title:
Scaling Properties of Saddle-Node Bifurcations on Fractal Basin
  Boundaries

abstract:
  We analyze situations where a saddle-node bifurcation occurs on a fractal
basin boundary. Specifically, we are interested in what happens when a system
parameter is slowly swept in time through the bifurcation. Such situations are
known to be indeterminate in the sense that it is difficult to predict the
eventual fate of an orbit that tracks the pre-bifurcation node attractor as the
system parameter is swept through the bifurcation. In this paper we investigate
the scaling of (1) the fractal basin boundary of the static (i.e., unswept)
system near the saddle-node bifurcation, (2) the dependence of the orbit's
final destination on the sweeping rate, (3) the dependence of the time it takes
for an attractor to capture a swept orbit on the sweeping rate, and (4) the
dependence of the final attractor capture probability on the noise level. With
respect to noise, our main result is that the effect of noise scales with the
5/6 power of the parameter drift rate. Our approach is to first investigate all
these issues using one-dimensional map models. The simplification of treatment
inherent in one dimension greatly facilitates analysis and numerical
experiment, aiding us in obtaining the new results listed above. Following our
one-dimensional investigations, we explain that these results can be applied to
two-dimensional systems. We show, through numerical experiments on a
periodically forced second order differential equation example, that the
scalings we have found also apply to systems that result in two dimensional
maps.


title: Finite-time Lyapunov exponents for products of random transformations

abstract:
  It is shown how continuous products of random transformations constrained by
a generic group structure can be studied by using Iwasawa's decomposition into
``angular'', ``diagonal'' and ``shear'' degrees of freedom. In the case of a
Gaussian process a set of variables, adapted to the Iwasawa decomposition and
still having a Gaussian distribution, is introduced and used to compute the
statistics of the finite-time Lyapunov spectrum of the process. The variables
also allow to show the exponential freezing of the ``shear'' degrees of
freedom, which contain information about the Lyapunov eigenvectors.


title: Interaction of pulses in nonlinear Schroedinger model

abstract:
  The interaction of two rectangular pulses in nonlinear Schroedinger model is
studied by solving the appropriate Zakharov-Shabat system. It is shown that two
real pulses may result in appearance of moving solitons. Different limiting
cases, such as a single pulse with a phase jump, a single chirped pulse,
in-phase and out-of-phase pulses, and pulses with frequency separation, are
analyzed. The thresholds of creation of new solitons and multi-soliton states
are found.


title: On the parametric dependences of a class of non-linear singular maps

abstract:
  We discuss a two-parameter family of maps that generalize piecewise linear,
expanding maps of the circle. One parameter measures the effect of a
non-linearity which bends the branches of the linear map. The second parameter
rotates points by a fixed angle. For small values of the nonlinearity
parameter, we compute the invariant measure and show that it has a singular
density to first order in the nonlinearity parameter. Its Fourier modes have
forms similar to the Weierstrass function. We discuss the consequences of this
singularity on the Lyapunov exponents and on the transport properties of the
corresponding multibaker map. For larger non-linearities, the map becomes
non-hyperbolic and exhibits a series of period-adding bifurcations.


title:
Quasi-classical DBAR-dressing approach to the weakly dispersive KP
  hierarchy

abstract:
  Recently proposed quasi-classical DBAR-dressing method provides a systematic
approach to study the weakly dispersive limit of integrable systems. We apply
the quasi-classical DBAR-dressing method to describe dispersive corrections of
any order. We show how calculate the DBAR problems at any order for a rather
general class of integrable systems, presenting explicit results for the KP
hierarchy case. We demonstrate the stability of the method at each order. We
construct an infinite set of commuting flows at first order which allow a
description analogous to the zero order (purely dispersionless) case,
highlighting a Whitham type structure. Obstacles for the construction of the
higher order dispersive corrections are also discussed.


title: Synchronization of Random Linear Maps

abstract:
  We study synchronization of random one-dimensional linear maps for which the
Lyapunov exponent can be calculated exactly. Certain aspects of the dynamics of
these maps are explained using their relation with a random walk. We confirm
that the Lyapunov exponent changes sign at the complete synchronization
transition. We also consider partial synchronization of nonidentical systems.
It turns out that the way partial synchronization manifests depends on the type
of differences (in Lyapunov exponent or in contraction points) between the
systems. The crossover from partial synchronization to complete synchronization
is also examined.


title:
Complex behavior from a simple physical system: A numerical
  investigation

abstract:
  In this paper, the familiar problems of free-fall motion and simple harmonic
motion (SHM) are combined. The novel composite system passes from regular to
chaotic behavior for increasing values of energy $E$. This system is a suitable
example to introduce undergraduates to some of the concepts of dynamical chaos
theory.


title:
Quiescent String Dominance Parameter F and Classification of
  One-Dimensional Cellular Automata

abstract:
  The mechanism which discriminates the pattern classes at the same $\lambda$,
is found. It is closely related to the structure of the rule table and
expressed by the numbers of the rules which break the strings of the quiescent
states. It is shown that for the N-neighbor and K-state cellular automata, the
class I, class II, class III and class IV patterns coexist at least in the
range, $\frac{1}{K} \le \lambda \le 1-\frac{1}{K} $. The mechanism is studied
quantitatively by introducing a new parameter $F$, which we call quiescent
string dominance parameter. It is taken to be orthogonal to $\lambda$. Using
the parameter F and $\lambda$, the rule tables of one dimensional 5-neighbor
and 4-state cellular automata are classified. The distribution of the four
pattern classes in ($\lambda$,F) plane shows that the rule tables of class III
pattern class are distributed in larger $F$ region, while those of class II and
class I pattern classes are found in the smaller $F$ region and the class IV
behaviors are observed in the overlap region between them. These distributions
are almost independent of $\lambda$ at least in the range $0.25 \leq \lambda
\leq 0.75$, namely the overlapping region in $F$, where the class III and class
II patterns coexist, has quite gentle $\lambda$ dependence in this $\lambda$
region. Therefore the relation between the pattern classes and the $\lambda$
parameter is not observed. PACS: 89.75.-k Complex Systems


title:
Non polynomial conservation law densities generated by the symmetry
  operators in some hydrodynamical models

abstract:
  New extra series of conserved densities for the polytropic gas model and
nonlinear elasticity equation are obtained without any references to the
recursion operator or to the Lax operator formalism. Our method based on the
utilization of the symmetry operators and allows us to obtain the densities of
arbitrary homogenuity dimensions. The nonpolynomial densities with logarithmics
behaviour are presented as an example. The special attention is paid for the
singular case $(\gamma=1)$ for which we found new non homogenious solutions
expressed in terms of the elementary functions.


title:
Wronskians, Generalized Wronskians and Solutions to the Korteweg-de
  Vries Equation

abstract:
  A bridge going from Wronskian solutions to generalized Wronskian solutions of
the Korteweg-de Vries equation is built. It is then shown that generalized
Wronskian solutions can be viewed as Wronskian solutions. The idea is used to
generate positons, negatons and their interaction solutions to the Korteweg-de
Vries equation. Moreover, general positons and negatons are constructed through
the Wronskian formulation. A few new exact solutions to the KdV equation are
explicitly presented as examples of Wronskian solutions.


title:
The $U_q(\hat{sl}(2/1))_1$-module $V(\Lambda_2)$ and a Corner Transfer
  Matrix at q=0

abstract:
  The north-west corner transfer matrix of an inhomogeneous integrable vertex
model constructed from the vector representation of $U_q\bigl(sl(2/1)\bigr)$
and its dual is investigated. In the limit $q\to0$, the spectrum can be
obtained. Based on an analysis of the half-infinite tensor products related to
all CTM-eigenvalues $\geq -4$, it is argued that the eigenvectors of the corner
transfer matrix are in one-to-one correspondance with the weight states of the
$U_q\bigl((\hat{sl}(2/1)\bigr)_1-$module $V(\Lambda_2)$ at level one. This is
supported by a comparison of the comlete set of eigenvectors with a
nondegenerate triple of eigenvalues of the CTM-Hamiltonian and the generators
of the Cartan-subalgebra of $U_q\bigl(sl(2|1)\bigr)$ to the weight states of
$V(\Lambda_2)$ with multiplicity one.


title: A Test of a Test for Chaos

abstract:
  A simple new binary test for chaos has been proposed by Gottwald and
Melbourne. We apply this test successfully to the Henon-Heiles and Lorenz
systems, demonstrating its applicability to conservative systems, as well as
dissipative systems. The binary test is effective for highly chaotic
Hamiltonian systems and orbits on a strange attractor and is particularly
useful as a marker of the transition from regularity to chaos. However, we find
it is not able to detect more subtle instances of transient chaos.


title: Anholonomy and Geometrical Localization in Dynamical Systems

abstract:
  We characterize the geometrical and topological aspects of a dynamical system
by associating a geometric phase with a phase space trajectory. Using the
example of a nonlinear driven damped oscillator, we show that this phase is
resilient to fluctuations, responds to all bifurcations in the system, and also
finds new geometric transitions. Enriching the phase space description is a
novel phenomenon of ``geometrical localization'' which manifests itself as a
significant deviation from planar dynamics over a short time interval.


title: Towards a general theory of nonlinear flow-distributed oscillations

abstract:
  We outline a general theory for the analysis of flow-distributed standing and
travelling wave patterns in one-dimensional, open plug-flows of oscillatory
chemical media. We treat both the amplitude and phase dynamics of small and
large-amplitude waves, considering both travelling and stationary waves on an
equal footing and emphasizing features that are generic to a variety of kinetic
models. We begin with a linear stability analysis for constant and periodic
boundary forcing, drawing attention to the implications for systems far from a
Hopf bifurcation. Among other results, we show that for systems far from a Hopf
bifurcation, the first absolutely unstable mode may be a stationary wave mode.
We then introduce a non-linear formalism for studying both travelling and
stationary waves and show that the wave forms and their amplitudes depend on a
single reduced transport parameter. Our formalism sheds light on cases where
neither the linearized analyis nor the kinematic theory of phase waves give an
adequate description, and it can be applied to study some of the more complex
types of bifurcations (Canards, period-doublings, etc.) in open flow systems.


title: New Expressions for Discrete Painlev\'e Equations

abstract:
  It is known that discrete Painlev\'e equations have symmetries of the affine
Weyl groups. In this paper we propose a new representation of discrete
Painlev\'e equations in which the symmetries become clearly visible. We know
how to obtain discrete Painlev\'e equations from certain rational surfaces in
connection with the extended affine Weyl groups. By means of this
representation, we clarify the relation between the equation and the surface.


