title: Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions

abstract:
  For finite-dimensional maps and periodic systems, Palmer rigorously proved
Smale horseshoe theorem using shadowing lemma in 1988. For infinite-dimensional
maps and periodic systems, such a proof was completed by Steinlein and Walther
in 1990, and Henry in 1994. For finite-dimensional autonomous systems, such a
proof was accomplished by Palmer in 1996. For infinite-dimensional autonomous
systems, the current article offers such a proof. First we prove an Inclination
Lemma to set up a coordinate system around a pseudo-orbit. Then we utilize
graph transform and the concept of persistence of invariant manifold, to prove
the existence of a shadowing orbit.


title: Group theory approach to combination tones

abstract:
  Traditional approaches to combination tones based on Helmholtz theory
encounter essential interpreting difficulties, which the most known example is
the anomalous behaviour of the combination tone 2f1-f2. Without doubt the
phenomenon of combination tones is connected with nonlinear phenomena in the
human ear. Therefore, it is generally assumed that the explanation of this
phenomenon is possible on the level of the ear. This paper proposes different
approach, which assumes the existence of an additional mechanism beyond the
ear. The aim of this mechanism is to diminish sensitivity of the auditory
process to nonlinear distortions by appropriate decoding of sound information.
The general rule of construction and the mathematical model of such a mechanism
is proposed, which allows for the explanation of many problems associated with
combination tones. As a consequence an unexpected result is obtained with
regard to a combination tone 2f1-f2 that it can originate already from the
nonlinear distortion of the second order. This mathematical model is formulated
in terms of finite groups.


title: Moving discrete breathers in a Klein--Gordon chain with an impurity

abstract:
  We analyze the influence of an impurity in the movement of discrete breathers
in Klein--Gordon chains. We observe that the moving breather can cross the
impurity, can be reflected by it, or can be trapped originating a
quasi-periodic breather. We find that resonance with a nonlinear localised mode
centred in the impurity is a necessary condition in order to observe the
trapping phenomenon, as a difference with the resonance condition with a linear
localised mode when this problem is studied within the Nonlinear
Schr\"{o}dinger Equation approximation.


title: The autocorrelation function for spectral determinants of quantum graphs

abstract:
  The paper considers the spectral determinant of quantum graph families with
chaotic classical limit and no symmetries. The secular coefficients of the
spectral determinant are found to follow distributions with zero mean and
variance approaching a constant in the limit of large network size. This
constant is in general different from the random matrix result and depends on
the classical limit. A closed expression for this system dependent constant is
given here explicitly in terms of the spectrum of an underlying Markov process.


title:
Linearisable third order ordinary differential equations and generalised
  Sundman transformations

abstract:
  We calculate in detail the conditions which allow the most general third
order ordinary differential equation to be linearised in X'''(T)=0 under the
transformation X(T)=F(x,t), dT=G(x,t)dt. Further generalisations are
considered.


title:
The extended Weyl group $\widetilde{W}(D_5^{(1)})$ as an extension of
  KNY's birational representation of $\widetilde{W}(A_1^{(1)}\times A_3^{(1)})$

abstract:
  We study the birational representation of $\wt{W}(A_1^{(1)}\times A_3^{(1)})$
proposed by Kajiwara-Noumi-Yamada (KNY) in the case of $m=2$ and $n=4$. It is
shown that the equation can be lifted to an automorphism of a family of
$A_3^{(1)}$ surfaces and therefore the group of Cremona isometries is
$\wt{W}(D_5^{(1)})$ ($\supset \wt{W}(A_1^{(1)}\times A_3^{(1)})$). The equation
can be decomposed into two mappings which are conjugate to the $q$-$P_{VI}$
equation. It is also shown that the subgroup of Cremona isometries which
commute with the original translation is isomorphic to $\mz \times
\wt{W}(A_3^{(1)}) \times \wt{W}(A_1^{(1)})$.


title: Baxter Q-operators for integrable DST chain

abstract:
  Following the procedure, described in the paper nlin.SI/0003002, for the
integrable DST chain we construct Baxter Q-operators as the traces of monodromy
of some M-operators, that act in quantum and auxiliary spaces. Within this
procedure we obtain two basic M-operators and derive some functional relations
between them such as intertwining relations and wronskian-type relations
between two basic Q-operators.


title:
The Lyapunov stability of first order dynamic equations with respect to
  time-dependent Riemannian metrics. An example

abstract:
  A simple example that I have been requested illustrates the statement in
E-print nlin.CD/0201060 that solutions of a smooth first order dynamic equation
can be made Lyapunov stable at will by the choice of an appropriate
time-dependent Riemannian metric.


title:
Phase-locked and phase drift solutions of phase oscillators with
  asymmetric coupling strengths

abstract:
  Phase-locked solutions of coupled oscillators are studied with asymmetric
coupling strengths or inhomogeneous natural frequencies. The solutions show
remarkable profiles of phase lags from the pacemaker corresponding to the ratio
of upward and downward coupling strengths. By means of the existence condition
of phase-locked solutions, the transition points from phase-locked to phase
drift states are estimated. The application of the existence condition to the
case of the linear gradient of natural frequency illustrates some scaling
properties in the frequency diagrams.


title: Semiclassical quantization with bifurcating orbits

abstract:
  Bifurcations of classical orbits introduce divergences into semiclassical
spectra which have to be smoothed with the help of uniform approximations. We
develop a technique to extract individual energy levels from semiclassical
spectra involving uniform approximations. As a prototype example, the method is
shown to yield excellent results for photo-absorption spectra for the hydrogen
atom in an electric field in a spectral range where the abundance of
bifurcations would render the standard closed-orbit formula without uniform
approximations useless. Our method immediately applies to semiclassical trace
formulae as well as closed-orbit theory and offers a general technique for the
semiclassical quantization of arbitrary systems.


title:
Emergence of local and global spatio-temporal order on the way to
  equipartition: A nonlinear field model

abstract:
  This paper has been withdrawn. A much-improved version can be found at
hep-ph/0209176.


title: Generalised Wigner surmise for 2x2 random matrices

abstract:
  We present new analytical results concerning the spectral distributions for
2x2 random real symmetric matrices which generalise the Wigner surmise.


title:
A strange recursion operator for a new integrable system of coupled
  Korteweg - de Vries equations

abstract:
  A recursion operator is constructed for a new integrable system of coupled
Korteweg - de Vries equations by the method of gauge-invariant description of
zero-curvature representations. This second-order recursion operator is
characterized by unusual structure of its nonlocal part.


title:
Adaptive Time-delay Hyperchaos Synchronization in Laser Diodes Subject
  to Optical Feedback

abstract:
  In this paper a proposal is made of an adaptive coupling function for
achieving synchronization between two lasers subject to optical feedback.Such a
control scheme requires knowledge of the systems' parameters. We demonstate
that when these parameters are not available on-line parameter estimation can
be applied.


title:
Symbiotic Sympatric Speciation: Compliance with Interaction-driven
  Phenotype Differentiation from a Single Genotype

abstract:
  A mechanism of sympatric speciation is presented based on the
interaction-induced developmental plasticity of phenotypes. First, phenotypes
of individuals with identical genotypes split into a few groups, according to
instability in the developmental dynamics that are triggered with the
competitive interaction among individuals. Then, through mutational change of
genes, the phenotypic differences are fixed to genes, until the groups are
completely separated in genes as well as phenotypes. It is also demonstrated
that the proposed theory leads to hybrid sterility under sexual recombination,
and thus speciation is completed in the sense of reproductive isolation. As a
result of this post-mating isolation, the mating preference evolves later. When
there are two alleles, the correlation between alleles is formed, to
consolidate the speciation. When individuals are located in space, different
species are later segregated spatially, implying that the speciation so far
regarded to be allopatric may be a result of sympatric speciation.
Relationships with previous theories, frequency-dependent selection,
reinforcement, Baldwin's effect, phenotypic plasticity, and resource
competition are briefly discussed. Relevance of the results to natural
evolution are discussed, including punctuated equilibrium, incomplete
penetrance in mutants, and the change in flexibility in genotype-phenotype
correspondence. Finally, it is discussed how our theory is confirmed both in
field and in laboratory (in an experiment with the use of E coli.).

