical and
numerical investigations in the case of periodically poled lithium niobate bulk
crystals.


title: Escaping from nonhyperbolic chaotic attractors

abstract:
  We study the noise-induced escape process from chaotic attractors in
nonhyperbolic systems. We provide a general mechanism of escape in the low
noise limit, employing the theory of large fluctuations. Specifically, this is
achieved by solving the variational equations of the auxiliary Hamiltonian
system and by incorporating the initial conditions on the chaotic attractor
unambiguously. Our results are exemplified with the H{\'e}non and the Ikeda map
and can be implemented straightforwardly to experimental data.


title:
Exact Phase Solutions of Nonlinear Oscillators on Two-dimensional
  Lattice

abstract:
  We present various exact solutions of a discrete complex Ginzburg-Landau
(CGL) equation on a plane lattice, which describe target patterns and spiral
patterns and derive their stability criteria. We also obtain similar solutions
to a system of van der Pol's oscillators.


title:
Geometric features of the mixing of passive scalars at high Schmidt
  numbers

abstract:
  The mixing of passive scalars of decreasing diffusivity, advected in each
case by the same three-dimensional Navier-Stokes turbulence, is studied. The
mixing becomes more isotropic with decreasing diffusivity. The local flow in
the vicinity of steepest negative and positive scalar gradients are in general
different, and its behavior is studied for various values of the scalar
diffusivity. Mixing approaches monofractal properties with diminishing
diffusivity. We consider these results in the context of possible singularities
of scalar dissipation.


title: A bi-Hamiltonian approach to the sine-Gordon and Liouville hierarchies

abstract:
  In this paper we study the sine-Gordon and the Liouville hierarchies in
laboratory coordinates from a bi-Hamiltonian point of view. Besides the
well-known local structure these hierarchies possess a second compatible
non-local Poisson structure.


title:
Propagation of exremely short pulses in non-resonant media: the total
  Maxwell-Duffing model

abstract:
  Propagation of extremely short pulses of electromagnetic field
(electromagnetic spikes) is considered in the framework of the total
Maxwell-Duffing model where anharmonic oscillators with cubic nonlinearities
(Duffing model) represent the material medium and wave propagation is governed
by the 1-d bidirectional Maxwell equations. This system of equations has a one
parameter family of exact analytical solutions representing an electromagnetic
spike propagating on a zero or a nonzero background. We find that the total
Maxwell-Duffing equations can be written as a system in bilinear form and that
the one-soliton solution of this system coincides with the steady state
solution obtained previously.


title:
Symmetry induced Dynamics in four-dimensional Models deriving from the
  van der Pol Equation

abstract:
  Different models of self-excited oscillators which are four-dimensional
extensions of the van der Pol system are reported. Their symmetries are
analyzed. Three of them were introduced to model the release of vortices behind
circular cylinders with a possible transition from a symmetric to an
antisymmetric Benard-von Karman vortex street. The fourth reported self-excited
oscillator is a new model which implements the breaking of the inversion
symmetry. It presents the phenomenon of second harmonic generation in a natural
way. The parallelism with second harmonic generation in nonlinear optics is
discussed. There is also a small region in the parameter space where the
dynamics of this system is quasiperiodic or chaotic.


title: Stable embedded solitons

abstract:
  Stable embedded solitons are discovered in the generalized third-order
nonlinear Schroedinger equation. When this equation can be reduced to a
perturbed complex modified KdV equation, we developed a soliton perturbation
theory which shows that a continuous family of sech-shaped embedded solitons
exist and are nonlinearly stable. These analytical results are confirmed by our
numerical simulations. These results establish that, contrary to previous
beliefs, embedded solitons can be robust despite being in resonance with the
linear spectrum.


title:
Multistability, noise and attractor-hopping: The crucial role of chaotic
  saddles

abstract:
  We investigate the hopping dynamics between different attractors in a
multistable system under the influence of noise. Using symbolic dynamics we
find a sudden increase of dynamical entropies, when a system parameter is
varied. This effect is explained by a novel bifurcation involving two chaotic
saddles. We also demonstrate that the transient lifetimes on the saddle obey a
scaling law in analogy to crisis.


title:
Enhancement of Noise-induced Escape through the Existence of a Chaotic
  Saddle

abstract:
  We study the noise-induced escape process in a prototype dissipative
nonequilibrium system, the Ikeda map. In the presence of a chaotic saddle
embedded in the basin of attraction of the metastable state, we find the novel
phenomenon of a strong enhancement of noise-induced escape. This result is
established by employing the theory of quasipotentials. Our finding is of
general validity and should be experimentally observable.


title:
Bihamiltonian structures and quadratic algebras in hydrodynamics and on
  non-commutative torus

abstract:
  We demonstrate the common bihamiltonian nature of several integrable systems.
The first one is an elliptic rotator that is an integrable Euler-Arnold top on
the complex group GL(N) for any $N$, whose inertia ellipsiod is related to a
choice of an elliptic curve. Its bihamiltonian structure is provided by the
compatible linear and quadratic Poisson brackets, both of which are governed by
the Belavin-Drinfeld classical elliptic $r$-matrix. We also generalize this
bihamiltonian construction of integrable Euler-Arnold tops to several
infinite-dimensional groups, appearing as certain large $N$ limits of GL(N).
These are the group of a non-commutative torus (NCT) and the group of
symplectomorphisms $SDiff(T^2)$ of the two-dimensional torus. The elliptic
rotator on symplectomorphisms gives an elliptic version of an ideal 2D
hydrodynamics, which turns out to be an integrable system. In particular, we
define the quadratic Poisson algebra on the space of Hamiltonians on $T^2$
depending on two irrational numbers. In conclusion, we quantize the
infinite-dimensional quadratic Poisson algebra in a fashion similar to the
corresponding finite-dimensional case.


title:
Observation of discrete vortex solitons in optically-induced photonic
  lattices

abstract:
  We report on the frst experimental observation of discrete vortex solitons in
two-dimensional optically-induced photonic lattices. We demonstrate strong
stabilization of an optical vortex by the lattice in a self-focusing nonlinear
medium and study the generation of the discrete vortices from a broad class of
singular beams.


title: Passive Scalar Evolution in Peripheral Region

abstract:
  We consider evolution of a passive scalar (concentration of pollutants or
temperature) in a chaotic (turbulent) flow. A universal asymptotic behavior of
the passive scalar decay (homogenization) related to peripheral regions (near
walls) is established. The passive scalar moments and its pair correlation
function in the peripheral region are analyzed. A special case investigated in
our paper is the passive scalar decay along a pipe.