title: Some Bi-Hamiltonian Equations in $R^3$

abstract:
  Hamiltonian formulation of N=3 systems is considered in general. The Jacobi
equation is solved in three classes. Compatible Poisson structures in these
classes are determined and explicitly given. The corresponding bi-Hamiltonian
systems are constructed and some explicit examples are given.


title:
Cavity Light Bullets: 3D Localized Structures in a Nonlinear Optical
  Resonator

abstract:
  We consider the paraxial model for a nonlinear resonator with a saturable
absorber beyond the mean-field limit and develop a method to study the
modulational instabilities leading to pattern formation in all three spatial
dimensions. For achievable parametric domains we observe total radiation
confinement and the formation of 3D localised bright structures. At difference
from freely propagating light bullets, here the self-organization proceeds from
the resonator feedback, combined with diffraction and nonlinearity. Such
"cavity" light bullets can be independently excited and erased by appropriate
pulses, and once created, they endlessly travel the cavity roundtrip. Also, the
pulses can shift in the transverse direction, following external field
gradients.


title:
Inverse scattering approach to coupled higher order nonlinear
  Schr\"odinger equation and N-soliton solutions

abstract:
  A generalized inverse scattering method has been applied to the linear
problem associated with the coupled higher order nonlinear schr\"odinger
equation to obtain it's $N$-soliton solution. An infinite number of conserved
quantities have been obtained by solving a set of coupled Riccati equations. It
has been shown that the coupled system admits two different class of solutions,
characterised by the number of local maxima of amplitude of the soliton.


title: An Integrable Version of Burgers Equation in "Magnetohydrodynamics"

abstract:
  It is pointed out that for the case of (compressible) magnetohydrodynamics
(MHD) with the fields $v_y(y,t)$ and $B_x(y,t)$ one can have equations of the
Burgers type which are integrable. We discuss the solutions. It turns out that
the propagation of the non-linear effects is governed by the initial velocity
(as in Burgers case) as well as by the initial Alfv\'en velocity. Many results
previously obtained for the Burgers equation can be transferred to the MHD
case. We also discuss equipartition $v_y=\pm B_x$. It is shown that an initial
localized small scale magnetic field will end up in fields moving to the left
and the right, thus transporting energy from smaller to larger distances.


title: Determining possible avenues of approach using ANTS

abstract:
  Threat assessment is an important part of level 3 data fusion. Here we study
a subproblem of this, worst-case risk assessment. Inspired by agent-based
models used for simulation of trail formation for urban planning, we use ant
colony optimization (ANTS) to determine possible avenues of approach for the
enemy, given a situation picture.
  One way of determining such avenues would be to calculate the ``potential
field'' caused by placing sources at possible goals for the enemy. This
requires postulating a functional form for the potential, and also takes long
time. Here we instead seek a method for quickly obtaining an effective
potential. ANTS, which has previously been used to obtain approximate solutions
to various optimization problems, is well suited for this. The output of our
method describes possible avenues of approach for the enemy, i.e, areas where
we should be prepared for attack. (The algorithm can also be run ``reversed''
to instead get areas of opportunity for our forces to exploit.)
  Using real geographical data, we found that our method gives a fast and
reliable way of determining such avenues. Our method can be used in a
computer-based command and control system to replace the first step of human
intelligence analysis.


title: On dispersionless coupled modified KP hierarchy

abstract:
  We define and study dispersionless coupled modified KP hierarchy, which
incorporates two different versions of dispersionless modified KP hierarchies.


title: Band-gap solitons in nonlinear optically-induced lattices

abstract:
  We introduce novel optical solitons that consist of a periodic and a
spatially localized components coupled nonlinearly via cross-phase modulation.
The spatially localized optical field can be treated as a gap soliton supported
by the optically-induced nonlinear grating. We find different types of these
band-gap composite solitons and demonstrate their dynamical stability.


title:
Separability and the Birkhoff-Gustavson Normalization of the Perturbed
  Harmonic Oscillators with Homogeneous Polynomial Potentials

abstract:
  In this paper, separability of the perturbed two-dimensional isotropic
harmonic oscillators with homogeneous polynomial potentials is characterized
from their Birkhoff-Gustavson (BG) normalization, one of the conventional
methods for non-integrable Hamiltonian systems.


title: Sato-Crutchfield formulation for some Evolutionary Games

abstract:
  The Sato-Crutchfield equations are studied analytically and numerically. The
Sato-Crutchfield formulation is corresponding to losing memory. Then
Sato-Crutchfield formulation is applied for some different types of games
including hawk-dove, prisoner's dilemma and the battle of the sexes games. The
Sato-Crutchfield formulation is found not to affect the evolutionarily stable
strategy of the ordinary games. But choosing a strategy becomes purely random
independent on the previous experiences, initial conditions, and the rules of
the game itself. Sato-Crutchfield formulation for the prisoner's dilemma game
can be considered as a theoretical explanation for the existence of cooperation
in a population of defectors.


title: On kink-dynamics of stacked-Josephson junctions

abstract:
  Dynamics of a fluxon in a stack of coupled long Josephson junctions is
studied numerically. Based on the numerical simulations, we show that the
dependence of the propagation velocity $c$ on the external bias current
$\gamma$ is determined by the ratio of the critical currents of the two
junctions $J$.


title:
Recursion operators for a class of integrable third-order evolution
  equations

abstract:
  We consider $u_t=u^{\alpha} u_{xxx}+n(u)u_xu_{xx}+m(u)u_x^3+ r(u)u_{xx}
+p(u)u_x^2 + q(u)u_x+s(u)$ with $\alpha=0$ and $\alpha=3$, for those functional
forms of $m, n, p, q, r, s$ for which the equation is integrable in the sense
of an infinite number of Lie-B\"acklund symmetries. Local $x$- and
$t$-independent recursion operators that generate these infinite sets of
symmetries are obtained for the equations. A combination of potential forms,
hodograph transformations and $x$-generalised hodograph transformations are
applied to the obtained equations.


title: On Treves' Algebraic Characterization of the KdV Hierarchy

abstract:
  We have found a possibility to streamline the proof of the Treves' theorem
(Duke Math. J., 108, 251-295, 2001) on an algebraic characterization of the KdV
hierarchy which makes it significantly shorter, following essentially the logic
of the original proof.


title:
Functional sensitivity of Burgers and related equations to initial
  conditions

abstract:
  In this paper, we apply sensitivity methods to nonlinear PDEs like Burgers
and KPZ equations. These equations are known to have analytical solutions which
make easier the analysis of the sensitivity of their solutions to initial
conditions. The main result stands in the fact that the most the solution is
sensitive to the initial condition, the most it is decorrelated in space, i.e.
the values of the initial condition participate to the solution at all
distances of the wave front. This finally reveals a particular aspect of the
Burgers turbulence.


title:
Negative Volterra Flows and Mixed Volterra Flows and Their Infinitely
  Many Conservation Laws

abstract:
  In this article, by means of considering an isospectral operator equation
which corresponds to the Volterra lattice, and constructing opportune time
evolution problems with negative powers of spectral parameter, and using
discrete zero curvature representation, negative Volterra flows are proposed.
We also propose the mixed Volterra flows, which come from positive and negative
volterra flows. From the Lax representation, we demonstrate the existence of
infinitely many conservation laws for the two flows and give the corresponding
conserved densities and the associated fluxes formulaically. Thus their
integrability is further confirmed.


title:
Heteroclinic Contours in Neural Ensembles and the Winnerless Competition
  Principle

abstract:
  The ability of nonlinear dynamical systems to process incoming information is
a key problem of many fundamental and applied sciences. Information processing
by computation with attractors (steady states, limit cycles and strange
attractors) has been a subject of many publications. In this paper we discuss a
new direction in information dynamics based on neurophysiological experiments
that can be applied for the explanation and prediction of many phenomena in
living biological systems and for the design of new paradigms in neural
computation. This new concept is the Winnerless Competition (WLC) principle.
The main point of this principle is the transformation of the incoming identity
or spatial inputs into identity-temporal output based on the intrinsic
switching dynamics of the neural system. In the presence of stimuli the
sequence of the switching, whose geometrical image in the phase space is a
heteroclinic contour, uniquely depends on the incoming information. The key
problem in the realization of the WLC principle is the robustness against noise
and, simultaneously, the sensitivity of the switching to the incoming input. In
this paper we prove two theorems about the stability of the sequential
switching and give several examples of WLC networks that illustrate the
coexistence of sensitivity and robustness.


title: On nonlocally coupled complex Ginzburg-Landau equation

abstract:
  A Ginzburg-Landau type equation with nonlocal coupling is derived
systematically as a reduced form of a universal class of reaction-diffusion
systems near the Hopf bifurcation point and in the presence of another small
parameter. The reaction-diffusion systems to be reduced are such that the
chemical components constituting local oscillators are non-diffusive or hardly
diffusive, so that the oscillators are almost uncoupled, while there is an
extra diffusive component which introduces effective nonlocal coupling over the
oscillators. Linear stability analysis of the reduced equation about the
uniform oscillation is also carried out. This revealed that new types of
instability which can never arise in the ordinary complex Ginzburg-Landau
equation are possible, and their physical implication is briefly discussed.


title: Wavefunction Statistics using Scar States

abstract:
  We describe the statistics of chaotic wavefunctions near periodic orbits
using a basis of states which optimise the effect of scarring. These states
reflect the underlying structure of stable and unstable manifolds in phase
space and provide a natural means of characterising scarring effects in
individual wavefunctions as well as their collective statistical properties. In
particular, these states may be used to find scarring in regions of the
spectrum normally associated with antiscarring and suggest a characterisation
of templates for scarred wavefunctions which vary over the spectrum. The
results are applied to quantum maps and billiard systems.


title: Randomness Relative to Cantor Expansions

abstract:
  Imagine a sequence in which the first letter comes from a binary alphabet,
the second letter can be chosen on an alphabet with 10 elements, the third
letter can be chosen on an alphabet with 3 elements and so on. When such a
sequence can be called random? In this paper we offer a solution to the above
question using the approach to randomness proposed by Algorithmic Information
Theory.