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


title:
Kinetic Origin of Heredity in a Replicating System with a Catalytic
  Network

abstract:
  The origin of heredity is studied as a recursive state in a replicating
proto-cell consisting of many molecule species in mutually catalyzing reaction
networks. Protocells divide when the number of molecules, increasing due to
replication, exceeds a certain threshold. We study how the chemicals in a
catalytic network can form recursive production states in the presence of
errors in the replication process. Depending on the balance between the total
number of molecules in a cell and the number of molecule species, we have found
three phases; a phase without a recursive production state, a phase with
itinerancy over a few recursive states, and a phase with fixed recursive
production states. Heredity is realized in the latter two phases where molecule
species that are population-wise in the minority are preserved and control the
phenotype of the cell. It is shown that evolvability is realized in the
itinerancy phase, where a change in the number of minority molecules controls a
change of the chemical state.


title: From Coupled Dynamical Systems to Biological Irreversibility

abstract:
  In the first half of the paper, some recent advances in coupled dynamical
systems, in particular, a globally coupled map are surveyed. First, dominance
of Milnor attractors in partially ordered phase is demonstrated. Second,
chaotic itinerancy in high-dimensional dynamical systems is briefly reviewed,
with discussion on a possible connection with a Milnor attractor network.
Third, infinite-dimensional collective dynamics is studied, in the
thermodynamic limit of the globally coupled map, where bifurcation to
lower-dimensional attractors by the addition of noise is briefly reviewed.
Following the study of coupled dynamical systems, a scenario for developmental
process of cell society is proposed, based on numerical studies of a system
with interacting units with internal dynamics and reproduction. Differentiation
of cell types is found as a natural consequence of such a system. "Stem cells"
that either proliferate or differentiate to different types generally appear in
the system, where irreversible loss of multipotency is demonstrated. Robustness
of the developmental process against microscopic and macroscopic perturbations
is found and explained, while irreversibility in developmental process is
analyzed in terms of the gain of stability, loss of diversity and chaotic
instability. Construction of a phenomenology theory for development is
discussed in comparison with the thermodynamics.

categories: nlin.CD cond-mat.stat-mech physics.bio-ph q-bio.CB


title:
Deformation and Recursion for the N=2 $\alpha=1$ Supersymmetric
  KdV-hierarchy

abstract:
  A detailed description is given for the construction of the deformation of
the N=2 supersymmetric $\alpha=1$ KdV-equation, leading to the recursion
operator for symmetries and the zero-th Hamiltonian structure; the solution to
a longstanding problem.


title: Angular momentum localization in oval billiards

abstract:
  Angular momentum ceases to be the preferred basis for identifying dynamical
localization in an oval billiard at large excentricity. We give reasons for
this, and comment on the classical phase-space structure that is encoded in the
wave functions of ``leaky'' dielectric resonators with oval cross section.


title:
Scalar second order evolution equations possessing an irreducible
  sl$_2$-valued zero curvature representation

abstract:
  We find all scalar second order evolution equations possessing an
sl$_2$-valued zero curvature representation that is not reducible to a proper
subalgebra of sl$_2$. None of these zero-curvature representations admits a
parameter.


title: Inverse cascade in Charney-Hasegawa-Mima turbulence

abstract:
  The inverse energy cascade in Charney-Hasegawa-Mima turbulence is
investigated. Kolmogorov law for the third order velocity structure function is
shown to be independent on the Rossby number, at variance with the energy
spectrum, as shown by high resolution direct numerical simulations. In the
asymptotic limit of strong rotation, coherent vortices are observed to form at
a dynamical scale which slowly grows with time. These vortices form an almost
quenched pattern and induce strong deviation form Gaussianity in the velocity
field.


title: Some Applications of the Difference Analysis for Stochastic Systems

abstract:
  The work relates to a new way for analysis of one-dimensional stochastic
systems, based on consideration of its higher order difference structure. From
this point of view, the deterministic and random processes are analyzed. A new
numerical characteristic for one-dimensional stochastic systems is introduced.
The applications to single neuron models and neural networks are given.


title:
Entropy production of diffusion in spatially periodic deterministic
  systems

abstract:
  This paper presents an {\it ab initio} derivation of the expression given by
irreversible thermodynamics for the rate of entropy production for different
classes of diffusive processes. The first class are Lorentz gases, where
non-interacting particles move on a spatially periodic lattice, and collide
elastically with fixed scatterers. The second class are periodic systems where
$N$ particles interact with each other, and one of them is a tracer particle
which diffuses among the cells of the lattice. We assume that, in either case,
the dynamics of the system is deterministic and hyperbolic, with positive
Lyapunov exponents. This work extends methods originally developed for a
chaotic two-dimensional model of diffusion, the multi-baker map, to higher
dimensional, continuous time dynamical systems appropriate for systems with one
or more moving particles. Here we express the rate of entropy production in
terms of hydrodynamic measures that are determined by the fractal properties of
microscopic hydrodynamic modes that describe the slowest decay of the system to
an equilibrium state.


title: Nambu system associated with n-dimensional maps

abstract:
  We studied that arbitrary 2-dimensional maps are Hamilton system if a initial
value of map is a "time" variable. In this paper, we generalize this
correspondence, and show that an n-dimensional map is a Nambu system in which
one of initial values of the map play a role of "time" variable.


title: Transitions to Measure Synchronization in Coupled Hamiltonian Systems

abstract:
  Transitions to measure synchronization in two coupled $\phi ^{4}$ lattices
are investigated based on numerical simulations. The relationship between
measure synchronization (MS), phase locking and system's total energy is
studied both for periodic and chaotic states. Two different scalings are
discovered during the process to MS according to phase locking. Random walk
like phase synchronization in chaotic measure synchronization is found, and
phase locking interrupted by phase slips irregularly is also investigated.
Meanwhile, related analysis is qualitative given to explain this phenomenon.


title:
Comment on ``Analysis of chaotic motion and its shape dependence in a
  generalized piecewise linear map''

abstract:
  Rajagopalan and Sabir [nlin.CD/0104021 and Phys. Rev. E 63, 057201 (2001)]
recently discussed deterministic diffusion in a piecewise linear map using an
approach developed by Fujisaka et al. We first show that they rederived the
random walk formula for the diffusion coefficient, which is known to be the
exact result for maps of Bernoulli type since the work of Fujisaka and
Grossmann [Z. Physik B {\bf 48}, 261 (1982)]. However, this correct solution is
at variance to the diffusion coefficient curve presented in their paper.
Referring to another existing approach based on Markov partitions, we answer
the question posed by the authors regarding solutions for more general
parameter values by recalling the finding of a fractal diffusion coefficient.
We finally argue that their model is not suitable for studying intermittent
behavior, in contrast to what was suggested in their paper.