title:
Geometric Analysis of Bifurcation and Symmetry Breaking in a
  Gross-Pitaevskii equation

abstract:
  Gross-Pitaevskii and nonlinear Hartree equations are equations of nonlinear
Schroedinger type, which play an important role in the theory of Bose-Einstein
condensation. Recent results of Aschenbacher et. al. [AFGST] demonstrate, for a
class of 3- dimensional models, that for large boson number (squared L^2 norm),
N, the ground state does not have the symmetry properties as the ground state
at small N. We present a detailed global study of the symmetry breaking
bifurcation for a 1-dimensional model Gross-Pitaevskii equation, in which the
external potential (boson trap) is an attractive double-well, consisting of two
attractive Dirac delta functions concentrated at distinct points. Using
dynamical systems methods, we present a geometric analysis of the symmetry
breaking bifurcation of an asymmetric ground state and the exchange of
dynamical stability from the symmetric branch to the asymmetric branch at the
bifurcation point.


title:
Liouville Operator Approach to Symplecticity-Preserving Renormalization
  Group Method

abstract:
  We present a method to construct symplecticity-preserving renormalization
group maps by using the Liouville operator, and obtain correctly reduced
symplectic maps describing their long-time behavior even when a resonant island
chain appears.


title: Universal spectral form factor for chaotic dynamics

abstract:
  We consider the semiclassical limit of the spectral form factor $K(\tau)$ of
fully chaotic dynamics. Starting from the Gutzwiller type double sum over
classical periodic orbits we set out to recover the universal behavior
predicted by random-matrix theory, both for dynamics with and without time
reversal invariance. For times smaller than half the Heisenberg time
$T_H\propto \hbar^{-f+1}$, we extend the previously known $\tau$-expansion to
include the cubic term. Beyond confirming random-matrix behavior of individual
spectra, the virtue of that extension is that the ``diagrammatic rules'' come
in sight which determine the families of orbit pairs responsible for all orders
of the $\tau$-expansion.


title:
A remark on nonlocal symmetries for the
  Calogero-Degasperis-Ibragimov-Shabat equation

abstract:
  We consider the Calogero-Degasperis-Ibragimov-Shabat (CDIS) equation and find
the complete set of its nonlocal symmetries depending on the local variables
and on the integral of the only local conserved density of the equation in
question. The Lie algebra of these symmetries turns out to be a central
extension of that of local generalized symmetries.


title: Optimal switching policies using coarse timesteppers

abstract:
  We present a computer-assisted approach to approximating coarse optimal
switching policies for systems described by microscopic/stochastic evolution
rules. The coarse timestepper constitutes a bridge between the underlying
kinetic Monte Carlo simulation and traditional, continuum numerical
optimization techniques formulated in discrete time. The approach is
illustrated through a simplified kinetic Monte Carlo simulation of NO reduction
on a Pt catalyst: a switch between two coexisting stable steady states is
implemented by minimal manipulation of a system parameter.


title: Nonlinear optical beams carrying phase dislocations

abstract:
  We describe different types of self-trapped optical beams carrying phase
dislocations, including vortex solitons and ring-like soliton clusters. We
demonstrate numerically how to create such nonlinear singular beams by the
interaction of several fundamental optical solitons. Mutual trapping of several
solitons can be regarded as a synthesis of `soliton molecules', and it
corresponds to a transfer of an initial orbital angular momentum of a system of
solitons to a spin momentum of an optical vortex.


title:
Gibbs attractor: a chaotic nearly Hamiltonian system, driven by external
  harmonic force

abstract:
  A chaotic autonomous Hamiltonian systems, perturbed by small damping and
small external force, harmonically dependent on time, can acquire a strange
attractor with properties similar to that of the canonical distribution - the
Gibbs attractor. The evolution of the energy in such systems can be described
as the energy diffusion. For the nonlinear Pullen - Edmonds oscillator with two
degrees of freedom the properties of the Gibbs attractor and their dependence
on parameters of the perturbation are studied both analytically and
numerically.


title: Sticky behavior of fluid particles in the compressible Kraichnan model

abstract:
  We consider the compressible Kraichnan model of turbulent advection with
small molecular diffusivity and velocity field regularized at short scales to
mimic the effects of viscosity. As noted in ref.[5], removing those two
regularizations in two opposite orders for intermediate values of
compressibility gives Lagrangian flows with quite different properties.
Removing the viscous regularization before diffusivity leads to the explosive
separation of trajectories of fluid particles whereas turning the
regularizations off in the opposite order results in coalescence of Lagrangian
trajectories. In the present paper we re-examine the situation first addressed
in ref.[6] in which the Prandtl number is varied when the regularizations are
removed. We show that an appropriate fine-tuning leads to a sticky behavior of
trajectories which hit each other on and off spending a positive amount of time
together. We examine the effect of such a trajectory behavior on the passive
transport showing that it induces anomalous scaling of the stationary 2-point
structure function of an advected tracer and influences the rate of
condensation of tracer energy in the zero wavenumber mode.


title: Partially incoherent optical vortices in self-focusing nonlinear media

abstract:
  We observe stable propagation of spatially localized single- and
double-charge optical vortices in a self-focusing nonlinear medium. The
vortices are created by self-trapping of partially incoherent light carrying a
phase dislocation, and they are stabilized when the spatial incoherence of
light exceeds a certain threshold. We confirm the vortex stabilization effect
by numerical simulations and also show that the similar mechanism of
stabilization applies to higher-order vortices.


title: Stochastic resonance between noise-sustained patterns

abstract:
  We study an extended system that without noise shows a spatially homogeneous
state, but when submitted to an adequate multiplicative noise, some
"noise-induced patterns" arise. The stochastic resonance between these
structures is investigated theoretically in terms of a two-state approximation.
The knowledge of the exact nonequilibrium potential allows us to obtain the
output signal-to-noise ratio. Its maximum is predicted in the symmetric case
for which both stable attractors have the same nonequilibrium potential value.


title: Q4

abstract:
  One of the most fascinating and technically demanding parts of the theory of
two-dimensional integrable systems constitute the models with the spectral
parameter on an elliptic curve, including Landau-Lifshitz and Krichever-Novikov
equations, as well as elliptic Toda and Ruijsenaars-Toda lattices, elliptic
Volterra lattice and Shabat-Yamilov lattice. We explain how all these models
can be unified on the basis of a single equation Q4 on a quad-graph. This
discrete model plays the role of the ``master equation'' in the (sl(2) part of)
2D integrability: most of other models in this area can be obtained from this
one by certain limiting procedures. More precisely, discrete versions of all of
the above mentioned models appear from Q4 imposed on a multi-dimensional cubic
lattice. Zero curvature representations for all models appear as a byproduct of
the general construction.