title:
Solitary pulses and periodic waves in the parametrically driven complex
  Ginzburg-Landau equation

abstract:
  A one-dimensional model of a dispersive medium with intrinsic loss,
compensated by a parametric drive, is proposed. It is a combination of the
well-known parametrically driven nonlinear Schr\"{o}dinger (NLS) and complex
cubic Ginzburg-Landau equations, and has various physical applications (in
particular, to optical systems). For the case when the zero background is
stable, we elaborate an analytical approximation for solitary-pulse (SP)
states. The analytical results are found to be in good agreement with numerical
findings. Unlike the driven NLS equation, in the present model SPs feature a
nontrivial phase structure. Combining the analytical and numerical methods, we
identify a stability region for the SP solutions in the model's parameter
space. Generally, the increase of the diffusion and nonlinear-loss parameters,
which differ the present model from its driven-NLS counterpart, lead to
shrinkage of the stability domain. At one border of the stability region, the
SP is destabilized by the Hopf bifurcation, which converts it into a localized
breather. Subsequent period doublings make internal vibrations of the breather
chaotic. In the case when the zero background is unstable, hence SPs are
irrelevant, we construct stationary periodic solutions, for which a very
accurate analytical approximation is developed too. Stability of the periodic
waves is tested by direct simulations.


title: Rotating Spirals without Phase Singularity in Reaction-Diffusion Systems

abstract:
  Rotating spiral waves without phase singularity are found to arise in a
certain class of three-component reaction-diffusion systems of biological
relevance. It is argued that this phenomenon is universal when some chemical
components involved are diffusion-free. Some more detailed mathematical and
numerical analyses are carried out on a complex Ginzburg-Landau equation with
non-local coupling to which the original system is reduced close to a
codimension-two parameter set.


title:
Bifurcation structure of two Coupled Periodically driven double-well
  Duffing Oscillators

abstract:
  The bifurcation structure of coupled periodically driven double-well Duffing
oscillators is investigated as a function of the strength of the driving force
$f$ and its frequency $\Omega$. We first examine the stability of the steady
state in linear response, and classify the different types of bifurcation
likely to occur in this model. We then explore the complex behaviour associated
with these bifurcations numerically. Our results show many striking departures
from the behaviour of coupled driven Duffing Oscillators with single
well-potentials, as characterised by Kozlowski et al \cite{k1}. In addition to
the well known routes to chaos already encountered in a one-dimensional Duffing
oscillator, our model exhibits imbricated period-doubling of both types,
symmetry-breaking, sudden chaos and a great abundance of Hopf bifurcations,
many of which occur more than once for a given driving frequency. We explore
the chaotic behaviour of our model using two indicators, namely Lyapunov
exponents and the power spectrum. Poincar\'e cross-sections and phase portraits
are also plotted to show the manifestation of coexisting periodic and chaotic
attractors including the destruction of $T^2$ tori doubling.


title: Formation of a Standing-Light Pulse through Collision of Gap Solitons

abstract:
  Results of a systematic theoretical study of collisions between moving
solitons in a fiber grating are presented. Various outcomes of the collision
are identified, the most interesting one being merger of the solitons into a
single zero-velocity pulse, which suggests a way to create pulses of ``standing
light". The merger occurs with the solitons whose energy takes values between
0.15 and 0.35 of the limit value, while their velocity is limited by about 0.2
of the limit light velocity in the fiber. If the energy is larger, another
noteworthy outcome is acceleration of the solitons as a result of the
collision. In the case of mutual passage of the solitons, inelasticity of the
collision is quantified by the energy-loss share. Past the soliton's stability
limit, the collision results in strong deformation and subsequent destruction
of the solitons. Simulations of multiple collisions of two solitons in a
fiber-loop configuration are performed too. In this case, the maximum velocity
admitting the merger increases to about 0.4 of the limit velocity. Influence of
an attractive local defect on the collision is also studied, with a conclusion
that the defect does not alter the overall picture, although it traps a
small-amplitude pulse. Related effects in single-soliton dynamics are
considered too, such as transformation of an input sech signal into a gap
soliton (which is quantified by the share of lost energy), and the rate of
decay of a quiescent gap soliton in a finite fiber grating, due to energy
leakage through loose edges.


title: Similarity reduction of the modified Yajima-Oikawa equation

abstract:
  We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The
hierarchy is associated with a non-standard Heisenberg subalgebra in the affine
Lie algebra of type A_2^{(1)}. The system of equations for self-similar
solutions is presented as a Hamiltonian system of degree of freedom two, and
admits a group of B\"acklund transformations isomorphic to the affine Weyl
group of type A_2^{(1)}. We show that the system is equivalent to a
two-parameter family of the fifth Painlev\'e equation.


title:
The Sasa-Satsuma higher order nonlinear Schrodinger equation and its
  bilinearization and multi-soliton solutions

abstract:
  Higher order and multicomponent generalizations of the nonlinear Schrodinger
equation are important in various applications, e.g., in optics. One of these
equations, the integrable Sasa-Satsuma equation, has particularly interesting
soliton solutions. Unfortunately the construction of multi-soliton solutions to
this equation presents difficulties due to its complicated bilinearization. We
discuss briefly some previous attempts and then give the correct
bilinearization based on the interpretation of the Sasa-Satsuma equation as a
reduction of the three-component Kadomtsev-Petvishvili hierarchy. In the
process we also get bilinearizations and multi-soliton formulae for a two
component generalization of the Sasa-Satsuma equation (the
Yajima-Oikawa-Tasgal-Potasek model), and for a (2+1)-dimensional
generalization.


title: Optical vortices in dispersive nonlinear Kerr type media

abstract:
  The applied method of slowly varying amplitudes of the electrical and magnet
vector fields give us the possibility to reduce the nonlinear vector
integro-differential wave equation to the amplitude vector nonlinear
differential equations. It can be estimated different orders of dispersion of
the linear and nonlinear susceptibility using this approximation. The critical
values of parameters to observe different linear and nonlinear effects are
determinate. The obtained amplitude equation is a vector version of 3D+1
Nonlinear Schredinger Equation (VNSE) describing the evolution of slowly
varying amplitudes of electrical and magnet fields in dispersive nonlinear Kerr
type media. We show that VNSE admit exact vortex solutions with classical
orbital momentum $\ell=1$ and finite energy. Dispersion region and medium
parameters necessary for experimental observation of these vortices, are
determinate.
  PACS 42.81.Dp;05.45.Yv;42.65.Tg


title: Recurrence Time Statistics for Finite Size Intervals

abstract:
  We investigate the statistics of recurrences to finite size intervals for
chaotic dynamical systems. We find that the typical distribution presents an
exponential decay for almost all recurrence times except for a few short times
affected by a kind of memory effect. We interpret this effect as being related
to the unstable periodic orbits inside the interval. Although it is restricted
to a few short times it changes the whole distribution of recurrences. We show
that for systems with strong mixing properties the exponential decay converges
to the Poissonian statistics when the width of the interval goes to zero.
However, we alert that special attention to the size of the interval is
required in order to guarantee that the short time memory effect is negligible
when one is interested in numerically or experimentally calculated Poincar{\'e}
recurrence time statistics.


title:
Localized behavior in the Lyapunov vectors for quasi-one-dimensional
  many-hard-disk systems

abstract:
  We introduce a definition of a "localization width" whose logarithm is given
by the entropy of the distribution of particle component amplitudes in the
Lyapunov vector. Different types of localization widths are observed, for
example, a minimum localization width where the components of only two
particles are dominant. We can distinguish a delocalization associated with a
random distribution of particle contributions, a delocalization associated with
a uniform distribution and a delocalization associated with a wave-like
structure in the Lyapunov vector. Using the localization width we show that in
quasi-one-dimensional systems of many hard disks there are two kinds of
dependence of the localization width on the Lyapunov exponent index for the
larger exponents: one is exponential, and the other is linear. Differences, due
to these kinds of localizations also appear in the shapes of the localized
peaks of the Lyapunov vectors, the Lyapunov spectra and the angle between the
spatial and momentum parts of the Lyapunov vectors. We show that the Krylov
relation for the largest Lyapunov exponent $\lambda\sim-\rho\ln\rho$ as a
function of the density $\rho$ is satisfied (apart from a factor) in the same
density region as the linear dependence of the localization widths is observed.
It is also shown that there are asymmetries in the spatial and momentum parts
of the Lyapunov vectors, as well as in their $x$ and $y$-components.


title: Travelling waves in pipe flow

abstract:
  A family of three-dimensional travelling waves for flow through a pipe of
circular cross section is identified. The travelling waves are dominated by
pairs of downstream vortices and streaks. They originate in saddle-node
bifurcations at Reynolds numbers as low as 1250. All states are immediately
unstable. Their dynamical significance is that they provide a skeleton for the
formation of a chaotic saddle that can explain the intermittent transition to
turbulence and the sensitive dependence on initial conditions in this shear
flow.


title:
Negaton and Positon solutions of the soliton equation with
  self-consistent sources

abstract:
  The KdV equation with self-consistent sources (KdVES) is used as a model to
illustrate the method. A generalized binary Darboux transformation (GBDT) with
an arbitrary time-dependent function for the KdVES as well as the formula for
$N$-times repeated GBDT are presented. This GBDT provides non-auto-B\"{a}cklund
transformation between two KdV equations with different degrees of sources and
enable us to construct more general solutions with $N$ arbitrary $t$-dependent
functions. By taking the special $t$-function, we obtain multisoliton,
multipositon, multinegaton, multisoliton-positon, multinegaton-positon and
multisoliton-negaton solutions of KdVES. Some properties of these solutions are
discussed.


title: On the maximal scarring for quantum cat map eigenstates

abstract:
  We consider the quantized hyperbolic automorphisms on the 2-dimensional torus
(or generalized quantum cat maps), and study the localization properties of
their eigenstates in phase space, in the semiclassical limit. We prove that if
the semiclassical measure corresponding to a sequence of normalized eigenstates
has a pure point component (phenomenon of ``strong scarring''), then the weight
of this component cannot be larger than the weight of the Lebesgue component,
and therefore admits the sharp upper bound 1/2.


title: Decay of the classical Loschmidt echo in integrable systems

abstract:
  We study both analytically and numerically the decay of fidelity of classical
motion for integrable systems. We find that the decay can exhibit two
qualitatively different behaviors, namely an algebraic decay, that is due to
the perturbation of the shape of the tori, or a ballistic decay, that is
associated with perturbing the frequencies of the tori. The type of decay
depends on initial conditions and on the shape of the perturbation but, for
small enough perturbations, not on its size. We demonstrate numerically this
general behavior for the cases of the twist map, the rectangular billiard, and
the kicked rotor in the almost integrable regime.