title: M\"obius Symmetry of Discrete Time Soliton Equations

abstract:
  We have proposed, in our previous papers, a method to characterize integrable
discrete soliton equations. In this paper we generalize the method further and
obtain a $q$-difference Toda equation, from which we can derive various
$q$-difference soliton equations by reductions.


title:
Conservation laws for multidimensional systems and related linear
  algebra problems

abstract:
  We consider multidimensional systems of PDEs of generalized evolution form
with t-derivatives of arbitrary order on the left-hand side and with the
right-hand side dependent on lower order t-derivatives and arbitrary space
derivatives. For such systems we find an explicit necessary condition for
existence of higher conservation laws in terms of the system's symbol. For
systems that violate this condition we give an effective upper bound on the
order of conservation laws. Using this result, we completely describe
conservation laws for viscous transonic equations, for the Brusselator model,
and the Belousov-Zhabotinskii system. To achieve this, we solve over an
arbitrary field the matrix equations SA=A^tS and SA=-A^tS for a quadratic
matrix A and its transpose A^t, which may be of independent interest.


title:
Decay of the Loschmidt Echo for quantum states with sub-Planck scale
  structures

abstract:
  Quantum states extended over a large volume in phase space have oscillations
from quantum interferences in their Wigner distribution on scales smaller than
$\hbar$ [W.H. Zurek, Nature {\bf 412}, 712 (2001)]. We investigate the
influence of those sub-Planck scale structures on the sensitivity to an
external perturbation of the state's time evolution. While we do find an
accelerated decay of the Loschmidt Echo for an extended state in comparison to
a localized wavepacket, the acceleration is described entirely by the classical
Lyapunov exponent and hence cannot originate from quantum interference.


title:
Applying the DFT representation for describing the order-chaos
  transition in H\'enon mapping dynamics

abstract:
  The DFT representation is applied, to a ordered-chaotic finite-duration
sequences set generated by the H\'enon mapping, for describing the order-chaos
transition. This representation has the advantage that it may be applied to a
relatively shorter discret-time series. The bifurcation diagram is produced and
the largest Lyapounov exponent is calculated for each time series of the set,
showing a good agreement with the results obtained by the DFT representation.
The threshold value of the control parameter for the order-chaos transition was
determined to be A=1.0532.


title:
Instabilities and Spatio-temporal Chaos of Long-wave Hexagon Patterns in
  Rotating Marangoni Convection

abstract:
  We consider surface-tension driven convection in a rotating fluid layer. For
nearly insulating boundary conditions we derive a long-wave equation for the
convection planform. Using a Galerkin method and direct numerical simulations
we study the stability of the steady hexagonal patterns with respect to general
side-band instabilities. In the presence of rotation steady and oscillatory
instabilities are identified. One of them leads to stable, homogeneously
oscillating hexagons. For sufficiently large rotation rates the stability
balloon closes, rendering all steady hexagons unstable and leading to
spatio-temporal chaos.


title: Perturbative Symmetry Approach

abstract:
  Perturbative Symmetry Approach is formulated in symbolic representation.
Easily verifiable integrability conditions of a given equation are constructed
in the frame of the approach. Generalisation for the case of non-local and
non-evolution equations is disscused. Application of the theory to the
Benjamin-Ono and Camassa-Holm type equations is considered.


title:
Cuspons, peakons and regular gap solitons between three dispersion
  curves

abstract:
  A general wave model with the cubic nonlinearity is introduced to describe a
situation when the linear dispersion relation has three branches, which would
intersect in the absence of linear couplings between the three waves. Actually,
the system contains two waves with a strong linear coupling between them, to
which a third wave is then coupled. This model has two gaps in its linear
spectrum. Realizations of this model can be made in terms of temporal or
spatial evolution of optical fields in, respectively, a planar waveguide or a
bulk-layered medium resembling a photonic-crystal fiber. Another physical
system described by the same model is a set of three internal wave modes in a
density-stratified fluid. A nonlinear analysis is performed for solitons which
have zero velocity in the reference frame in which the group velocity of the
third wave vanishes. Disregarding the self-phase modulation (SPM) term in the
equation for the third wave, we find two coexisting families of solitons:
regular ones, which may be regarded as a smooth deformation of the usual gap
solitons in a two-wave system, and cuspons with a singularity in the first
derivative at their center. Even in the limit when the linear coupling of the
third wave to the first two vanishes, the soliton family remains drastically
different from that in the linearly uncoupled system; in this limit, regular
solitons whose amplitude exceeds a certain critical value are replaced by
peakons. While the regular solitons, cuspons, and peakons are found in an exact
analytical form, their stability is tested numerically, which shows that they
all may be stable. If the SPM terms are retained, we find that there again
coexist two different families of generic stable soliton solutions, namely,
regular ones and peakons.


title: Strong NLS Soliton-Defect Interactions

abstract:
  We consider the interaction of a nonlinear Schrodinger soliton with a
localized (point) defect in the medium through which it travels. Using
numerical simulations, we find parameter regimes under which the soliton may be
reflected, transmitted, or captured by the defect. We propose a mechanism of
resonant energy transfer to a nonlinear standing wave mode supported by the
defect. Following Forinash et al, we derive a finite-dimensional model for the
interaction of the soliton with the defect via a collective coordinates method.
The system thus derived is a three degree-of-freedom Hamiltonian with an
additional conserved quantity. We study this system using the tools of
dynamical systems theory, and find that it exhibits a variety of interesting
behaviors, largely determined by the structures of stable and unstable
manifolds of special classes of periodic orbits. We use this geometrical
understanding to interpret the simulations.


title: Progressive motion of an ac-driven kink in an annular damped system

abstract:
  A novel dynamical effect is presented: systematic drift of a topological
soliton in ac-driven weakly damped systems with periodic boundary conditions.
The effect is demonstrated in detail for a long annular Josephson junction.
Unlike earlier considered cases of the ac-driven motion of fluxons (kinks), in
the present case the long junction is_spatially uniform_. Numerical simulations
reveal that progressive motion of the fluxon commences if the amplitude of the
ac drive exceeds a threshold value. The direction of the motion is randomly
selected by initial conditions, and a strong hysteresis is observed. An
analytical approach to the problem is based on consideration of the interaction
between plasma waves emitted by the fluxon under the action of the ac drive and
the fluxon itself, after the waves complete round trip in the annular junction.
The analysis predicts instability of the zero-average-velocity state of the
fluxon interacting with its own radiation tails, provided that the drive's
amplitude exceeds an explicitly found threshold. The predicted threshold
amplitude strongly depends on the phase shift gained by the wave after the
round trip. A very similar dependence is found in the simulations, testifying
to the relevance of the analytical consideration.