title: Traveling solitons in the damped driven nonlinear Schr\"odinger equation

abstract:
  The well known effect of the linear damping on the moving nonlinear
Schr\"odinger soliton (even when there is a supply of energy via the spatially
homogeneous driving) is to quench its momentum to zero. Surprisingly, the zero
momentum does not necessarily mean zero velocity. We show that two or more
parametrically driven damped solitons can form a complex traveling with zero
momentum at a nonzero constant speed.
  All traveling complexes we have found so far, turned out to be unstable.
Thus, the parametric driving is capable of sustaining the uniform motion of
damped solitons, but some additional agent is required to stabilize it.


title:
Multistable Pulse-like Solutions in a Parametrically Driven
  Ginzburg-Landau Equation

abstract:
  It is well known that pulse-like solutions of the cubic complex
Ginzburg-Landau equation are unstable but can be stabilised by the addition of
quintic terms. In this paper we explore an alternative mechanism where the role
of the stabilising agent is played by the parametric driver. Our analysis is
based on the numerical continuation of solutions in one of the parameters of
the Ginzburg-Landau equation (the diffusion coefficient $c$), starting from the
nonlinear Schr\"odinger limit (for which $c=0$). The continuation generates,
recursively, a sequence of coexisting stable solutions with increasing number
of humps. The sequence "converges" to a long pulse which can be interpreted as
a bound state of two fronts with opposite polarities.


title:
The Interactive Minority Game: a Web-based investigation of human market
  interactions

abstract:
  The unprecedented access offered by the World Wide Web brings with it the
potential to gather huge amounts of data on human activities. Here we exploit
this by using a toy model of financial markets, the Minority Game (MG), to
investigate human speculative trading behaviour and information capacity.
Hundreds of individuals have played a total of tens of thousands of game turns
against computer-controlled agents in the Web-based_Interactive Minority Game_.
The analytical understanding of the MG permits fine-tuning of the market
situations encountered, allowing for investigation of human behaviour in a
variety of controlled environments. In particular, our results indicate a
transition in players' decision-making, as the markets become more difficult,
between deductive behaviour making use of short-term trends in the market, and
highly repetitive behaviour that ignores entirely the market history, yet
outperforms random decision-making.
  PACS: 02.50.Le; 89.65.Gh; 89.70.+c
  Keywords: Decision theory and game theory; Economics and financial markets;
Information theory

categories: nlin.AO cond-mat.stat-mech cs.HC physics.soc-ph


title:
Shear dispersion along circular pipes is affected by bends, but the
  torsion of the pipe is negligible

abstract:
  The flow of a viscous fluid along a curving pipe of fixed radius is driven by
a pressure gradient. For a generally curving pipe it is the fluid flux which is
constant along the pipe and so I correct fluid flow solutions of Dean (1928)
and Topakoglu (1967) which assume constant pressure gradient. When the pipe is
straight, the fluid adopts the parabolic velocity profile of Poiseuille flow;
the spread of any contaminant along the pipe is then described by the shear
dispersion model of Taylor (1954) and its refinements by Mercer, Watt et al
(1994,1996). However, two conflicting effects occur in a generally curving
pipe: viscosity skews the velocity profile which enhances the shear dispersion;
whereas in faster flow centrifugal effects establish secondary flows that
reduce the shear dispersion. The two opposing effects cancel at a Reynolds
number of about 15. Interestingly, the torsion of the pipe seems to have very
little effect upon the flow or the dispersion, the curvature is by far the
dominant influence. Lastly, curvature and torsion in the fluid flow
significantly enhance the upstream tails of concentration profiles in
qualitative agreement with observations of dispersion in river flow.


title: Binding Social and Cultural Networks: A Model

abstract:
  Until now, most studies carried onto social or semantic networks have
considered each of these networks independently. Our goal here is to bring a
formal frame for studying both networks empirically as well as to point out
stylized facts that would explain their reciprocal influence and the emergence
of clusters of agents, which may also be regarded as ''cultural cliques''. We
show how to apply the Galois lattice theory to the modeling of the coevolution
of social and conceptual networks, and the characterization of cultural
communities. Basing our approach on Barabasi-Albert's models, we however extend
the usual preferential attachment probability in order to take into account the
reciprocal influence of both networks, therefore introducing the notion of dual
distance. In addition to providing a theoretic frame we draw here a program of
empirical tests which should give root to a more analytical model and the
consequent simulation and validation. In a broader view, adopting and actually
implementing the paradigm of cultural epidemiology, we could proceed further
with the study of knowledge diffusion and explain how the social network
structure affects concept propagation and in return how concept propagation
affects the social network.


title:
Onset and universality of drag reduction in the turbulent Kolmogorov
  flow

abstract:
  We investigate the phenomenon of drag reduction in a viscoelastic fluid model
of dilute polymer solutions. By means of direct numerical simulations of the
three-dimensional turbulent Kolmogorov flow we show that drag reduction takes
place above a critical Reynolds number $Re_c$. An explicit expression for the
dependence of $Re_c$ on polymer elasticity and diffusivity is derived. The
values of the drag coefficient obtained for different fluid parameters collapse
onto a universal curve when plotted as a function of the rescaled Reynolds
number $Re/Re_c$. The analysis of the momentum budget allows to gain some
insight on the physics of drag reduction, and suggests the existence of a
maximum drag reduction asymptote.


title: The Varieties of Dynamic Multiscaling in Fluid Turbulence

abstract:
  We show that different ways of extracting time scales from time-dependent
velocity structure functions lead to different dynamic-multiscaling exponents
in fluid turbulence. These exponents are related to equal-time multiscaling
exponents by different classes of bridge relations which we derive. We check
this explicitly by detailed numerical simulations of the GOY shell model for
fluid turbulence. Our results can be generalized to any system in which both
equal-time and time-dependent structure functions show multiscaling.


title: A simple theory for the Raman spike

abstract:
  The classical stimulated Raman scattering system describing resonant
interaction between two electromagnetic waves and a fast relaxing medium wave
is studied by introducting a systematic perturbation approach in powers of the
relaxation time. We separate amplitude and phase effects for these complex
fields. The analysis of the former shows the existence of a stagnation distance
after which monotonous energy transfer begins from one electromagnetic wave to
the other and this quantity is calculated. Concerning phase effects we give the
conditions for the formation of a Raman spike from an initial fast and large
phase jump in one of the waves. The spike evolution and width estimated from
the reduced model agree with the results from numerical simulations of the
original system.