title: Poisson maps and integrable deformations of Kowalevski top

abstract:
  We construct a Poisson map between manifolds with linear Poisson brackets
corresponding to the Lie algebras $e(3)$ and $so(4)$. Using this map we
establish a connection between the deformed Kowalevski top on $e(3)$ proposed
by Sokolov and the Kowalevski top on $so(4)$. The connection between these
systems leads to the separation of variables for the deformed system on $e(3)$
and yields the natural $5\times 5$ Lax pair for the Kowalevski top on $so(4)$.


title: Intermediate wave-function statistics

abstract:
  We calculate statistical properties of the eigenfunctions of two quantum
systems that exhibit intermediate spectral statistics: star graphs and Seba
billiards. First, we show that these eigenfunctions are not quantum ergodic,
and calculate the corresponding limit distribution. Second, we find that they
can be strongly scarred by short periodic orbits, and construct sequences of
states which have such a limit. Our results are illustrated by numerical
computations.


title: Reversors and Symmetries for Polynomial Automorphisms of the Plane

abstract:
  We obtain normal forms for symmetric and for reversible polynomial
automorphisms (polynomial maps that have polynomial inverses) of the plane. Our
normal forms are based on the generalized \Henon normal form of Friedland and
Milnor. We restrict to the case that the symmetries and reversors are also
polynomial automorphisms. We show that each such reversor has finite-order, and
that for nontrivial, real maps, the reversor has order 2 or 4. The normal forms
are shown to be unique up to finitely many choices. We investigate some of the
dynamical consequences of reversibility, especially for the case that the
reversor is not an involution.


title: Chaotic filtering of moving atoms in pulsed optical lattices

abstract:
  We propose a mechanism for a velocity-selective device, which exploits the
fundamental phenomenon of dynamical localization. It would allow packets of
cold atoms travelling through a pulsed optical lattice in one direction to pass
undisturbed, while dispersing atoms travelling in the opposite direction. This
mechanism is based on a chaotic diffusion process, with a momentum-dependent
diffusion coefficient. We analyse this classical effect, and we show that it is
enhanced by dynamical localization.


title: Nonlinear Amplitude Maxwell-Dirac Equations. Optical Leptons

abstract:
  We apply the method of slowly-varying amplitudes of the electrical and magnet
fields to integro-differential system of nonlinear Maxwell equations. The
equations are reduced to system of differential Nonlinear Maxwell amplitude
Equations (NME). Different orders of dispersion of the linear and nonlinear
susceptibility can be estimated. This method allow us to investigate also the
optical pulses with time duration equal or shorter than the relaxation time of
the media. The electric and magnetic fields are presented as sums of circular
and linear components. Thus, NME is written as a set of Nonlinear Dirac
Equations (NDE). Exact solutions of NDE with classical orbital momentum $\ell =
1$ and opposite directions of the spin (opposite charge) $j = \pm1 / 2$ are
obtained. The possible generalization of NME to higher number of optical
components and higher number of $\ell$ and $j$ is discussed. Two kind of Kerr
type media: with and without linear dispersion of the electric and the magnet
susceptibility are consider. The vortex solutions in case of media with
dispersion admit finite energy while the solutions in case of a media without
dispersion admit infinite energy.


title: Riemann zeros, prime numbers and fractal potentials

abstract:
  Using two distinct inversion techniques, the local one-dimensional potentials
for the Riemann zeros and prime number sequence are reconstructed. We establish
that both inversion techniques, when applied to the same set of levels, lead to
the same fractal potential. This provides numerical evidence that the potential
obtained by inversion of a set of energy levels is unique in one-dimension. We
also investigate the fractal properties of the reconstructed potentials and
estimate the fractal dimensions to be $D=1.5$ for the Riemann zeros and $D =
1.8$ for the prime numbers. This result is somewhat surprising since the
nearest-neighbour spacings of the Riemann zeros are known to be chaotically
distributed whereas the primes obey almost poisson-like statistics. Our
findings show that the fractal dimension is dependent on both the
level-statistics and spectral rigidity, $\Delta_3$, of the energy levels.


title:
Deviations from Berry--Robnik Distribution Caused by Spectral
  Accumulation

abstract:
  By extending the Berry--Robnik approach for the nearly integrable quantum
systems,\cite{[1]} we propose one possible scenario of the energy level spacing
distribution that deviates from the Berry--Robnik distribution. The result
described in this paper implies that deviations from the Berry--Robnik
distribution would arise when energy level components show strong accumulation,
and otherwise, the level spacing distribution agrees with the Berry--Robnik
distribution.


title: Control of chaotic transport in Hamiltonian systems

abstract:
  It is shown that a relevant control of Hamiltonian chaos is possible through
suitable small perturbations whose form can be explicitly computed. In
particular, it is possible to control (reduce) the chaotic diffusion in the
phase space of a Hamiltonian system with 1.5 degrees of freedom which models
the diffusion of charged test particles in a ``turbulent'' electric field
across the confining magnetic field in controlled thermonuclear fusion devices.
Though still far from practical applications, this result suggests that some
strategy to control turbulent transport in magnetized plasmas, in particular
tokamaks, is conceivable.

categories: nlin.CD math-ph math.MP physics.plasm-ph


title:
Elastica solutions in viscous fingering. The nonlinear Saffman-Taylor
  instability

abstract: This paper has been withdrawn.


title: Supporting random wave models: a quantum mechanical approach

abstract:
  We show how two-point correlation functions derived within non-isotropic
random wave models are in fact quantum results that are obtained in the
appropriate limit in terms of the exact Green function of the quantum system.
Since no statistical model is required for this derivation, this shows that
taking the wave functions as Gaussian processes is the only assumption of those
random wave models. We also show how for clean systems the two-point
correlation function defined through an energy average defines a Gaussian
theory which substantially reduces the spurious contributions coming from the
normalisation problem.


title:
Using the Magnitude-Squared Coherence for Determining Order-Chaos
  Transition in a System Governed by Logistic Equation Dynamics

abstract:
  This paper is devoted to show the results obtained by using the
magnitude-squared coherence for determining order-chaos transition in a system
described by the logistic equation dynamics. For determining the power spectral
density of a chaotic finite-duration discrete-time sequence the Welch average
periodogram method was used. This method has the advantage that can be applied
to any stationary signal by using the discrete Fourier transform (DFT)
representation of a discrete-time series which allows an effective computation
via fast Fourier transform (FFT) algorithm, and that can be applied to a
discrete-time series shorter than that required by nonlinear dynamical analysis
methods. The estimate of the Inverse Average Magnitude-Squared Coherence Index
(IAMSCI) for each discrete-time series in the set obtained from the logistic
mapping was calculated. The control parameter, {\bf r}, ranges in the interval
[2.8,4] for producing the discrete-time series set. When the condition $r \geq
3.57$ is satisfied, each discrete-time series exhibited a positive value for
IAMSCI estimate, indicating a high level of coherence loss of the signal and
corresponding to a chaotic behavior of the dynamical system. Its effectiveness
was demonstrated by comparing the results with those obtained by calculating
the largest Lyapounov exponent of the time series set obtained from the
logistic equation.


title:
Applying the Inverse Average Magnitude Squared Coherence Index for
  Determining Order-Chaos Transition in a System Governed by H\'enon Mapping
  Dynamics

abstract:
  The quantitative determination of the order-chaos transition in a nonlinear
dynamical system described by H\'enon mapping defined as
$x[n+1]=1.0-A*x[n]^2+B*y[n], y[n+1]=B*x[n]$, where $B=0.3$, and $A$ is an
adjustable control parameter, was made. This was achieved by applying the
Inverse Average Magnitude-Squared Coherence Index (IAMSCI). This method is
based on the Welch average periodogram technique and it has the advantage
respect to nonlinear dynamical methods that it may be applied to any stationary
signal by using discrete Fourier transform (DFT) representation which allows to
operate on a short discrete-time series. Its effectiveness was demonstrated by
comparing the results obtained by applying IAMSCI method with those obtained by
calculating the largest Lyapounov exponent (LLE), both applied to the same
discrete-time series set derived from the H\'enon mapping.


title: Creation of twistless circles in a model of stellar pulsations

abstract:
  In the present paper, we study the Poincare map associated to a periodic
perturbation, both in space and time, of a linear Hamiltonian system. The
dynamical system embodies the essential physics of stellar pulsations and
provides a global and qualitative explanation of the chaotic oscillations
observed in some stars. We show that this map is an area preserving one with an
oscillating rotation number function. The nonmonotonic property of the rotation
number function induced by the triplication of the elliptic fixed point is
superposed on the nonmonotonic character due to the oscillating perturbation.
This superposition leads to the co-manifestation of generic phenomena such as
reconnection and meandering, with the nongeneric scenario of creation of
vortices. The nonmonotonic property due to the triplication bifurcation is
shown to be different from that exhibited by the cubic Henon map, which can be
considered as the prototype of area preserving maps which undergo a
triplication followed by the twistless bifurcation. Our study exploits the
reversibility property of the initial system, which induces the time-reversal
symmetry of the Poincare map.


title: The spin contribution to the form factor of quantum graphs

abstract:
  Following the quantisation of a graph with the Dirac operator (spin-1/2) we
explain how additional weights in the spectral form factor K(\tau) due to spin
propagation around orbits produce higher order terms in the small-\tau
asymptotics in agreement with symplectic random matrix ensembles. We determine
conditions on the group of spin rotations sufficient to generate CSE
statistics.


title: Supersymmetric Extensions of the Harry Dym Hierarchy

abstract:
  We study the supersymmetric extensions of the Harry Dym hierarchy of
equations. We obtain the susy-B extension, the doubly susy-B extension as well
as the N=1 and the N=2 supersymmetric extensions for this system. The N=2
supersymmetric extension is particularly interesting, since it leads to new
classical integrable systems in the bosonic limit. We prove the integrability
of these systems through the bi-Hamiltonian formulation of integrable models
and through the Lax description. We also discuss the supersymmetric extension
of the Hunter-Zheng equation which belongs to the Harry Dym hierarchy of
equations.


title: Global Superdiffusion of Weak Chaos

abstract:
  A class of kicked rotors is introduced, exhibiting accelerator-mode islands
(AIs) and {\em global} superdiffusion for {\em arbitrarily weak} chaos. The
corresponding standard maps are shown to be exactly related to generalized web
maps taken modulo an ``oblique cylinder''. Then, in a case that the web-map
orbit structure is periodic in the phase plane, the AIs are essentially {\em
normal} web islands folded back into the cylinder. As a consequence, chaotic
orbits sticking around the AI boundary are accelerated {\em only} when they
traverse tiny {\em ``acceleration spots''}. This leads to chaotic flights
having a quasiregular {\em steplike} structure. The global weak-chaos
superdiffusion is thus basically different in nature from the strong-chaos one
in the usual standard and web maps.