title: Intermittency in the Joint Cascade of Energy and Helicity

abstract:
  The statistics of the energy and helicity fluxes in isotropic turbulence are
studied using high resolution direct numerical simulation. The scaling
exponents of the energy flux agree with those of the transverse velocity
structure functions through refined similarity hypothesis, consistent with
Kraichnan's prediction \cite{Kr74}. The helicity flux is even more intermittent
than the energy flux and its scaling exponents are closer to those of the
passive scalar. Using Waleffe's helical decomposition, we demonstrate that the
existence of positive mean helicity flux inhibits the energy transfer in the
negative helical modes, a non-passive effect.


title: Noise-induced dephasing of an ac-driven Josephson junction

abstract:
  We consider phase-locked dynamics of a Josephson junction driven by
finite-spectral-linewidth ac current. By means of a transformation, the effect
of frequency fluctuations is reduced to an effective additive noise, the
corresponding (large) dephasing time being determined, in the logarithmic
approximation, by the Kramers' expression for the lifetime. For sufficiently
small values of the drive's amplitude, direct numerical simulations show
agreement of the dependence of the dephasing activation energy on the
ac-drive's spectral linewidth and amplitude with analytical predictions.
Solving the corresponding Fokker-Planck equation analytically, we find a
universal dependence of a critical value of the effective phase-diffusion
parameter on the drive's amplitude at a point of a sharp transition from the
phase-locked state to an unlocked one. For large values of the drive amplitude,
saturation and subsequent decrease of the activation energy are revealed by
simulations, which cannot be accounted for by the perturbative analysis. The
same new effect is found for a previously studied case of ac-driven Josephson
junctions with intrinsic thermal noise. The work was performed in the framework
of a cooperation agreement between Consiglio Nazionale di Ricerca (Italy) and
the Israeli Ministry of Science and Technology.


title:
Parameter Mismatches and Perfect Anticipating Synchronization in
  bi-directionally coupled external cavity laser diodes

abstract:
  We study perfect chaos synchronization between two bi-directionally coupled
external cavity semiconductor lasers and demonstrate for the first time that
mismatches in laser photon decay rates can explain the experimentally observed
anticipating time in synchronization.


title: Stochastic ionization through noble tori: Renormalization results

abstract:
  We find that chaos in the stochastic ionization problem develops through the
break-up of a sequence of noble tori. In addition to being very accurate, our
method of choice, the renormalization map, is ideally suited for analyzing
properties at criticality. Our computations of chaos thresholds agree closely
with the widely used empirical Chirikov criterion.


title:
Renormalization for cubic frequency invariant tori in Hamiltonian
  systems with two degrees of freedom

abstract:
  We consider the break-up of invariant tori in Hamiltonian systems with two
degrees of freedom with a frequency which belongs to a cubic field. We define
and construct renormalization-group transformations in order to determine the
threshold of the break-up of these tori. A first transformation is defined from
the continued fraction expansion of the frequency, and a second one is defined
with a fixed frequency vector in a space of Hamiltonians with three degrees of
freedom.


title:
Breathers in a Discrete Nonlinear Schr\"{o}dinger type Model : Exact
  Stability Results

abstract:
  Exact stability analysis of 1-site breathers in an NLS type model (ref. [4],
see below) indicates destabilisation through a growth rate becoming positive as
a stability border is crossed, while above a critical spatial decay rate
($\lambda_c$) the breather is always unstable. Similar analysis on 2-site
breathers with phase difference $\pi$ (see below) indicates destabilisation
through {\it Krein collision} of a pair of growth rates. All eigenmodes can be
exactly calculated and conform to numerical estimates.


title: Surface Quasigeostrophic Turbulence : The Study of an Active Scalar

abstract:
  We study the statistical and geometrical properties of the potential
temperature (PT) field in the Surface Quasigeostrophic (SQG) system of
equations. In addition to extracting information in a global sense via tools
such as the power spectrum, the g-beta spectrum and the structure functions we
explore the local nature of the PT field by means of the wavelet transform
method. The primary indication is that an initially smooth PT field becomes
rough (within specified scales), though in a qualitatively sparse fashion.
Similarly, initially 1D iso-PT contours (i.e., PT level sets) are seen to
acquire a fractal nature. Moreover, the dimensions of the iso-PT contours
satisfy existing analytical bounds. The expectation that the roughness will
manifest itself in the singular nature of the gradient fields is confirmed via
the multifractal nature of the dissipation field. Following earlier work on the
subject, the singular and oscillatory nature of the gradient field is
investigated by examining the scaling of a probability measure and a sign
singular measure respectively. A physically motivated derivation of the
relations between the variety of scaling exponents is presented, the aim being
to bring out some of the underlying assumptions which seem to have gone
unnoticed in previous presentations. Apart from concentrating on specific
properties of the SQG system, a broader theme of the paper is a comparison of
the diagnostic inertial range properties of the SQG system with both the 2D and
3D Euler equations.


title:
Master equation approach to the conjugate pairing rule of Lyapunov
  spectra for many-particle thermostatted systems

abstract:
  The master equation approach to Lyapunov spectra for many-particle systems is
applied to non-equilibrium thermostatted systems to discuss the conjugate
pairing rule. We consider iso-kinetic thermostatted systems with a shear flow
sustained by an external restriction, in which particle interactions are
expressed as a Gaussian white randomness. Positive Lyapunov exponents are
calculated by using the Fokker-Planck equation to describe the tangent vector
dynamics. We introduce another Fokker-Planck equation to describe the
time-reversed tangent vector dynamics, which allows us to calculate the
negative Lyapunov exponents. Using the Lyapunov exponents provided by these two
Fokker-Planck equations we show the conjugate pairing rule is satisfied for
thermostatted systems with a shear flow in the thermodynamic limit. We also
give an explicit form to connect the Lyapunov exponents with the
time-correlation of the interaction matrix in a thermostatted system with a
color field.


title:
Bailout Embeddings and Neutrally Buoyant Particles in Three-Dimensional
  Flows

abstract:
  We use the bailout embeddings of three-dimensional volume-preserving maps to
study qualitatively the dy- namics of small spherical neutrally buoyant
impurities suspended in a time-periodic incompressible fluid flow. The
accumulation of impurities in tubular vortical structures, the detachment of
particles from fluid trajectories near hyperbolic invariant lines, and the
formation of nontrivial three-dimensional structures in the distribution of
particles are predicted.


title: Generalization of the Goraychev--Chaplygin Case

abstract:
  In this paper we present a generalization of the Goraychev--Chaplygin
integrable case on a bundle of Poisson brackets, and on Sokolov terms in his
new integrable case of Kirchhoff equations. We also present a new analogous
integrable case for the quaternion form of rigid body dynamics' equations. This
form of equations is recently developed and we can use it for the description
of rigid body motions in specific force fields, and for the study of different
problems of quantum mechanics. In addition we present new invariant relations
in the considered problems.


title: Orbit bifurcations and wavefunction autocorrelations

abstract:
  It was recently shown (Keating & Prado, {\it Proc. R. Soc. Lond. A} {\bf
457}, 1855-1872 (2001)) that, in the semiclassical limit, the scarring of
quantum eigenfunctions by classical periodic orbits in chaotic systems may be
dramatically enhanced when the orbits in question undergo bifurcation.
Specifically, a bifurcating orbit gives rise to a scar with an amplitude that
scales as $\hbar^{\alpha}$ and a width that scales as $\hbar^{\omega}$, where
$\alpha$ and $\omega$ are bifurcation-dependent scar exponents whose values are
typically smaller than those ($\alpha=\omega=1/2$) associated with isolated and
unstable periodic orbits. We here analyze the influence of bifurcations on the
autocorrelation function of quantum eigenstates, averaged with respect to
energy. It is shown that the length-scale of the correlations around a
bifurcating orbit scales semiclassically as $\hbar^{1-\alpha}$, where $\alpha$
is the corresponding scar amplitude exponent. This imprint of bifurcations on
quantum autocorrelations is illustrated by numerical computations for a family
of perturbed cat maps.