title: Self-organizing Traffic Control: First Results

abstract:
  We developed a virtual laboratory for traffic control where agents use
different strategies in order to self-organize on the road. We present our
first results where we compare the performance and behaviour promoted by
environmental constrains and five different simple strategies: three inspired
in flocking behaviour, one selfish, and one inspired in the minority game.
Experiments are presented for comparing the strategies. Different issues are
discussed, such as the important role of environmental constrains and the
emergence of traffic lanes.


title: Weak-wave advancement in nearly collinear four-wave mixing

abstract:
  We identify a new four-wave mixing process in which two nearly collinear pump
beams produce phase-dependent gain into a weak bisector signal beam in a
self-defocusing Kerr medium. Phase matching is achieved by weak-wave
advancement caused by cross-phase modulation between the pump and signal beams.
We relate this process to the inverse of spatial modulational instability and
suggest a time-domain analog.


title: Synchronization in model networks of class I neurons

abstract:
  We study a modification of the canonical model for networks of class I
neurons, presented by Hoppensteadt and Izhikevich, in which the 'pulse' emitted
by a neuron is smooth rather than a delta-function. We prove two types of
results about synchronization and desynchronization of such networks, the first
type pertaining to 'pulse' functions which are symmetric, and the other type in
the regime in which each neuron is connected to many other neurons.

categories: nlin.AO cond-mat.dis-nn nlin.PS q-bio.NC


title: Trapping in quantum chains

abstract:
  A quantum breather on a translationally invariant one-dimensional anharmonic
lattice is an extended Bloch state with two or more particles in a strongly
correlated state. We discuss several effects that break the lattice symmetry
and lead to spatial localization of the breather.


title: Towards Modeling of Consciousness

abstract:
  An approach to nonlinear dynamical modeling of interaction between conscious
and automatic processes in the brain is proposed. Illustration of this approach
on the nonlinear equation for the current density in the cortex is presented.
Nonlinearity is determined by the sigmoidal firing rate of neurons. The current
density is considered as complex field with real and imaginary parts
representing automatic and conscious processes. The interaction is due to the
nonlinearity of the system. From this approach it follows that the nonlinear
dynamics for pure imaginary process (a dream) is quite different from the
dynamics of automatic process.


title:
Systematic construction of separable systems with quadratic in momenta
  first integrals

abstract:
  Liouville integrable separable systems with quadratic in momenta first
integrals are considered. Particular attention is paid to the systems generated
by the so-called special conformal Killing tensors, i.e. Benenti systems. Then,
infinitely many new classes of separable systems are constructed by appropriate
deformations of Benenti class systems.


title:
Ecology of active and passive players and their impact on information
  selection

abstract:
  Is visitors' attendance a fair indicator of a web site's quality? Internet
sub-domains are usually characterized by power law distributions of visits,
thus suggesting a richer-get-richer process. If this is the case, the number of
visits is not a relevant measure of quality. If, on the other hand, there are
active players, i.e. visitors who can tell the value of the information
available, better sites start getting richer after a crossover time.

categories: nlin.AO cond-mat.dis-nn cond-mat.stat-mech nlin.SI physics.soc-ph


title:
Avalanche of Bifurcations and Hysteresis in a Model of Cellular
  Differentiation

abstract:
  Cellular differentiation in a developping organism is studied via a discrete
bistable reaction-diffusion model. A system of undifferentiated cells is
allowed to receive an inductive signal emenating from its environment.
Depending on the form of the nonlinear reaction kinetics, this signal can
trigger a series of bifurcations in the system. Differentiation starts at the
surface where the signal is received, and cells change type up to a given
distance, or under other conditions, the differentiation process propagates
through the whole domain. When the signal diminishes hysteresis is observed.


title:
On computational irreducibility and the predictability of complex
  physical systems

abstract:
  Using elementary cellular automata (CA) as an example, we show how to
coarse-grain CA in all classes of Wolfram's classification. We find that
computationally irreducible (CIR) physical processes can be predictable and
even computationally reducible at a coarse-grained level of description. The
resulting coarse-grained CA which we construct emulate the large-scale behavior
of the original systems without accounting for small-scale details. At least
one of the CA that can be coarse-grained is irreducible and known to be a
universal Turing machine.


title: Stackel separability for Newton systems of cofactor type

abstract:
  A conservative Newton system (d/dt)^2 q = -grad V(q) in R^n is called
separable when the Hamilton--Jacobi equation for the natural Hamiltonian H =
(1/2) p^2 + V(q) can be solved through separation of variables in some
curvilinear coordinates. If these coordinates are orhogonal, the Newton system
admits n first integrals, which all have separable Stackel form with quadratic
dependence on p.
  We study here separability of the more general class of Newton systems
(d/dt)^2 q = -cof G^(-1) grad W(q) that admit n quadratic first integrals. We
prove that a related system with the same integrals can be transformed through
a non-canonical transformation into a Stackel separable Hamiltonian system and
solved by qudratures, providing a solution to the original system.
  The separation coordinates, which are defined as characteristic roots of a
linear pencil (G - mu G~) of elliptic coordinates matrices, generalize the well
known elliptic and parabolic coordinates. Examples of such new coordinates in
two and three dimensions are given.
  These results extend, in a new direction, the classical separability theory
for natural Hamiltonians developed in the works of Jacobi, Liouville, Stackel,
Levi-Civita, Eisenhart, Benenti, Kalnins and Miller.


title:
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star
  graphs I: construction and numerical results

abstract:
  In a series of two papers we investigate the universal spectral statistics of
chaotic quantum systems in the ten known symmetry classes of quantum mechanics.
In this first paper we focus on the construction of appropriate ensembles of
star graphs in the ten symmetry classes. A generalization of the
Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry
classes. The conjecture is supported by numerical results that demonstrate the
fidelity of the spectral statistics of star graphs to the corresponding
Gaussian random-matrix theories.


title:
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star
  graphs II: semiclassical approach

abstract:
  A semiclassical approach to the universal ergodic spectral statistics in
quantum star graphs is presented for all known ten symmetry classes of quantum
systems. The approach is based on periodic orbit theory, the exact
semiclassical trace formula for star graphs and on diagrammatic techniques. The
appropriate spectral form factors are calculated upto one order beyond the
diagonal and self-dual approximations. The results are in accordance with the
corresponding random-matrix theories which supports a properly generalized
Bohigas-Giannoni-Schmit conjecture.