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


title: Multi-frequency Craik-Criminale solutions of the Navier-Stokes equations

abstract:
  An exact Craik-Criminale (CC) solution to the incompressible Navier-Stokes
(NS) equations describes the instability of an elliptical columnar flow
interacting with a single Kelvin wave. These CC solutions are extended to allow
multi-harmonic Kelvin waves to interact with any exact ``base'' solution of the
NS equations. The interaction is evaluated along an arbitrarily chosen flowline
of the base solution, so exact nonlinear instability in this context is locally
convective, rather than absolute. Furthermore, an iterative method called
``WKB-bootstrapping'' is introduced which successively adds Kelvin waves with
incommensurate phases to the extended CC solutions. This is illustrated by
constructing an extended CC solution consisting of several Kelvin waves with
incommensurate phases interacting with an elliptical columnar flow.


title: Chaos and Shadowing Around a Homoclinic Tube

abstract:
  Let $F$ be a $C^3$ diffeomorphism on a Banach space $B$. $F$ has a homoclinic
tube asymptotic to an invariant manifold. Around the homoclinic tube, Bernoulli
shift dynamics of submanifolds is established through shadowing lemma. This
work removes an uncheckable condition of Silnikov [Equation (11), page 625 of
L. P. Silnikov, Soviet Math. Dokl., vol.9, no.3, (1968), 624-628]. Also, the
result of Silnikov does not imply Bernoulli shift dynamics of a single map,
rather only provides a labeling of all invariant tubes around the homoclinic
tube. The work of Silnikov was done in ${\mathbb R}^n$, and the current work is
done in a Banach space.


title: Homoclinic Tubes and Chaos in Perturbed Sine-Gordon Equation

abstract:
  In an early work, Bernoulli shift dynamics of submanifolds was established in
a neighborhood of a homoclinic tube. In this article, we will present a
concrete example: sine-Gordon equation under a quasi-periodic perturbation.


title: Experimental synchronization of spatiotemporal chaos

abstract:
  We report the first experimental evidence of a successful synchronization of
spatiotemporal chaos. The experiments were performed on two unidirectionally
coupled, nonlinear-optical systems of single-feedback type. The synchronization
was investigated for different degrees of complexity of the spontaneous
structures. In all cases, the cross-correlation between the two system states
was found to increase with the strength of the coupling. Numerical simulations
yield comparable results and throw a light on the role of spatial
inhomogeneities, which hamper a perfect synchronization.


title: Analytical analysis of ground states on 0-$\pi$ long Josephson junctions

abstract:
  We investigate analytically a long Josephson 0-$\pi$-junction with several 0
and $\pi$ facets which are comparable to the Josephson penetration length
$\lambda_J$. Such junctions can be fabricated exploiting (a) the d-wave order
parameter symmetry of cuprate superconductors; (b) the spacial oscillations of
the order parameter in superconductor-insulator-ferromagnet-superconductor
structures with different thicknesses of ferromagnetic layer to produce 0 or
$\pi$ coupling or (c) the structure of the corresponding sine-Gordon equations
and substituting the phase $\pi$-discontinuities by the artificial current
injectors. We investigate analytically the possible ground states in such a
system and show that there is a critical facet length $a_c$, which separates
the states with half-integer flux quanta (semifluxons) from the trivial ``flat
phase state'' without magnetic flux. We analyze different branches of the
bifurcation diagram, derive a system of transcendental equations which can be
effectively solved to find the crossover distance $a_c$ (bifurcation point) and
present the solutions for different number of facets and the edge facets
length. We show that the edge facets may drastically affect the state of the
system.


title:
Estimating good discrete partitions from observed data: symbolic false
  nearest neighbors

abstract:
  A symbolic analysis of observed time series data requires making a discrete
partition of a continuous state space containing observations of the dynamics.
A particular kind of partition, called ``generating'', preserves all dynamical
information of a deterministic map in the symbolic representation, but such
partitions are not obvious beyond one dimension, and existing methods to find
them require significant knowledge of the dynamical evolution operator or the
spectrum of unstable periodic orbits. We introduce a statistic and algorithm to
refine empirical partitions for symbolic state reconstruction. This method
optimizes an essential property of a generating partition: avoiding topological
degeneracies. It requires only the observed time series and is sensible even in
the presence of noise when no truly generating partition is possible. Because
of its resemblance to a geometrical statistic frequently used for
reconstructing valid time-delay embeddings, we call the algorithm ``symbolic
false nearest neighbors''.


title:
Decay of metastable patterns for the Rosensweig instability: revisiting
  the dispersion relation

abstract:
  The decay of metastable patterns in the form of magnetic liquid ridges is
studied in the frame of a linear stability analysis where the ridges are the
most unstable linear pattern of the Rosensweig instability. The analysis of the
corresponding dispersion relation reveals that different sets of solutions
exist. Two of them are associated with an oscillatory and a pure exponential
decay, respectively.


title: Generalized dynamical entropies in weakly chaotic systems

abstract:
  A large class of technically non-chaotic systems, involving scatterings of
light particles by flat surfaces with sharp boundaries, is nonetheless
characterized by complex random looking motion in phase space. For these
systems one may define a generalized, Tsallis type dynamical entropy that
increases linearly with time. It characterizes a maximal gain of information
about the system that increases as a power of time. However, this entropy
cannot be chosen independently from the choice of coarse graining lengths and
it assigns positive dynamical entropies also to fully integrable systems. By
considering these dependencies in detail one usually will be able to
distinguish weakly chaotic from fully integrable systems.


title: All Poisson Structures in $R^3$

abstract:
  Hamiltonian formulation of N=3 systems is considered in general. The most
general solution of the Jacobi equation in ${\mathbb R}^3$ is proposed.
Compatible Poisson structures and the corresponding bi-Hamiltonian N=3 systems
are also discussed.


title: Energy thresholds for discrete breathers

abstract:
  Discrete breathers are time-periodic, spatially localized solutions of the
equations of motion for a system of classical degrees of freedom interacting on
a lattice. An important issue, not only from a theoretical point of view but
also for their experimental detection, are their energy properties. We
considerably enlarge the scenario of possible energy properties presented by
Flach, Kladko, and MacKay [Phys. Rev. Lett. 78, 1207 (1997)]. Breather energies
have a positive lower bound if the lattice dimension is greater than or equal
to a certain critical value d_c. We show that d_c can generically be greater
than two for a large class of Hamiltonian systems. Furthermore, examples are
provided for systems where discrete breathers exist but do not emerge from the
bifurcation of a band edge plane wave. Some of these systems support breathers
of arbitrarily low energy in any spatial dimension.


title: Basin Bifurcations in a Two-Dimensional Logistic Map

abstract:
  Previous works have been devoted to the study of two-dimensional
noninvertible maps, obtained using a coupling between one-dimensional logistic
maps. This paper is devoted to the study of a specific one, in order to
complete previous results (Ref. 5 and 7), regarding the evolution of basins and
attractors, when considering the tool of critical manifolds.


title: Nonlinear elastodynamics and energy transport in biopolymers

abstract:
  We present an exact analysis of the intrinsic dynamics of two types of
biopolymers, described by the wormlike chain (WLC) and the wormlike rod chain
(WLRC) models, respectively. We map each model to a classical field theory for
an appropriate unit vector field defined on the space curve formed by the axis
of the biopolymer. Conformation dynamics and energy transport are shown to be
described, respectively, by the soliton-bearing Landau-Lifshitz equation and
the nonlinear Schr\"odinger equation. This enables us to identify the soliton
representing the ``conformon'' hypothesized in biopolymers.


title:
Hypergeometric tau functions $\tau({\bf t},T,{\bf t}^*)$ as
  $\infty$-soliton tau function in T variables

abstract:
  We consider KP tau function of hypergeometric type $\tau({\bf t},T,{\bf
t}^*)$, where the set ${\bf t}$ is the KP higher times and $T,{\bf t}^*$ are
sets of parameters. Fixing ${\bf t}^*$, we find that $\tau({\bf t},T,{\bf
t}^*)$ is an infinite-soliton solution of different (dual) multi-component KP
(and TL) hierarchy, where the roles of the variables ${\bf t}$ and $T$ are
interchanged. When $\tau({\bf t},T,{\bf t}^*)$ is a polynomial in ${\bf t}$, we
obtain a $N$-soliton solution of the dual hierarchy. Parameters of the solitons
are related to the Frobenius coordinates of partitions in the Schur function
development of $\tau({\bf t},T,{\bf t}^*)$.


title:
Elimination of spiral chaos by periodic force for the Aliev-Panfilov
  model

abstract:
  Spiral chaos appears in the two dimensional Aliev-Panfilov model. The
generation mechanism of the spiral chaos is related to the breathing
instability of pulse trains. The spiral chaos can be eliminated by applying
periodic force uniformly. The elimination of spiral chaos is most effective,
when the frequency of the periodic force is close to that of the breathing
motion.


title:
Self-organization of hierarchical structures in nonlocally coupled
  replicator models

abstract:
  We study a simple replicator model with non-symmetric and nonlocal
interactions. Hierarchical structures with prey-predator relations are
self-organized from a homogeneous state, induced by the dynamical instability
of nonlinear interactions.


title: Phase separation in coupled chaotic maps on fractal networks

abstract:
  The phase ordering dynamics of coupled chaotic maps on fractal networks are
investigated. The statistical properties of the systems are characterized by
means of the persistence probability of equivalent spin variables that define
the phases. The persistence saturates and phase domains freeze for all values
of the coupling parameter as a consequence of the fractal structure of the
networks, in contrast to the phase transition behavior previously observed in
regular Euclidean lattices. Several discontinuities and other features found in
the saturation persistence curve as a function of the coupling are explained in
terms of changes of stability of local phase configurations on the fractals.


title:
Asymptotic behaviour of multiple scattering on infinite number of
  parallel demi-planes

abstract:
  The exact solution for the scattering of electromagnetic waves on an infinite
number of parallel demi-planes has been obtained by J.F. Carlson and A.E. Heins
in 1947 using the Wiener-Hopf method. We analyze their solution in the
semiclassical limit of small wavelength and find the asymptotic behaviour of
the reflection and transmission coefficients. The results are compared with the
ones obtained within the Kirchhoff approximation.