title: ROSAT/Asca Observations of the Mixed-Morphology Supernova Remnant W28

abstract:
  We present ROSAT PSPC and ASCA observations of the supernova remnant (SNR)
W28. The overall shape of X-ray emission in W28 is elliptical, dominated by a
centrally-concentrated interior emission, sharply peaked at the center. The
ASCA spectra reveal emission lines of Ne, Mg, Si, and Fe K$\alpha$ and
continuum extending at least up to 7 keV, showing thermal origin with a hot
thermal component. We found that spectral variations are present in W28. The
southwestern shell can be fit well by a plane-shock model with a temperature of
1.5 keV, and the northeastern shell, with a lower temperature of 0.56 keV.
Unlike for the southwestern and northeastern shells, the central emission
requires a two-temperature components with 0.6 keV and 1.8 keV. The low
temperature component is similar to those seen in other Mixed-morphology SNRs.
The X-ray luminosity of W28 is 6x 10^34 ergs/s, and the estimated X-ray mass is
only ~20 - 25 solar mass. A comparison of W28 with other typical
Mixed-morphology SNRs reveals significant differences in its X-ray properties;
W28 has a significantly higher temperature and noticeable spectral variations.
W28 belongs to a class of SNRs considered by Chevalier (1999), with a radiative
shell interacting with clumpy molecular clouds. X-ray emission at its center is
a ``fossil'' radiation from gas which was shocked early in the evolution of the
remnant, and its centrally-peaked morphology could have been caused by
processes such as evaporation, electron thermal conduction, and mixing induced
by various hydrodynamical instabilities. But W28 poses a challenge for existing
models of X-ray emission, because the evaporation model of White & Long (1991)
is in conflict with observations, while the presence of temperature variations
seems inconsistent with SNR models with efficient thermal conduction.


title:
Faraday instability on viscous ferrofluids in a horizontal magnetic
  field: Oblique rolls of arbitrary orientation

abstract:
  A linear stability analysis of the free surface of a horizontally unbounded
ferrofluid layer of arbitrary depth subjected to vertical vibrations and a
horizontal magnetic field is performed. A nonmonotonic dependence of the
stability threshold on the magnetic field is found at high frequencies of the
vibrations. The reasons of the decrease of the critical acceleration amplitude
caused by a horizontal magnetic field are discussed. It is revealed that the
magnetic field can be used to select the first unstable pattern of Faraday
waves. In particular, a rhombic pattern as a superposition of two different
oblique rolls can occur. A scaling law is presented which maps all data into
one graph for the tested range of viscosities, frequencies, magnetic fields and
layer thicknesses.


title: Thermomagnetic convection of magnetic fluids in a cylindrical geometry

abstract:
  The thermomagnetic convection of magnetic fluids in a cylindrical geometry
subjected to a homogeneous magnetic field is studied. The study is motivated by
a novel thermal instability [W. Luo et al., Phys. Rev. Lett. 82, 4134 (1999)].
As model system a composite cylinder with inner heating is considered which
reflects the symmetry of the experimentally setup. The general condition for
the existence of a potentially unstable stratification in the magnetic fluid is
derived. Within a linear stability analysis the critical external induction for
the onset of thermomagnetic convection is determined for dilute and nondilute
magnetic fluids. The difference between both thresholds allows to test
experimentally whether a test sample is a dilute fluid or not.


title:
A systematic correlation between two-dimensional flow topology and the
  abstract statistics of turbulence

abstract:
  Velocity differences in the direct enstrophy cascade of two-dimensional
turbulence are correlated with the underlying flow topology. The statistics of
the transverse and longitudinal velocity differences are found to be governed
by different structures. The wings of the transverse distribution are dominated
by strong vortex centers, whereas, the tails of the longitudinal differences
are dominated by saddles. Viewed in the framework of earlier theoretical work
this result suggests that the transfer of enstrophy to smaller scales is
accomplished in regions of the flow dominated by saddles.


title: Chaotic stirring by a mesoscale surface-ocean flow

abstract:
  The horizontal stirring properties of the flow in a region of the
East-Australian Current are calculated. A surface velocity field derived from
remotely sensed data, using the Maximum Cross Correlation method, is integrated
to derive the distribution of the finite-time Lyapunov exponents. For the
region studied (between latitudes 36 S and 41 S and longitudes 150 E and 156 E)
the mean Lyapunov exponent during 1997 is estimated to be 4 x 10^{-7} s^{-1}.
This is in close agreement with the few other measurements of stirring rates in
the surface ocean which are available. Recent theoretical results on the
multifractal spectra of advected reactive tracers are applied to an analysis of
a sea-surface temperature image of the study region. The spatial pattern seen
in the image compares well with the pattern seen in an advected tracer with a
first-order response to changes in surface forcing. The response timescale is
estimated to be 20 days.


title: Contact tracing and epidemics control in social networks

abstract:
  A generalization of the standard susceptible-infectious-removed (SIR)
stochastic model for epidemics in sparse random networks is introduced which
incorporates contact tracing in addition to random screening. We propose a
deterministic mean-field description which yields quantitative agreement with
stochastic simulations on random graphs. We also analyze the role of contact
tracing in epidemics control in small-world networks and show that its
effectiveness grows as the rewiring probability is reduced.


title: Magic Number 7 +- 2 in Globally Coupled Dynamical Systems?

abstract:
  The prevalence of Milnor attractors has recently been reported in a class of
high-dimensional dynamical systems. We study how this prevalence depends on the
number of degrees of freedom by using a globally coupled map and show that the
basin fraction of Milnor attractors increases drastically around 5-10 degrees
of freedom, saturating for higher numbers of degrees of freedom. It is argued
that this dominance of Milnor attractors in the basin arises from a
combinatorial explosion of the basin boundaries. In addition, the dominance is
also found in a system without permutation symmetry, i,e., a coupled dynamical
system of non-identical elements. Possible relevance to the magic number $7\pm
2$ in psychology is discussed.


title: Inverse velocity statistics in two dimensional turbulence

abstract:
  We present a numerical study of two-dimensional turbulent flows in the
enstrophy cascade regime, with different large-scale forcings and energy sinks.
In particular, we study the statistics of more-than-differentiable velocity
fluctuations by means of two recently introduced sets of statistical
estimators, namely {\it inverse statistics} and {\it second order differences}.
We show that the 2D turbulent velocity field, $\bm u$, cannot be simply
characterized by its spectrum behavior, $E(k) \propto k^{-\alpha}$. There
exists a whole set of exponents associated to the non-trivial smooth
fluctuations of the velocity field at all scales. We also present a numerical
investigation of the temporal properties of $\bm u$ measured in different
spatial locations.