title: Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

abstract:
  Two-dimensional random Lorentz gases with absorbing traps are considered in
which a moving point particle undergoes elastic collisions on hard disks and
annihilates when reaching a trap. In systems of finite spatial extension, the
asymptotic decay of the survival probability is exponential and characterized
by an escape rate, which can be related to the average positive Lyapunov
exponent and to the dimension of the fractal repeller of the system. For
infinite systems, the survival probability obeys a stretched exponential law of
the form P(c,t)~exp(-Ct^{1/2}). The transition between the two regimes is
studied and we show that, for a given trap density, the non-integer dimension
of the fractal repeller increases with the system size to finally reach the
integer dimension of the phase space. Nevertheless, the repeller remains
fractal. We determine the special scaling properties of this fractal.


title:
Rotating Spiral Waves with Phase-Randomized Core in Non-locally Coupled
  Oscillators

abstract:
  Rotating spiral waves with a central core composed of phase-randomized
oscillators can arise in reaction-diffusion systems if some of the chemical
components involved are diffusion-free. This peculiar phenomenon is
demonstrated for a paradigmatic three-component reaction-diffusion model. The
origin of this anomalous spiral dynamics is the effective non-locality in
coupling, whose effect is stronger for weaker coupling. There exists a critical
coupling strength which is estimated from a simple argument. Detailed
mathematical and numerical analyses are carried out in the extreme case of weak
coupling for which the phase reduction method is applicable. Under the
assumption that the mean field pattern keeps to rotate steadily as a result of
a statistical cancellation of the incoherence, we derive a functional
self-consistency equation to be satisfied by this space-time dependent
quantity. Its solution and the resulting effective frequencies of the
individual oscillators are found to agree excellently with the numerical
simulation.


title: Faster than Lyapunov decays of classical Loshcmidt echo

abstract:
  We show that in the classical interaction picture the echo-dynamics, namely
the composition of perturbed forward and unperturbed backward hamiltonian
evolution, can be treated as a time-dependent hamiltonian system. For strongly
chaotic (Anosov) systems we derive a cascade of exponential decays for the
classical Loschmidt echo, starting with the leading Lyapunov exponent, followed
by a sum of two largest exponents, etc. In the loxodromic case a decay starts
with the rate given as twice the largest Lyapunov exponent. For a class of
perturbations of symplectic maps the echo-dynamics exhibits a drift resulting
in a super-exponential decay of the Loschmidt echo.


title: Two-soliton collisions in a near-integrable lattice system

abstract:
  We examine collisions between identical solitons in a weakly perturbed
Ablowitz-Ladik (AL) model, augmented by either onsite cubic nonlinearity (which
corresponds to the Salerno model, and may be realized as an array of strongly
overlapping nonlinear optical waveguides), or a quintic perturbation, or both.
Complex dependences of the outcomes of the collisions on the initial phase
difference between the solitons and location of the collision point are
observed. Large changes of amplitudes and velocities of the colliding solitons
are generated by weak perturbations, showing that the elasticity of soliton
collisions in the AL model is fragile (for instance, the Salerno's perturbation
with the relative strength of 0.08 can give rise to a change of the solitons'
amplitudes by a factor exceeding 2). Exact and approximate conservation laws in
the perturbed system are examined, with a conclusion that the small
perturbations very weakly affect the norm and energy conservation, but
completely destroy the conservation of the lattice momentum, which is explained
by the absence of the translational symmetry in generic nonintegrable lattice
models. Data collected for a very large number of collisions correlate with
this conclusion. Asymmetry of the collisions (which is explained by the
dependence on the location of the central point of the collision relative to
the lattice, and on the phase difference between the solitons) is investigated
too, showing that the nonintegrability-induced effects grow almost linearly
with the perturbation strength. Different perturbations (cubic and quintic
ones) produce virtually identical collision-induced effects, which makes it
possible to compensate them, thus finding a special perturbed system with
almost elastic soliton collisions.


title: Symmetries and linearization of ordinary difference equations

abstract:
  The connection between symmetries and linearizations of discrete-time
dynamical systems is being inverstigated. It is shown, that existence of
semigroup structures related to the vector field and having linear
representations enables reduction of linearization problem to a system of first
order partial differential equations. By means of inverse of the Poincare map
one can relate symmetries in such linearizable systems to continuous and
discrete ones of the corresponding differential equations.


title: Painlev\'{e} test of coupled Gross-Pitaevskii equations

abstract:
  Painlev\'{e} test of the coupled Gross-Pitaevskii equations has been carried
out with the result that the coupled equations pass the P-test only if a
special relation containing system parameters (masses, scattering lengths) is
satisfied. Computer algebra is applied to evaluate j=4 compatibility condition
for admissible external potentials. Appearance of an arbitrary real potential
embedded in the external potentials is shown to be the consequence of the
coupling. Connection with recent experiments related to stability of
two-component Bose-Einstein condensates of Rb atoms is discussed.


title:
The reduced Maxwell-Duffing description of extremely short pulses in
  non-resonant media

abstract:
  The propagation of extremely short pulses of electromagnetic field
(electromagnetic spikes) is considered in the framework of a model where the
material medium is represented by anharmonic oscillators with cubic
nonlinearities (Duffing model) and waves can propagate only in the right
direction. The system of reduced Maxwell-Duffing equations admits two families
of exact analytical solutions in the form of solitary waves. These are bright
spikes propagating on a zero background, and bright and dark spikes,
propagating on a nonzero background. Direct simulations demonstrate that these
pulses are very robust against perturbations. We find that a high frequency
modulated electromagnetic pulse evolves into a breather-like one. Conversely a
low frequency pulse transforms into a quasi-harmonic wave.


title: Symmetries of Discrete Systems

abstract:
  In this series of lectures presented at the CIMPA Winter School on Discrete
Integrable Systems in Pondicherry, India, in February, 2003 we give a review of
the application of Lie point symmetries, and their generalizations to the study
of difference equations. The overall theme of these lectures could be called
"continuous symmetries of discrete equations".


title: Second-harmonic generation in vortex-induced waveguides

abstract:
  We study the second-harmonic generation and localization of light in a
reconfigurable waveguide induced by an optical vortex soliton in a defocusing
Kerr medium. We show that the vortex-induced waveguide greatly improves
conversion efficiency from the fundamental to the second harmonic field.