title: Strange Heat Flux in (An)Harmonic Networks

abstract:
  We study the heat transport in systems of coupled oscillators driven out of
equilibrium by Gaussian heat baths. We illustrate with a few examples that such
systems can exhibit ``strange'' transport phenomena. In particular, {\em
circulation} of heat flux may appear in the steady state of a system of three
oscillators only. This indicates that the direction of the heat fluxes can in
general not be "guessed" from the temperatures of the heat baths. Although we
primarily consider harmonic couplings between the oscillators, we explain why
this strange behavior persists under weak anharmonic perturbations.


title:
Separation of variables in multi-Hamiltonian systems: an application to
  the Lagrange top

abstract:
  Starting from the tri-Hamiltonian formulation of the Lagrange top in a
six-dimensional phase space, we discuss the reduction of the vector field and
of the Poisson tensors. We show explicitly that, after the reduction on each
one of the symplectic leaves, the vector field of the Lagrange top is separable
in the sense of Hamilton-Jacobi.


title: Cycle expansions for intermittent maps

abstract:
  In a generic dynamical system chaos and regular motion coexist side by side,
in different parts of the phase space. The border between these, where
trajectories are neither unstable nor stable but of marginal stability,
manifests itself through intermittency, dynamics where long periods of nearly
regular motions are interrupted by irregular chaotic bursts. We discuss the
Perron-Frobenius operator formalism for such systems, and show by means of a
1-dimensional intermittent map that intermittency induces branch cuts in
dynamical zeta functions. Marginality leads to long-time dynamical
correlations, in contrast to the exponentially fast decorrelations of purely
chaotic dynamics. We apply the periodic orbit theory to quantitative
characterization of the associated power-law decays.


title:
Form factor for large quantum graphs: evaluating orbits with
  time-reversal

abstract:
  It has been shown that for a certain special type of quantum graphs the
random-matrix form factor can be recovered to at least third order in the
scaled time \tau using periodic-orbit theory. Two types of contributing pairs
of orbits were identified, those which require time-reversal symmetry and those
which do not. We present a new technique of dealing with contribution from the
former type of orbits.
  The technique allows us to derive the third order term of the expansion for
general graphs. Although the derivation is rather technical, the advantages of
the technique are obvious: it makes the derivation tractable, it identifies
explicitly the orbit configurations which give the correct contribution, it is
more algorithmical and more system-independent, making possible future
applications of the technique to systems other than quantum graphs.


title: Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling

abstract:
  Small lattices of $N$ nearest neighbor coupled excitable FitzHugh-Nagumo
systems, with time-delayed coupling are studied, and compared with systems of
FitzHugh-Nagumo oscillators with the same delayed coupling. Bifurcations of
equilibria in N=2 case are studied analytically, and it is then numerically
confirmed that the same bifurcations are relevant for the dynamics in the case
$N>2$. Bifurcations found include inverse and direct Hopf and fold limit cycle
bifurcations. Typical dynamics for different small time-lags and coupling
intensities could be excitable with a single globally stable equilibrium,
asymptotic oscillatory with symmetric limit cycle, bi-stable with stable
equilibrium and a symmetric limit cycle, and again coherent oscillatory but
non-symmetric and phase-shifted. For an intermediate range of time-lags inverse
sub-critical Hopf and fold limit cycle bifurcations lead to the phenomenon of
oscillator death. The phenomenon does not occur in the case of FitzHugh-Nagumo
oscillators with the same type of coupling.


title:
Chaotic Synchronization of Symbolic Information in the Discrete
  Nonlinear Schroedinger Equation

abstract:
  We have studied the discrete nonlinear Schroedinger equation (DNLSE) with
on-site defects and periodic boundary conditions. When the array dynamics
becomes chaotic, the otherwise quasiperiodic amplitude correlations between the
oscillators are destroyed. However, we show that synchronization of symbolic
information of suitable amplitude signals is possible in this hamiltonian
system.


title:
A spatiotemporal-chaos-based cryptosystem taking advantages of both
  synchronous and self-synchronizing schemes

abstract:
  Two-dimensional one-way coupled map lattices are used for cryptograph where
multiple space units produce chaotic outputs in parallel. One of the outputs
plays the role of driving for synchronization of the decryption system while
the others perform the function of information encoding. With this separation
of functions the receiver can establish a self-checking and self-correction
mechanism, and enjoys the advantages of both synchronous and self-synchronizing
schemes. A comparison between the present system with the system of Advanced
Encryption Standard, AES, is presented in the aspect of channel noise
influence. Numerical investigations show that our system is much stronger than
AES against channel noise perturbations, and thus can be better used for secure
communications with large noise channel exists in open channels, e.g.,
mobile-phone secure communications.


title:
Reduction of spurious velocity in finite difference lattice Boltzmann
  models for liquid - vapor systems

abstract:
  The origin of the spurious interface velocity in finite difference lattice
Boltzmann models for liquid - vapor systems is related to the first order
upwind scheme used to compute the space derivatives in the evolution equations.
A correction force term is introduced to eliminate the spurious velocity. The
correction term helps to recover sharp interfaces and sets the phase diagram
close to the one derived using the Maxwell construction.


title: Velocity and velocity-difference distributions in Burgers turbulence

abstract:
  We consider the one-dimensional Burgers equation randomly stirred at large
scales by a Gaussian short-time correlated force. Using the method of
dissipative anomalies, we obtain velocity and velocity-difference probability
density functions and confirm the results with high-resolution numerical
simulations.

categories: nlin.CD astro-ph cond-mat hep-th physics.flu-dyn


title: Error Function Attack of chaos synchronization based encryption schemes

abstract:
  Different chaos synchronization based encryption schemes are reviewed and
compared from the practical point of view. As an efficient cryptanalysis tool
for chaos encryption, a proposal based on the Error Function Attack is
presented systematically and used to evaluate system security. We define a
quantitative measure (Quality Factor) of the effective applicability of a chaos
encryption scheme, which takes into account the security, the encryption speed,
and the robustness against channel noise. A comparison is made of several
encryption schemes and it is found that a scheme based on one-way coupled
chaotic map lattices performs outstandingly well, as judged from Quality
Factor.


title: Strange nonchaotic attractors in noise driven systems

abstract:
  Strange nonchaotic attractors (SNAs) in noise driven systems are
investigated. Before the transition to chaos, due to the effect of noise, a
typical trajectory will wander between the periodic attractor and its nearby
chaotic saddle in an intermittent way, forms a strange attractor gradually. The
existence of SNAs is confirmed by simulation results of various critera both in
map and continuous systems. Dimension transition is found and intermittent
behavior is studied by peoperties of local Lyapunov exponent. The universality
and generalization of this kind of SNAs are discussed and common features are
concluded.


title: Clustering of inertial particles in a turbulent flow

abstract:
  We analyzed formation of small-scale inhomogeneities of particle spatial
distribution (particle clustering) in a turbulent flow. The particle clustering
is a consequence of a spontaneous breakdown of their homogeneous space
distribution, and is caused by a combined effect of the particle inertia and a
finite correlation time of the turbulent velocity field. Theory of the particle
clustering is extended to the case when the particle Stokes time is larger than
the Kolmogorov time scale, but is much smaller than the correlation time at the
integral scale of turbulence. The criterion of the clustering instability is
obtained. Applications of the analyzed effects to the dynamics of inertial
particles in industrial turbulent flows are discussed.


title:
Theory of Concentration Dependence in Drag Reduction by Polymers and of
  the MDR asymptote

abstract:
  A simple model of the effect of polymer concentration on the amount of drag
reduction in turbulence is presented, simulated and analyzed. The qualitative
phase diagram of drag coefficient vs. Reynolds number (Re) is recaptured in
this model, including the theoretically elusive onset of drag reduction and the
Maximum Drag Reduction (MDR) asymptote. The Re-dependent drag and the MDR are
analytically explained, and the dependence of the amount of drag on material
parameters is rationalized.


title: Multi-Zone Shell Model for Turbulent Wall Bounded Flows

abstract:
  We suggested a \emph{Multi-Zone Shell} (MZS) model for wall-bounded flows
accounting for the space inhomogeneity in a "piecewise approximation", in which
cross-section area of the flow, $S$, is subdivided into "$j$-zones". The area
of the first zone, responsible for the core of the flow, $S_1\simeq S/2$, and
areas of the next $j$-zones, $S_j$, decrease towards the wall like $S_j\propto
2^{-j}$. In each $j$-zone the statistics of turbulence is assumed to be space
homogeneous and is described by the set of "shell velocities" $u_{nj}(t)$ for
turbulent fluctuations of the scale $\propto 2^{-n}$. The MZS-model includes a
new set of complex variables, $V_j(t)$, $j=1,2,... \infty$, describing the
amplitudes of the near wall coherent structures of the scale $s_j\sim 2^{-j}$
and responsible for the mean velocity profile. Suggested MZS-equations of
motion for $u_{nj}(t)$ and $V_j(t)$ preserve the actual conservations laws
(energy, mechanical and angular momenta), respect the existing symmetries
(including Galilean and scale invariance) and account for the type of the
non-linearity in the Navier-Stokes equation, dimensional reasoning, etc. The
MZS-model qualitatively describes important characteristics of the wall bounded
turbulence, e.g., evolution of the mean velocity profile with increasing
Reynolds number, $\RE$, from the laminar profile towards the universal
logarithmic profile near the flat-plane boundary layer as $\RE\to \infty$.


title: Nodal domains on quantum graphs

abstract:
  We consider the real eigenfunctions of the Schr\"odinger operator on graphs,
and count their nodal domains. The number of nodal domains fluctuates within an
interval whose size equals the number of bonds $B$. For well connected graphs,
with incommensurate bond lengths, the distribution of the number of nodal
domains in the interval mentioned above approaches a Gaussian distribution in
the limit when the number of vertices is large. The approach to this limit is
not simple, and we discuss it in detail. At the same time we define a random
wave model for graphs, and compare the predictions of this model with analytic
and numerical computations.