title: Lyapunov Exponents, Transport and the Extensivity of Dimensional Loss

abstract:
  An explicit relation between the dimensional loss ($\Delta D$), entropy
production and transport is established under thermal gradients, relating the
microscopic and macroscopic behaviors of the system. The extensivity of $\Delta
D$ in systems with bulk behavior follows from the relation. The maximum
Lyapunov exponents in thermal equilibrium and $\Delta D$ in non-equilibrium
depend on the choice of heat-baths, while their product is unique and
macroscopic. Finite size corrections are also computed and all results are
verified numerically.


title:
Does ohmic heating influence the flow field in thin-layer
  electrodeposition?

abstract:
  In thin-layer electrodeposition the dissipated electrical energy leads to a
substantial heating of the ion solution. We measured the resulting temperature
field by means of an infrared camera. The properties of the temperature field
correspond closely with the development of the concentration field. In
particular we find, that the thermal gradients at the electrodes act like a
weak additional driving force to the convection rolls driven by concentration
gradients.


title:
Lorenz-like systems and classical dynamical equations with memory
  forcing: a new point of view for singling out the origin of chaos

abstract:
  A novel view for the emergence of chaos in Lorenz-like systems is presented.
For such purpose, the Lorenz problem is reformulated in a classical mechanical
form and it turns out to be equivalent to the problem of a damped and forced
one dimensional motion of a particle in a two-well potential, with a forcing
term depending on the ``memory'' of the particle past motion. The dynamics of
the original Lorenz system in the new particle phase space can then be
rewritten in terms of an one-dimensional first-exit-time problem. The emergence
of chaos turns out to be due to the discontinuous solutions of the
transcendental equation ruling the time for the particle to cross the
intermediate potential wall. The whole problem is tackled analytically deriving
a piecewise linearized Lorenz-like system which preserves all the essential
properties of the original model.


title: New Darboux Transformation for Hirota-Satsuma coupled KdV System

abstract:
  A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV
system. It is shown that this Darboux transformation can be constructed by
means of two methods: Painlev\'{e} analysis and reduction of a binary Darboux
transformation. By iteration of the Darboux transformation, the Grammian type
solutions are found for the coupled KdV system.


title: Sliding on the inside of a Conical Surface

abstract:
  We analyze the frictionless motion of a point-like particle that slides under
gravity on an inverted conical surface. This motion is studied for arbitrary
initial conditions and a general relation, valid within 13%, between the
periods of radial and angular oscillations, that holds for both small and high
energy trajectories, is obtained. This relation allows us to identify the
closed orbits of the system. The virtues of this model to illustrate
pedagogically, how in a physical system, the energy is transferred between
different modes, are also emphasized. Two easy identification criteria for this
type of motion with potential interest in industrial design are obtained.


title: Lorenz deterministic diffusion

abstract:
  The Lorenz 1963 dynamical system is known to reduce in the steady state to a
one-dimensional motion of a classical particle subjected to viscous damping in
a past history-dependent potential field. If the potential field is substituted
by a periodic function of the position, the resulting system shows a rich
dynamics where (standard) diffusive behaviours, ballistc motions and trapping
take place by varying the model control parameters. This system permits to
highlight the intimate relation between chaos and long-time deterministic
diffusion.


title: Symmetry Classification of Diatomic Molecular Chains

abstract:
  A symmetry classification of possible interactions in a diatomic molecular
chain is provided. For nonlinear interactions the group of Lie point
transformations, leaving the lattice invariant and taking solutions into
solutions, is at most five-dimensional. An example is considered in which
subgroups of the symmetry group are used to reduce the dynamical
differential-difference equations to purely difference ones.


title:
The clustering instability of inertial particles spatial distribution in
  turbulent flows

abstract:
  A theory of clustering of inertial particles advected by a turbulent velocity
field caused by an instability of their spatial distribution is suggested. The
reason for the clustering instability is a combined effect of the particles
inertia and a finite correlation time of the velocity field. The crucial
parameter for the clustering instability is a size of the particles. The
critical size is estimated for a strong clustering (with a finite fraction of
particles in clusters) associated with the growth of the mean absolute value of
the particles number density and for a weak clustering associated with the
growth of the second and higher moments. A new concept of compressibility of
the turbulent diffusion tensor caused by a finite correlation time of an
incompressible velocity field is introduced. In this model of the velocity
field, the field of Lagrangian trajectories is not divergence-free. A mechanism
of saturation of the clustering instability associated with the particles
collisions in the clusters is suggested. Applications of the analyzed effects
to the dynamics of droplets in the turbulent atmosphere are discussed. An
estimated nonlinear level of the saturation of the droplets number density in
clouds exceeds by the orders of magnitude their mean number density. The
critical size of cloud droplets required for clusters formation is more than
$20 \mu$m.


title:
Features of the Extension of a Statistical Measure of Complexity to
  Continuous Systems

abstract:
  We discuss some aspects of the extension to continuous systems of a
statistical measure of complexity introduced by Lopez-Ruiz, Mancini and Calbet
(LMC) [Phys. Lett. A 209 (1995) 321]. In general, the extension of a magnitude
from the discrete to the continuous case is not a trivial process and requires
some choice. In the present study, several possibilities appear available. One
of them is examined in detail. Some interesting properties desirable for any
magnitude of complexity are discovered on this particular extension.


title:
Coupled map gas: structure formation and dynamics of interacting motile
  elements with internal dynamics

abstract:
  A model of interacting motile chaotic elements is proposed. The chaotic
elements are distributed in space and interact with each other through
interactions depending on their positions and their internal states. As the
value of a governing parameter is changed, the model exhibits successive phase
changes with novel pattern dynamics, including spatial clustering, fusion and
fission of clusters and intermittent diffusion of elements. We explain the
manner in which the interplay between internal dynamics and interaction leads
to this behavior by employing certain quantities characterizing diffusion,
correlation, and the information cascade of synchronization. Keywords:
collective motion, coupled map system, interacting motile elements


title: Stadium Billiard with Moving Walls

abstract:
  We study the evolution of the energy distribution for a stadium with moving
walls. We consider one period driving cycle, which is characterized by an
amplitude $A$ and wall velocity $V$. This evolving energy distribution has both
"parametric" and "stochastic" components. The latter are important for the
theory of quantum irreversibility and dissipation in driven mesoscopic devices
(eg in the context of quantum computation). For extremely slow wall velocity
$V$ the spreading mechanism is dominated by transitions between neighboring
levels, while for larger (non-adiabatic) velocities the spreading mechanism has
both perturbative and non-perturbative features. We present, for the first
time, a numerical study which is aimed in identifying the latter features. A
procedure is developed for the determination of the various $V$ regimes. The
possible implications on linear response theory are discussed.


title:
Comment on "On two-dimensional magnetohydrodynamic turbulence" [Phys.
  Plasmas, 8, 3282 (2001)]

abstract:
  Biskamp and Schwarz [Phys. Plasmas, 8, 3282 (2001)] have reported that the
energy spectrum of two-dimensional magnetohydrodynamic turbulence is
proportional to $k^{-3/2}$, which is a prediction of Iroshnikov-Kraichnan
phenomenology. In this comment we report some earlier results which
conclusively show that for two-dimensional magnetohydrodynamic turbulence,
Kolmogorov-like phenomenology (spectral index 5/3) is better model than
Iroshnikov-Kraichnan phenomenology; these results are based on energy flux
analysis.