title: Hybrid dynamics of two coupled oscillators that can impact a fixed stop

abstract:
  We consider two linearly coupled masses, where one mass can have inelastic
impacts with a fixed, rigid stop. This leads to the study of a two degree of
freedom, piecewise linear, frictionless, unforced, constrained mechanical
system. The system is governed by three types of dynamics: coupled harmonic
oscillation, simple harmonic motion and discrete rebounds. Energy is dissipated
discontinuously in discrete amounts, through impacts with the stop. We prove
the existence of a nonzero measure set of orbits that lead to infinite impacts
with the stop in a finite time. We show how to modify the mathematical model so
that forward existence and uniqueness of solutions for all time is guaranteed.
Existence of hybrid periodic orbits is shown. A geometrical interpretation of
the dynamics based on action coordinates is used to visualize numerical
simulation results for the asymptotic dynamics.


title: Point perturbations of circle billiards

abstract:
  The spectral statistics of the circular billiard with a point-scatterer is
investigated. In the semiclassical limit, the spectrum is demonstrated to be
composed of two uncorrelated level sequences. The first corresponds to states
for which the scatterer is located in the classically forbidden region and its
energy levels are not affected by the scatterer in the semiclassical limit
while the second sequence contains the levels which are affected by the
point-scatterer. The nearest neighbor spacing distribution which results from
the superposition of these sequences is calculated analytically within some
approximation and good agreement with the distribution that was computed
numerically is found.


title: Soliton Gas in Space-Charge Dominated Beams

abstract:
  Based on the Vlasov-Maxwell equations describing the self-consistent
nonlinear beam dynamics and collective processes, the evolution of an intense
sheet beam propagating through a periodic focusing field has been studied. It
has been shown that in the case of a beam with uniform phase space density the
Vlasov-Maxwell equations can be replaced exactly by the hydrodynamic equations
with a triple adiabatic pressure law coupled to the Maxwell equations. We
further demonstrate that starting from the system of hydrodynamic and Maxwell
equations a set of coupled nonlinear Schrodinger equations for the slowly
varying amplitudes of density waves can be derived. In the case where a
parametric resonance between a certain mode of density waves and the external
focusing occurs, the slow evolution of the resonant amplitudes in the cold-beam
limit is shown to be governed by a system of coupled Gross-Pitaevskii
equations. Properties of the nonlinear Schrodinger equation as well as
properties of the Gross-Pitaevskii equation are discussed, together with
soliton and condensate formation in intense particle beams.


title: Closed dressing chains of D=1 and D=2 Toda lattice

abstract:
  We apply the method of dressing chains to reproduction of Toda lattice in the
case of D=1 and D=2. On the example of modified equations $m_0TL$ and $m_1TL$
it is shown that combination of the Darboux and Schlesinger transformations
results in closed dressing chains.


title:
Nonlinear spherical gravitational downfall of gas onto a solid ball:
  analytic and numerical results

abstract:
  The process of downfall of initially homogeneous gas onto a solid ball due to
the ball's gravity (relevant in astrophysical situations) is studied with a
combination of analytic and numerical methods. The initial explicit solution
soon becomes discontinuous and gives rise to a shock wave. Afterwards, there is
a crossover between two intermediate asymptotic similarity regimes, where the
shock wave propagates outwards according to two self-similar laws, initially
accelerating and eventually decelerating and vanishing, leading to a static
state. The numerical study allows one to investigate in detail this dynamical
problem and its time evolution, verifying and complementing the analytic
results on the initial solution, intermediate self-similar laws and static
long-term solution.


title: Statistics of finite-time Lyapunov exponents in the Ulam map

abstract:
  The statistical properties of finite-time Lyapunov exponents at the Ulam
point of the logistic map are investigated. The exact analytical expression for
the autocorrelation function of one-step Lyapunov exponents is obtained,
allowing the calculation of the variance of exponents computed over time
intervals of length $n$. The variance anomalously decays as $1/n^2$. The
probability density of finite-time exponents noticeably deviates from the
Gaussian shape, decaying with exponential tails and presenting $2^{n-1}$ spikes
that narrow and accumulate close to the mean value with increasing $n$. The
asymptotic expression for this probability distribution function is derived. It
provides an adequate smooth approximation to describe numerical histograms
built for not too small $n$, where the finiteness of bin size trimmes the sharp
peaks.

categories: nlin.CD cond-mat.stat-mech math-ph math.MP


title: Comment on ''Exceptional points and double poles of the S matrix''

abstract:
  In a recent paper [Phys. Rev. E 67, 026204 (2003)], Rotter calculates the
geometric phases that are picked up by the eigenvectors of a two-state quantum
system when an exceptional point is encircled. In addition to the geometric
phases observed by Dembowski et al. in a microwave cavity experiment [Phys.
Rev. Lett. 86, 787 (2001)], she finds a phase factor of i. We show that Rotters
results are inconsistent.


title: Self-sustained activity in a small-world network of excitable neurons

abstract:
  We study the dynamics of excitable integrate-and-fire neurons in a
small-world network. At low densities $p$ of directed random connections, a
localized transient stimulus results in either self-sustained persistent
activity or in a brief transient followed by failure. Averages over the
quenched ensemble reveal that the probability of failure changes from 0 to 1
over a narrow range in $p$; this failure transition can be described
analytically through an extension of an existing mean-field result. Exceedingly
long transients emerge at higher densities $p$; their activity patterns are
disordered, in contrast to the mostly periodic persistent patterns observed at
low $p$. The times at which such patterns die out are consistent with a
stretched-exponential distribution, which depends sensitively on the
propagation velocity of the excitation.


title: Multifractality of the Feigenbaum attractor and fractional derivatives

abstract:
  It is shown that fractional derivatives of the (integrated) invariant measure
of the Feigenbaum map at the onset of chaos have power-law tails in their
cumulative distributions, whose exponents can be related to the spectrum of
singularities $f(\alpha)$. This is a new way of characterizing multifractality
in dynamical systems, so far applied only to multifractal random functions
(Frisch and Matsumoto (J. Stat. Phys. 108:1181, 2002)). The relation between
the thermodynamic approach (Vul, Sinai and Khanin (Russian Math. Surveys 39:1,
1984)) and that based on singularities of the invariant measures is also
examined. The theory for fractional derivatives is developed from a heuristic
point view and tested by very accurate simulations.