title:
Dynamical Instabilities of Quasi-static Crack Propagation Under Thermal
  Stress

abstract:
  We address the theory of quasi-static crack propagation in a strip of glass
that is pulled from a hot oven towards a cold bath. This problem had been
carefully studied in a number of experiments that offer a wealth of data to
challenge the theory. We improve upon previous theoretical treatments in a
number of ways. First, we offer a technical improvement of the discussion of
the instability towards the creation of a straight crack. This improvement
consists of employing Pad\'e approximants to solve the relevant Weiner-Hopf
factorization problem that is associated with this transition. Next we improve
the discussion of the onset of oscillatory instability towards an undulating
crack. We offer a novel way of considering the problem as a sum of solutions of
a finite strip without a crack and an infinite medium with a crack. This allows
us to present a closed form solution of the stress intensity factors in the
vicinity of the oscillatory instability. Most importantly we develop a {\em
dynamical} description of the actual trajectory in the regime of oscillatory
crack. This theory is based on the dynamical law for crack propagation proposed
by Hodgdon and Sethna. We show that this dynamical law results in a solution of
the actual track trajectory in post critical conditions; we can compute from
first principles the critical value of the control parameters, and the
characteristics of the solution like the wavelength of the oscillations. We
present detailed comparison with experimental measurements without any free
parameter. The comparison appears quite excellent. Lastly we show that the
dynamical law can be translated to an equation for the amplitude of the
oscillatory crack; this equation predicts correctly the scaling exponents
observed in experiments.


title: Nasca Lines: A Mystery wrapped in an Enigma

abstract:
  We analyze the geometrical structure of the astonishing Nasca geoglyphs in
terms of their fractal dimension with the idea of dating these manifestations
of human cultural engagements in relation to one another. Our findings suggest
that the first delineated images consist of straight, parallel lines and that
having sophisticated their abilities, Nasca artist moved on to the design of
more complex structures.


title: On classification of non-abelian Toda systems

abstract:
  A simple procedure to enumerate all Toda systems associated with complex
classical Lie groups is given.


title: Turbulence with Pressure: Anomalous Scaling of a Passive Vector Field

abstract:
  The field-theoretic renormalization group (RG) and the operator product
expansion are applied to the model of a divergence-free vector quantity,
passively advected by the ``synthetic'' turbulent flow with a finite
correlation time. The vector field is described by the stochastic
advection-diffusion equation with the most general form of the inertial
nonlinearity. The statistics of the advecting velocity field is Gaussian with
powerlike energy spectrum and dispersion law. The inertial-range behavior of
the model is described by seven regimes that correspond to nontrivial fixed
points of the RG equations and exhibit anomalous scaling. The corresponding
anomalous exponents are associated with the critical dimensions of tensor
composite operators built solely of the passive vector field, which allows to
construct a regular perturbation expansion. The actual calculation is performed
in one-loop approximation, including the anisotropic sectors. Universality of
the exponents, their (in)dependence on the forcing, effects of the large-scale
anisotropy, compressibility and pressure are discussed. In particular, for all
the scaling regimes the exponents obey a hierarchy related to the degree of
anisotropy: the more anisotropic is the contribution of a composite operator to
a correlation function, the faster it decays in the inertial-range. Relevance
of the results for the real developed turbulence described by the stochastic
Navier-Stokes equation is discussed.


title: On the Stability of Knots in Excitable Media

abstract:
  Through extensive numerical simulations we investigate the evolution of
knotted and linked vortices in the FitzHugh-Nagumo model. On medium time
scales, of the order of a hundred times the vortex rotation period, knots
simultaneously translate and precess with very little change of shape. However,
on long time scales we find that knots evolve in a more complicated manner,
with particular arcs expanding and contracting, producing substantial
variations in the total length. The topology of a knot is preserved during the
evolution and after several thousand vortex rotation periods the knot appears
to approach an asymptotic state. Furthermore, this asymptotic state is
dependent upon the initial conditions and suggests that, even within a given
topology, a host of meta-stable configurations exist, rather than a unique
stable solution. We discuss a possible mechanism for the observed evolution,
associated with the impact of higher frequency wavefronts emanating from parts
of the knot which are more twisted than the expanding arcs.


title:
(Non)local Hamiltonian and symplectic structures, recursions, and
  hierarchies: a new approach and applications to the N=1 supersymmetric KdV
  equation

abstract:
  Using methods of math.DG/0304245 and [I.S.Krasil'shchik and P.H.M.Kersten,
Symmetries and recursion operators for classical and supersymmetric
differential equations, Kluwer, 2000], we accomplish an extensive study of the
N=1 supersymmetric Korteweg-de Vries equation. The results include: a
description of local and nonlocal Hamiltonian and symplectic structures, five
hierarchies of symmetries, the corresponding hierarchies of conservation laws,
recursion operators for symmetries and generating functions of conservation
laws. We stress that the main point of the paper is not just the results on
super-KdV equation itself, but merely exposition of the efficiency of the
geometrical approach and of the computational algorithms based on it.


title:
Complex Dynamics of Piecewise Linear Model of the Van der Pol Oscillator
  Under External Periodic Force

abstract:
  The electronic model of Van der Pol oscillator with piecewise linear
V-I--characteristics of nonlinear element is proposed. It is carried out the
experimental investigations of behaviour of the Van der Pol oscillator under
external periodic force. The parameter plane "the amplitude of external force
-- the frequency of external force" for the dynamical regimes of the oscillator
is experimentally plotted. It is shown in experiments and numerical simulations
that transition to chaos in the oscillator takes place through the period
doubling cascade. It is revealed that the internal structure of the
synchronization tongues has the characteristic view as for the crossroad area.


title:
Chaos assisted instanton tunneling in one dimensional perturbed periodic
  potential

abstract:
  For the system with one-dimensional spatially periodic potential we
demonstrate that small periodic in time perturbation results in appearance of
chaotic instanton solutions. We estimate parameter of local instability, width
of stochastic layer and correlator for perturbed instanton solutions.
Application of the instanton technique enables to calculate the amplitude of
the tunneling, the form of the spectrum and the lower bound for width of the
ground quasienergy zone.


title: Coherent Tunneling and Instantons in Presence of Classical Chaos

abstract:
  We consider the instanton approach to the problem of chaos assisted tunneling
in the context of existing analytical and numerical results obtained in this
field. We provide the estimation for the range of validity of this method and
briefly discuss possible applications of chaos assisted tunneling for quantum
computations.


title:
Stabilization of solitons of the multidimensional nonlinear Schrodinger
  equation: Matter-wave breathers

abstract:
  We demonstrate that stabilization of solitons of the multidimensional
Schrodinger equation with a cubic nonlinearity may be achieved by a suitable
periodic control of the nonlinear term. The effect of this control is to
stabilize the unstable solitary waves which belong to the frontier between
expanding and collapsing solutions and to provide an oscillating solitonic
structure, some sort of breather-type solution. We obtain precise conditions on
the control parameters to achieve the stabilization and compare our results
with accurate numerical simulations of the nonlinear Schrodinger equation.
Because of the application of these ideas to matter waves these solutions are
some sort of matter breathers.


title:
Theta function parameterization and fusion for 3-D integrable Boltzmann
  weights

abstract:
  We report progress in constructing Boltzmann weights for integrable
3-dimensional lattice spin models. We show that a large class of vertex
solutions to the modified tetrahedron equation can be conveniently
parameterized in terms of N-th roots of theta-functions on the Jacobian of a
compact algebraic curve. Fay's identity guarantees the Fermat relations and the
classical equations of motion for the parameters determining the Boltzmann
weights. Our parameterization allows to write a simple formula for fused
Boltzmann weights R which describe the partition function of an arbitrary open
box and which also obey the modified tetrahedron equation. Imposing periodic
boundary conditions we observe that the R satisfy the normal tetrahedron
equation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov model
and the Chessboard model as special cases.


title: Classical Coulomb three-body problem in collinear eZe configuration

abstract:
  Classical dynamics of two-electron atom and ions H$^{-}$, He, Li$^{+}$,
Be$^{2+}$,... in collinear eZe configuration is investigated. It is revealed
that the mass ratio $\xi$ between necleus and electron plays an important role
for dynamical behaviour of these systems. With the aid of analytical tool and
numeircal computation, it is shown that thanks to large mass ratio $\xi$,
classical dynamics of these systems is fully chaotic, probably hyperbolic.
Experimental manifestation of this finding is also proposed.


title: (Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation

abstract:
  The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian
system with two degrees of freedom have frequency ratio 1:1 (saddle-centre) and
1:2 (period-doubling). The twist, which is the derivative of the rotation
number with respect to the action, is studied near these bifurcations. When the
twist vanishes the nondegeneracy condition of the (isoenergetic) KAM theorem is
not satisfied, with interesting consequences for the dynamics. We show that
near the saddle-centre bifurcation the twist always vanishes. At this
bifurcation a ``twistless'' torus is created, when the resonance is passed. The
twistless torus replaces the colliding periodic orbits in phase space. We
explicitly derive the position of the twistless torus depending on the
resonance parameter, and show that the shape of this curve is universal. For
the period doubling bifurcation the situation is different. Here we show that
the twist does not vanish in a neighborhood of the bifurcation.


title: Weyl formulas for annular ray-splitting billiards

abstract:
  We consider the distribution of eigenvalues for the wave equation in annular
(electromagnetic or acoustic) ray-splitting billiards. These systems are
interesting in that the derivation of the associated smoothed spectral counting
function can be considered as a canonical problem. This is achieved by
extending a formalism developed by Berry and Howls for ordinary (without
ray-splitting) billiards. Our results are confirmed by numerical computations
and permit us to infer a set of rules useful in order to obtain Weyl formulas
for more general ray-splitting billiards.


title: A Toy Model of Flying Snake's Glide

abstract:
  We have developed a toy model of flying snake's glide [J.J. Socha, Nature
vol. 418 (2002) 603.] by modifying a model for a falling paper. We have found
that asymmetric oscillation is a key about why snake can glide. Further
investigation for snake's glide will provide us details about how it can glide
without a wing.


title:
Verifying relationship between Height and Spacing, in Barchan Dunes
  simulated by the Coupled Map Lattice Model

abstract:
  We have investigated the relationship between height and spacing of Barchan
dunes which the coupled map lattice model numerically generates. There is a
scaling relation between them and the values of the scaling exponents agree
well with real dunes' values. The values of these scaling exponents are same
for both steady states and transient states.