title:
Local shell-to-shell energy transfer via nonlocal Interactions in fluid
  turbulence

abstract:
  In this paper we analytically compute the strength of nonlinear interactions
in a triad, and the energy exchanges between wavenumber shells in
incompressible fluid turbulence. The computation has been done using
first-order perturbative field theory. In three dimension, magnitude of triad
interactions is large for nonlocal triads, and small for local triads. However,
the shell-to-shell energy transfer rate is found to be local and forward. This
result is due to the fact that the nonlocal triads occupy much less Fourier
space volume than the local ones. The analytical results on three-dimensional
shell-to-shell energy transfer match with their numerical counterparts. In
two-dimensional turbulence, the energy transfer rates to the near-by shells are
forward, but to the distant shells are backward; the cumulative effect is an
inverse cascade of energy.


title:
Complex patterns on the plane: different types of basin fractalization
  in a two-dimensional mapping

abstract:
  Basins generated by a noninvertible mapping formed by two symmetrically
coupled logistic maps are studied when the only parameter \lambda of the system
is modified. Complex patterns on the plane are visualised as a consequence of
basins' bifurcations. According to the already established nomenclature in the
literature, we present the relevant phenomenology organised in different
scenarios: fractal islands disaggregation, finite disaggregation, infinitely
disconnected basin, infinitely many converging sequences of lakes, countable
self-similar disaggregation and sharp fractal boundary. By use of critical
curves, we determine the influence of zones with different number of first rank
preimages in the mechanisms of basin fractalization.


title: Separation of variables for bi-Hamiltonian systems

abstract:
  We address the problem of the separation of variables for the Hamilton-Jacobi
equation within the theoretical scheme of bi-Hamiltonian geometry. We use the
properties of a special class of bi-Hamiltonian manifolds, called omega-N
manifolds, to give intrisic tests of separability (and Staeckel separability)
for Hamiltonian systems. The separation variables are naturally associated with
the geometrical structures of the omega-N manifold itself. We apply these
results to bi-Hamiltonian systems of the Gel'fand-Zakharevich type and we give
explicit procedures to find the separated coordinates and the separation
relations.


title: Focusing Revisited: an MN-dynamics Approach

abstract:
  The nonlinear Schr{\"o}dinger (NLS) equation is a ubiquitous example of an
envelope wave equation for conservative, dispersive systems. We revisit here
the problem of self-similar focusing of waves in the case of the focusing NLS
equation through the prism of a dynamic renormalization technique (MN dynamics)
that factors out self-similarity and yields a bifurcation view of the onset of
focusing. As a result, identifying the focusing self-similar solution becomes a
steady state problem. The discretized steady states are subsequently obtained
and their linear stability is numerically examined. The calculations are
performed in the setting of variable index of refraction, in which the onset of
focusing appears as a supercritical bifurcation of a novel type of mixed
Hamiltonian-dissipative dynamical system (reminiscent, to some extent, of a
pitchfork bifurcation).


title: Stability borders of feedback control of delayed measured systems

abstract:
  When stabilization of unstable periodic orbits or fixed points by the method
given by Ott, Grebogi and Yorke (OGY) has to be based on a measurement delayed
by $\tau$ orbit lengths, the performance of unmodified OGY method is expected
to decline. For experimental considerations, it is desired to know the range of
stability with minimal knowledge of the system. We find that unmodified OGY
control fails beyond a maximal Ljapunov number of
$\lambda_{max}=1+\frac{1}{\tau}$. In this paper the area of stability is
investigated both for OGY control of known fixed points and for difference
control of unknown or inaccurately known fixed points. An estimated value of
the control gain is given. Finally we outline what extensions have to be
considered if one wants to stabilize fixed points with Ljapunov numbers above
$\lambda_{max}$.


title:
Breathers on a Background: Periodic and Quasiperiodic Solutions of
  Extended Discrete Nonlinear Wave Systems

abstract:
  In this paper we investigate the emergence of time-periodic and and
time-quasiperiodic (sometimes infinitely long lived and sometimes very long
lived or metastable) solutions of discrete nonlinear wave equations: discrete
sine Gordon, discrete $\phi^4$ and discrete nonlinear Schr\"odinger. The
solutions we consider are periodic oscillations on a kink or standing wave
breather background. The origin of these oscillations is the presence of
internal modes, associated with the static ground state. Some of these modes
are associated with the breaking of translational invariance, in going from a
spatially continuous to a spatially discrete system. Others are associated with
discrete modes which bifurcate from the continuous spectrum. It is also
possible that such modes exist in the continuum limit and persist in the
discrete case. The regimes of existence, stability and metastability of states
as the lattice spacing is varied are investigated analytically and numerically.
A consequence of our analysis is a class of spatially localized, time
quasiperiodic solutions of the discrete nonlinear Schr\"odinger equation. We
demonstrate, however, that this class of quasiperiodic solution is rather
special and that its natural generalizations yield only metastable
quasiperiodic solutions.


title:
Thresholds to Chaos and Ionization for the Hydrogen Atom in Rotating
  Fields

abstract:
  We analyze the classical phase space of the hydrogen atom in crossed magnetic
and circularly polarized microwave fields in the high frequency regime, using
the Chirikov resonance overlap criterion and the renormalization map. These
methods are used to compute thresholds to large scale chaos and to ionization.
The effect of the magnetic field is a strong stabilization of a set of
invariant tori which bound the trajectories and prevent stochastic ionization.
In order to ionize, larger amplitudes of the microwave field are necessary in
the presence of a magnetic field.


title:
Effective Viscosity and Time Correlation for the Kuramoto-Sivashinsky
  Equation

abstract:
  A shock-like structure appears in a time-averaged pattern produced by the
Kuramoto-Sivashinsky equation and the noisy Burgers equation with fixed
boundary conditions. We show that the effective viscosity computed from the
width of the time-averaged shock structure is consistent with that computed
from the time correlation of the fluctuations. The effective viscosity depends
on the lengthscale, although our system size is not sufficiently large to
satisfy the asymptotic dynamic scaling law. We attempt to determine the
effective viscosity in a finite size system with the projection operator
method.


title:
Chaotic diffusion of particles with finite mass in oscillating
  convection flows

abstract:
  Deterministic diffusion in temporally oscillating convection is studied for
particles with finite mass. The particles are assumed to obey a simple
dissipative dynamical system and the particle diffusion is induced by the
strange attractor. The diffusion constants are numerically calculated for
convection models with free and rigid boundary conditions.


title:
Higher Order Potential Expansion for the Continuous Limits of the Toda
  Hierarchy

abstract:
  A method for introducing the higher order terms in the potential expansion to
study the continuous limits of the Toda hierarchy is proposed in this paper.
The method ensures that the higher order terms are differential polynomials of
the lower ones and can be continued to be performed indefinitly. By introducing
the higher order terms, the fewer equations in the Toda hierarchy are needed in
the so-called recombination method to recover the KdV hierarchy. It is shown
that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda
hierarchy tend towards the corresponding ones of the KdV hierarchy in
continuous limit.


title: Trigonometric osp(1|2) Gaudin model

abstract:
  The problems connected with Gaudin models are reviewed by analyzing model
related to the trigonometric osp(1|2) classical r-matrix. The eigenvectors of
the trigonometric osp(1|2) Gaudin Hamiltonians are found using explicitly
constructed creation operators. The commutation relations between the creation
operators and the generators of the trigonometric loop superalgebra are
calculated. The coordinate representation of the Bethe states is presented. The
relation between the Bethe vectors and solutions to the Knizhnik-Zamolodchikov
equation yields the norm of the eigenvectors. The generalized
Knizhnik-Zamolodchikov system is discussed both in the rational and in the
trigonometric case.