title: Microscopic chaos and transport in thermostated dynamical systems

abstract:
  A fundamental challenge is to understand nonequilibrium statistical mechanics
starting from microscopic chaos in the equations of motion of a many-particle
system. In this review we summarize recent theoretical advances along these
lines. Particularly, we are concerned with nonequilibrium situations created by
external electric fields and by temperature or velocity gradients. These
constraints pump energy into a system, hence there must be some thermal
reservoir that prevents the system from heating up. About twenty years ago a
deterministic and time-reversible modeling of such thermal reservoirs was
proposed in form of Gaussian and Nose-Hoover thermostats. This approach yielded
simple relations between fundamental quantities of nonequilibrium statistical
mechanics and of dynamical systems theory. The main theme of our review is to
critically assess the universality of these results. As a vehicle of
demonstration we employ the driven periodic Lorentz gas, which is a toy model
for the classical dynamics of an electron in a metal under application of an
electric field. Applying different types of thermal reservoirs to this system
we compare the resulting nonequilibrium steady states with each other. Along
the same lines we discuss an interacting many-particle system under shear and
heat. Finally, we outline an unexpected relationship between deterministic
thermostats and active Brownian particles modeling biophysical cell motility.


title:
Near-Resonant, Steady Mode Interaction: Periodic, Quasi-Periodic and
  Localized Patterns

abstract:
  Motivated by the rich variety of complex periodic and quasi-periodic patterns
found in systems such as two-frequency forced Faraday waves, we study the
interaction of two spatially periodic modes that are nearly resonant. Within
the framework of two coupled one-dimensional Ginzburg-Landau equations we
investigate analytically the stability of the periodic solutions to general
perturbations, including perturbations that do not respect the periodicity of
the pattern, and which may lead to quasi-periodic solutions. We study the
impact of the deviation from exact resonance on the destabilizing modes and on
the final states. In regimes in which the mode interaction leads to traveling
waves our numerical simulations reveal localized waves in which the wavenumbers
are resonant and which drift through a steady background pattern that has an
off-resonant wavenumber ratio.


title:
Algebro-geometric solution of the discrete KP equation over a finite
  field out of a hyperelliptic curve

abstract:
  We transfer the algebro-geometric method of construction of solutions of the
discrete KP equation to the finite field case. We emphasize role of the
Jacobian of the underlying algebraic curve in construction of the solutions. We
illustrate in detail the procedure on example of a hyperelliptic curve.


title: Renormalization Analysis of Resonance Structure in 2-D Symplectic Map

abstract:
  A symplecticity-preserving RG analysis is carried out to study a resonance
structure near an elliptic fixed point of a proto-type symplectic map in two
dimensions. Through analyzing fixed points of a reduced RG map, a topology of
the resonance structure such as a chain of resonant islands can be determined
analytically. An application of this analysis to the Henon map is also
presented.


title:
Fourier form of the dressing method: simple example of integrable
  (2+1)-dimensional integral-differential equation

abstract:
  We represent the Fourier form of the dressing method, which is effective for
construction of multidimensional integral-differential equations together with
their solutions.
  Example of integrable (but non-physical) expansion of Intermediate Long Wave
equation in (2+1)-dimensions is considered.


title: Polariton effect in nonlinear pulse propagation

abstract:
  The joint influence of the polariton effect and Kerr-like nonlinearity on
propagation of optical pulses is studied. The existence of different families
of envelope solitary wave solutions in the vicinity of polariton gap is shown.
The properties of solutions depend strongly on the carrier wave frequency. In
particular, solitary waves inside and outside of the polariton gap exhibit
different velocity and amplitude dependencies on their duration.


title:
Nonlinear Bloch-wave interaction and Bragg scattering in
  optically-induced lattices

abstract:
  We study, both theoretically and experimentally, the Bragg scattering of
light in optically-induced photonic lattices and reveal the key physical
mechanisms which govern nonlinear self-action of narrow beams under the
combined effects of Bragg scattering and wave diffraction, allowing for
selecting bands with different effective dispersion.


title: On the Deformations of Dorfman's and Sokolov's Operators

abstract:
  We deform the Dorfman's and Sokolov's Hamiltonian operators by the
quasi-Miura transformation coming from the topological field theory and
investigate the deformed operators.


title:
Transformation of a generalized Harry Dym equation into the
  Hirota--Satsuma system

abstract:
  The new generalized Harry Dym equation, recently introduced by Z. Popowicz in
Phys. Lett. A 317, 260--264 (2003), is transformed into the Hirota--Satsuma
system of coupled KdV equations.


title:
Dessins d'Enfants, Their Deformations and Algebraic the Sixth Painlev\'e
  and Gauss Hypergeometric Functions

abstract:
  We consider an application of Grothendieck's dessins d'enfants to the theory
of the sixth Painlev\'e and Gauss hypergeometric functions: two classical
special functions of the isomonodromy type. It is shown that, higher order
transformations and the Schwarz table for the Gauss hypergeometric function are
closely related with some particular Belyi functions. Moreover, we introduce a
notion of deformation of the dessins d'enfants and show that one dimensional
deformations are a useful tool for construction of algebraic the sixth
Painlev\'e functions.


title: Universal Mechanisms in the Growth of Voluntary Organizations

abstract:
  We analyze the growth statistics of Swedish trade unions and find a universal
functional form for the probability distribution of growth rates of union size,
and a power law dependence of the standard deviation of this distribution on
the number of members of the union. We also find that the typical size and the
typical number of local chapters scales as a power law of the union size.
Intriguingly, our results are similar to results reported for other human
organizations of a quite different nature. Our findings are consistent with the
possibility that universal mechanisms may exist governing the growth patterns
of human organizations.


title: Factorization, reduction and embedding in integrable cellular automata

abstract:
  Factorized dynamics in soliton cellular automata with quantum group symmetry
is identified with a motion of particles and anti-particles exhibiting pair
creation and annihilation. An embedding scheme is presented showing that the
D^{(1)}_n-automaton contains, as certain subsectors, the box-ball systems and
all the other automata associated with the crystal bases of non-exceptional
affine Lie algebras. The results extend the earlier ones to higher
representations by a certain reduction and to a wider class of boundary
conditions.


title: Noise-induced synchronization for phase turbulence

abstract:
  Phase turbulence is suppressed by applying common noise additively to the
Kuramoto-Sivashinsky type equation, and the noise-induced phase synchronization
is realized. The noise strength necessary for the suppression of phase
turbulence is evaluated theoretically.


title:
Validity of numerical trajectories in the synchronization transition of
  complex systems

abstract:
  We investigate the relationship between the loss of synchronization and the
onset of shadowing breakdown {\it via} unstable dimension variability in
complex systems. In the neighborhood of the critical transition to strongly
non-hyperbolic behavior, the system undergoes on-off intermittency with respect
to the synchronization state. There are potentially severe consequences of
these facts on the validity of the computer-generated trajectories obtained
from dynamical systems whose synchronization manifolds share the same
non-hyperbolic properties.