title: Vanishing Twist in the Hamiltonian Hopf Bifurcation

abstract:
  The Hamiltonian Hopf bifurcation has an integrable normal form that describes
the passage of the eigenvalues of an equilibrium through the 1: -1 resonance.
At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium
turn into a complex quadruplet of eigenvalues and the equilibrium becomes a
linearly unstable focus-focus point. We explicitly calculate the frequency map
of the integrable normal form, in particular we obtain the rotation number as a
function on the image of the energy-momentum map in the case where the fibres
are compact. We prove that the isoenergetic non-degeneracy condition of the KAM
theorem is violated on a curve passing through the focus-focus point in the
image of the energy-momentum map. This is equivalent to the vanishing of twist
in a Poincar\'e map for each energy near that of the focus-focus point. In
addition we show that in a family of periodic orbits (the non-linear normal
modes) the twist also vanishes. These results imply the existence of all the
unusual dynamical phenomena associated to non-twist maps near the Hamiltonian
Hopf bifurcation.


title: A frequency measure robust to linear filtering

abstract:
  A definition of frequency (cycles per unit-time) based on an approximate
reconstruction of the phase-space trajectory of an oscillator from a signal is
introduced. It is shown to be invariant under linear filtering, and therefore
inaccessible by spectral methods. The effect of filtering on frequency in cases
where this definition does not perfectly apply is quantified.


title:
Data driven optimal filtering for phase and frequency of noisy
  oscillations: application to vortex flowmetering

abstract:
  A new method for extracting the phase of oscillations from noisy time series
is proposed. To obtain the phase, the signal is filtered in such a way that the
filter output has minimal relative variation in the amplitude (MIRVA) over all
filters with complex-valued impulse response. The argument of the filter output
yields the phase. Implementation of the algorithm and interpretation of the
result are discussed. We argue that the phase obtained by the proposed method
has a low susceptibility to measurement noise and a low rate of artificial
phase slips. The method is applied for the detection and classification of mode
locking in vortex flowmeters. A novel measure for the strength of mode locking
is proposed.


title: Turbulence and Tsallis Statistics

abstract:
  Fully-developed incompressible Navier-Stokes turbulence in three dimensions
is a dissipative dynamical system that exhibits strong departure from absolute
equilibrium. Nevertheless, several kinds of representation by Tsallis
equilibria have been proposed. The symmetry of the contributions to the kinetic
energy from the degrees of freedom of the flow must be broken in order to
construct turbulence applications. Tsallis representations of turbulence
involve the extrapolation of results for systems with a single degree of
freedom, a procedure that requires critical examination. Theories of Tsallis
statistics for acceleration of fluid particles are compared with computer
simulations of the pressure gradient field in the present paper. Applications
of Tsallis formulations to statistics of longitudinal velocity differences are
analyzed. No flavor of equilibrium statistics based on the kinetic energy can
encompass the dynamically fundamental turbulent energy cascade, which requires
correlation of triads of degrees of freedom at different scales.


title: The Generalized Harry Dym Equation

abstract:
  The Harry Dym equation is generalized to the system of equations in the
manner as the Korteweg - de Vries equation is generalized to the Hirota -
Satsuma equation . The Lax and Hamiltonian formulation for this generalization
is given.This generalized Lax operator gives the hierarchy of equations also.


title: An analysis of chaos via contact transformation

abstract:
  Transition from chaotic to quasi-periodic phase in modified Lorenz model is
analyzed by performing the contact transformation such that the trajectory in
${\Vec R}^3$ is projected on ${\Vec R}^2$. The relative torsion number and the
characteristics of the template are measured using the eigenvector of the
Jacobian instead of vectors on moving frame along the closed trajectory.
  Application to the circulation of a fluid in a convection loop and
oscillation of the electric field in single-mode laser system are performed.
The time series of the eigenvalues of the Jacobian and the scatter plot of the
trajectory in the transformed coordinate plane $X-Z$ in the former and
$|X|-|Z|$ in the latter, allow to visualize characteristic pattern change at
the transition from quasi-periodic to chaotic. In the case of single mode
laser, we observe the correlation between the critical movement of the
eigenvalues of the Jacobian in the complex plane and intermittency.


title: Diffusion of intrinsic localised modes by attractor hopping

abstract:
  Propagating intrinsic localised modes (discrete breathers) exist in the
dissipative driven sine-Gordon chain as attractors of the dynamics. Thermal
fluctuations induce random transitions between attracting configurations
corresponding to opposite signs of the propagation velocity, effecting a
diffusive motion of the breather. Results for the mean value and the variance
of the position from numerical simulations (Heun method) are compared with
predictions from a two-parameter model developed in this paper.


title: On integrability of (2+1)-dimensional quasilinear systems

abstract:
  A (2+1)-dimensional quasilinear system is said to be `integrable' if it can
be decoupled in infinitely many ways into a pair of compatible n-component
one-dimensional systems in Riemann invariants. Exact solutions described by
these reductions, known as nonlinear interactions of planar simple waves, can
be viewed as natural dispersionless analogs of n-gap solutions. It is
demonstrated that the requirement of the existence of 'sufficiently many'
n-component reductions provides the effective classification criterion. As an
example of this approach we classify integrable (2+1)-dimensional systems of
conservation laws possessing a convex quadratic entropy.


title: Response maxima in modulated turbulence

abstract:
  Isotropic and homogeneous turbulence driven by an energy input modulated in
time is studied within a variable range mean-field theory. The response of the
system, observed in the second order moment of the large-scale velocity
difference D(L,t)=<<(\u(x+L)-\u(x))^2>>~Re(t)^2$, is calculated for varying
modulation frequencies w and weak modulation amplitudes. For low frequencies
the system follows the modulation of the driving with almost constant
amplitude, whereas for higher driving frequencies the amplitude of the response
decreases on average 1/w. In addition, at certain frequencies the amplitude of
the response either almost vanishes or is strongly enhanced. These frequencies
are connected with the frequency scale of the energy cascade and multiples
thereof.


title: Response maxima in modulated turbulence Part II: Numerical simulations

abstract:
  Numerical simulations of fully developed turbulence driven by a modulated
energy input rate or driving force are performed within two dynamical cascade
models, the GOY shell model and a reduced wave vector set approximation of the
Navier-Stokes equation (REWA). The frequency behavior of the system response is
studied and compared with predictions from a variable range mean-field theory,
which excludes turbulent fluctuations. In agreement with the mean-field
approach we find a constant response amplitude for low driving frequencies and
a 1/w-decay of the amplitude for high frequencies. In the mean-field theory,
the finite cascade time scale had lead to an oscillating behavior of the
response amplitude as a function of the driving frequency. In the simulations
of both models we observe the main maximum. The higher maxima and minima are
completely washed out by fluctuations.


title:
Umbral Calculus, Difference Equations and the Discrete Schroedinger
  Equation

abstract:
  We discuss umbral calculus as a method of systematically discretizing linear
differential equations while preserving their point symmetries as well as
generalized symmetries. The method is then applied to the Schr\"{o}dinger
equation in order to obtain a realization of nonrelativistic quantum mechanics
in discrete space-time. In this approach a quantum system on a lattice has a
symmetry algebra isomorphic to that of the continuous case. Moreover, systems
that are integrable, superintegrable or exactly solvable preserve these
properties in the discrete case.


title: An invariant classification of cubic integrals of motion

abstract:
  We employ an isometry group invariants approach to study Killing tensors of
valence three defined in the Euclidean plane. The corresponding invariants are
found to be homogeneous polynomials of the parameters of the vector space of
the Killing tensors. The invariants are used to classify the non-trivial first
integrals of motion which are cubic in the momenta of Hamiltonian systems
defined in the Euclidean plane. The integrable cases isolated by Holt and
Fokas-Lagerstrom are investigated from this viewpoint.


title:
Algebraic Bethe ansatz method for the exact calculation of energy
  spectra and form factors: applications to models of Bose-Einstein condensates
  and metallic nanograins

abstract:
  In this review we demonstrate how the algebraic Bethe ansatz is used for the
calculation of the energy spectra and form factors (operator matrix elements in
the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As
examples we apply the theory to several models of current interest in the study
of Bose-Einstein condensates, which have been successfully created using
ultracold dilute atomic gases. The first model we introduce describes Josephson
tunneling between two coupled Bose-Einstein condensates. It can be used not
only for the study of tunneling between condensates of atomic gases, but for
solid state Josephson junctions and coupled Cooper pair boxes. The theory is
also applicable to models of atomic-molecular Bose-Einstein condensates, with
two examples given and analysed. Additionally, these same two models are
relevant to studies in quantum optics. Finally, we discuss the model of
Bardeen, Cooper and Schrieffer in this framework, which is appropriate for
systems of ultracold fermionic atomic gases, as well as being applicable for
the description of superconducting correlations in metallic grains with
nanoscale dimensions. In applying all of the above models to physical
situations, the need for an exact analysis of small scale systems is
established due to large quantum fluctuations which render mean-field
approaches inaccurate.


title: Exact solvability in contemporary physics

abstract:
  We review the theory for exactly solving quantum Hamiltonian systems through
the algebraic Bethe ansatz. We also demonstrate how this theory applies to
current studies in Bose-Einstein condensation and metallic grains which are of
nanoscale size.


title: Probabilistic cellular automata with conserved quantities

abstract:
  We demonstrate that the concept of a conservation law can be naturally
extended from deterministic to probabilistic cellular automata (PCA) rules. The
local function for conservative PCA must satisfy conditions analogous to
conservation conditions for deterministic cellular automata. Conservation
condition for PCA can also be written in the form of a current conservation
law. For deterministic nearest-neighbour CA the current can be computed
exactly. Local structure approximation can partially predict the equilibrium
current for non-deterministic cases. For linear segments of the fundamental
diagram it actually produces exact results.


title: Periodic Phase Synchronization in coupled chaotic oscillators

abstract:
  We investigate the characteristics of temporal phase locking states observed
in the route to phase synchronization. It is found that before phase
synchronization there is a periodic phase synchronization state characterized
by periodic appearance of temporal phase locking state and that the state leads
to local negativeness in one of the vanishing Lyapunov exponents. By taking a
statistical measure, we present the evidences of the phenomenon in
unidirectionally and mutually coupled chaotic oscillators, respectively. And it
is qualitatively discussed that the phenomenon is described by a nonuniform
oscillator model in the presence of noise.


title:
Chaos Synchronization of delayed systems in the presence of delay time
  modulation

abstract:
  We investigate synchronization in the presence of delay time modulation for
application to communication. We have observed that the robust synchronization
is established by a common delay signal and its threshold is presented using
Lyapunov exponents analysis. The influence of the delay time modulation in
chaotic oscillators is also discussed.