title: Real Hamiltonian forms of Hamiltonian systems

abstract:
  We introduce the notion of a real form of a Hamiltonian dynamical system in
analogy with the notion of real forms for simple Lie algebras. This is done by
restricting the complexified initial dynamical system to the fixed point set of
a given involution. The resulting subspace is isomorphic (but not
symplectomorphic) to the initial phase space. Thus to each real Hamiltonian
system we are able to associate another nonequivalent (real) ones. A crucial
role in this construction is played by the assumed analyticity and the
invariance of the Hamiltonian under the involution. We show that if the initial
system is Liouville integrable, then its complexification and its real forms
will be integrable again and this provides a method of finding new integrable
systems starting from known ones. We demonstrate our construction by finding
real forms of dynamics for the Toda chain and a family of Calogero--Moser
models. For these models we also show that the involution of the complexified
phase space induces a Cartan-like involution of their Lax representations.


title: Gap soliton formation by nonlinear supratransmission in Bragg media

abstract:
  A Bragg medium in the nonlinear Kerr regime, submitted to incident
cw-radiation at a frequency in a band gap, switches from total reflection to
transmission when the incident energy overcomes some threshold. We demonstrate
that this is a result of nonlinear supratransmission, which allows to prove
that i) the threshold incident amplitude is simply expressed in terms of the
deviation from the Bragg resonance, ii) the process is not the result of a
shift of the gap in the nonlinear dispersion relation, iii) the transmission
does occur by means of gap soliton trains, as experimentally observed [D.
Taverner et al., Opt Lett 23 (1998) 328], iv) the required energy tends to zero
close to the band edge.


title: Nonlinear supratransmission as a fundamental instability

abstract:
  The nonlinear supratransmission is the property of a nonlinear system
possessing a natural forbidden band gap to transmit energy of a signal with a
frequency in the gap by means of generation of nonlinear modes (gap solitons).
This process is shown to result from a generic instability of the evanescent
wave profile generated in a nonlinear medium by the incident signal.


title: No production of entropy in the Euler fluid

abstract:
  We derive the Euler equations as the hydrodynamic limit of a stochastic model
of a hard-sphere gas on a lattice. We show that the system does not produce
entropy.


title:
Absolute continuity of projected SRB measures of coupled Arnold cat map
  lattices

abstract:
  We study a d-dimensional coupled map lattice consisting of hyperbolic toral
automorphisms (Arnold cat maps) that are weakly coupled by an analytic coupling
map. We construct the Sinai-Ruelle-Bowen measure for this system and study its
marginals on the tori. We prove they are absolutely continuous with respect to
the Lebesgue measure if and only if the coupling satisfies a nondegeneracy
condition.


title: From dispersionless to soliton systems via Weyl-Moyal like deformations

abstract:
  The formalism of quantization deformation is reviewed and the Weyl-Moyal like
deformation is applied to systematic construction of the field and lattice
integrable soliton systems from Poisson algebras of dispersionless systems.


title:
Equation-free modeling of evolving diseases: Coarse-grained computations
  with individual-based models

abstract:
  We demonstrate how direct simulation of stochastic, individual-based models
can be combined with continuum numerical analysis techniques to study the
dynamics of evolving diseases. % Sidestepping the necessity of obtaining
explicit population-level models, the approach analyzes the (unavailable in
closed form) `coarse' macroscopic equations, estimating the necessary
quantities through appropriately initialized, short `bursts' of
individual-based dynamic simulation. % We illustrate this approach by analyzing
a stochastic and discrete model for the evolution of disease agents caused by
point mutations within individual hosts. % Building up from classical SIR and
SIRS models, our example uses a one-dimensional lattice for variant space, and
assumes a finite number of individuals. % Macroscopic computational tasks
enabled through this approach include stationary state computation, coarse
projective integration, parametric continuation and stability analysis.


title: A simple way of making a Hamiltonian system into a bi-Hamiltonian one

abstract:
  Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we
show that its Lie derivative $L_{\tau}(P)$ along a vector field $\tau$ defines
another Poisson structure, which is automatically compatible with $P$, if and
only if $[L_{\tau}^2(P),P]=0$, where $[\cdot,\cdot]$ is the Schouten bracket.
We further prove that if $\dim\ker P\leq 1$ and $P$ is of locally constant
rank, then all Poisson structures compatible with a given Poisson structure $P$
on a finite-dimensional manifold $M$ are locally of the form $L_{\tau}(P)$,
where $\tau$ is a local vector field such that
$L_{\tau}^2(P)=L_{\tilde\tau}(P)$ for some other local vector field
$\tilde\tau$. This leads to a remarkably simple construction of bi-Hamiltonian
dynamical systems. We also present a generalization of these results to the
infinite-dimensional case. In particular, we provide a new description for
pencils of compatible local Hamiltonian operators of Dubrovin--Novikov type and
associated bi-Hamiltonian systems of hydrodynamic type.
  Key words: compatible Poisson structures, Hamiltonian operators,
bi-Hamiltonian systems (= bihamiltonian systems), integrability, Schouten
bracket, master symmetry, Lichnerowicz--Poisson cohomology, hydrodynamic type
systems.
  MSC 2000: Primary: 37K10; Secondary: 37K05, 37J35

categories: nlin.SI hep-th math-ph math.DG math.MP math.SG


title: Shell Model of Two-dimensional Turbulence in Polymer Solutions

abstract:
  We address the effect of polymer additives on two dimensional turbulence, an
issue that was studied recently in experiments and direct numerical
simulations. We show that the same simple shell model that reproduced drag
reduction in three-dimensional turbulence reproduces all the reported effects
in the two-dimensional case. The simplicity of the model offers a
straightforward understanding of the all the major effects under consideration.


title: The thermodynamic limit of the Whitham equations

abstract:
  The infinite-genus limit of the KdV-Whitham equations is derived. The limit
involves special scaling for the associated spectral surface such that the
integrated density of states remains finite as $N \to \infty$ (the
thermodynamic type limit). The limiting integro-differential system describes
slow evolution of the density of states and can be regarded as the kinetic
equation for a soliton gas.


title:
Intermittent distribution of tracers advected by a compressible random
  flow

abstract:
  Multifractal properties of a tracer density passively advected by a
compressible random velocity field are characterized. A relationship is
established between the statistical properties of mass on the dynamical fractal
attractor towards which the trajectories converge and large deviations of the
stretching rates of the flow. In the framework of the compressible Kraichnan
model, this result is illustrated by analytical calculations and confirmed by
numerical simulations.