title: Neutrality: A Necessity for Self-Adaptation

abstract:
  Self-adaptation is used in all main paradigms of evolutionary computation to
increase efficiency. We claim that the basis of self-adaptation is the use of
neutrality. In the absence of external control neutrality allows a variation of
the search distribution without the risk of fitness loss.


title:
Periodic orbit quantization of the closed three-disk billiard as an
  example of a chaotic system with strong pruning

abstract:
  Classical chaotic systems with symbolic dynamics but strong pruning present a
particular challenge for the application of semiclassical quantization methods.
In the present study we show that the technique of periodic orbit quantization
by harmonic inversion of trace formulae, which does not rely on the existence
of a complete symbolic dynamics or other specific properties, lends itself
ideally to calculating semiclassical eigenvalues from periodic orbit data even
in strongly pruned systems. As the number of periodic orbits proliferates
exponentially in chaotic systems, we apply the harmonic inversion technique to
cross-correlated periodic orbit sums, which allows us to reduce the required
number of orbits. The power of the method is demonstrated for the closed
three-disk billiard as a prime example of a classically chaotic bound system
with strong pruning.


title:
Multiple mechanisms of spiral wave breakup in a model of cardiac
  electrical activity

abstract:
  It has become widely accepted that the most dangerous cardiac arrhythmias are
due to re- entrant waves, i.e., electrical wave(s) that re-circulate repeatedly
throughout the tissue at a higher frequency than the waves produced by the
heart's natural pacemaker (sinoatrial node). However, the complicated structure
of cardiac tissue, as well as the complex ionic currents in the cell, has made
it extremely difficult to pinpoint the detailed mechanisms of these
life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac
action potential (AP), which can be fitted to a wide variety of experimentally
and numerically obtained mesoscopic characteristics of cardiac tissue such as
AP shape and restitution of AP duration and conduction velocity, is used to
explain many different mechanisms of spiral wave breakup which in principle can
occur in cardiac tissue. Some, but not all, of these mechanisms have been
observed before using other models; therefore, the purpose of this paper is to
demonstrate them using just one framework model and to explain the different
parameter regimes or physiological properties necessary for each mechanism
(such as high or low excitability, corresponding to normal or ischemic tissue,
spiral tip trajectory types, and tissue structures such as rotational
anisotropy and periodic boundary conditions). Each mechanism is compared with
data from other ionic models or experiments to illustrate that they are not
model-specific phenomena. The fact that many different breakup mechanisms exist
has important implications for antiarrhythmic drug design and for comparisons
of fibrillation experiments using different species, electromechanical
uncoupling drugs, and initiation protocols.


title: Soliton Organization of Thermal Field in a Chain at High Temperature

abstract:
  Thermal field soliton self-organization arising due to absorption of
background atoms vibrations is observed in numerical experiment in nonlinear
chain with Lennard-Jones potential at high temperature. At some stage intensive
space-localized waves are formed and give additional peaks on high-energy tile
of energy distribution unlike of Gibbs one.


title:
Solution of the Dirichlet boundary value problem for the Sine-Gordon
  equation

abstract:
  The sine-Gordon equation in light cone coordinates is solved when Dirichlet
conditions on the L-shape boundaries of the strip [0,T]X[0,infinity) are
prescribed in a class of functions that vanish (mod 2 pi) for large x at
initial time. The method is based on the inverse spectral transform (IST) for
the Schroedinger spectral problem on the semi-line solved as a Hilbert boundary
value problem. Contrarily to what occurs when using the Zakharov-Shabat
eigenvalue problem, the spectral transform is regular and in particular the
discrete spectrum contains a finite number of eigenvalues (and no accumulation
point).


title:
Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili
  equation from Painlev\'e I and II

abstract:
  We present two constructions of new solutions to the dispersionless KP (dKP)
equation arising from the first two Painlev\'e transcendents. The first
construction is a hodograph transformation based on Einstein--Weyl geometry,
the generalised Nahm's equation and the isomonodromy problem. The second
construction, motivated by the first, is a direct characterisation of solutions
to dKP which are constant on a central quadric.
  We show how the solutions to the dKP equations can be used to construct some
three-dimensional Einstein--Weyl structures, and four--dimensional
anti-self-dual null-K\"ahler metrics.


title:
Anomalous scaling of a passive scalar advected by the turbulent velocity
  field with finite correlation time: Two-loop approximation

abstract:
  The renormalization group and operator product expansion are applied to the
model of a passive scalar quantity advected by the Gaussian self-similar
velocity field with finite, and not small, correlation time. The inertial-range
energy spectrum of the velocity is chosen in the form $E(k)\propto
k^{1-2\eps}$, and the correlation time at the wavenumber $k$ scales as
$k^{-2+\eta}$. Inertial-range anomalous scaling for the structure functions and
other correlation functions emerges as a consequence of the existence in the
model of composite operators with negative scaling dimensions, identified with
anomalous exponents. For $\eta>\eps$, these exponents are the same as in the
rapid-change limit of the model; for $\eta<\eps$, they are the same as in the
limit of a time-independent (quenched) velocity field. For $\eps=\eta$ (local
turnover exponent), the anomalous exponents are nonuniversal through the
dependence on a dimensionless parameter, the ratio of the velocity correlation
time and the scalar turnover time. The universality reveals itself, however,
only in the second order of the $\eps$ expansion, and the exponents are derived
to order $O(\eps^{2})$, including anisotropic contributions. It is shown that,
for moderate $n$, the order of the structure function, and $d$, the space
dimensionality, finite correlation time enhances the intermittency in
comparison with the both limits: the rapid-change and quenched ones. The
situation changes when $n$ and/or $d$ become large enough: the correction to
the rapid-change limit due to the finite correlation time is positive (that is,
the anomalous scaling is suppressed), it is maximal for the quenched limit and
monotonically decreases as the correlation time tends to zero.


title:
Structure of Rule Table and Phase Diagram of One Dimensional Cellular
  Automata

abstract:
  In addition to the $\lambda$ parameter, we have found another parameter which
characterize the class III, class II and class IV patterns more quantitatively.
It explains why the different classes of patterns coexist at the same
$\lambda$. With this parameter, the phase diagram for an one dimensional
cellular automata is obtained. Our result explains why the edge of chaos(class
IV) is scattered rather wide range in $\lambda$ around 0.5, and presents an
effective way to control the pattern classes. \noindent PACS: 89.75.-k Complex
Systems


title: Deterministic diffusion in flower shape billiards

abstract:
  We propose a flower shape billiard in order to study the irregular parameter
dependence of chaotic normal diffusion. Our model is an open system consisting
of periodically distributed obstacles of flower shape, and it is strongly
chaotic for almost all parameter values. We compute the parameter dependent
diffusion coefficient of this model from computer simulations and analyze its
functional form by different schemes all generalizing the simple random walk
approximation of Machta and Zwanzig. The improved methods we use are based
either on heuristic higher-order corrections to the simple random walk model,
on lattice gas simulation methods, or they start from a suitable Green-Kubo
formula for diffusion. We show that dynamical correlations, or memory effects,
are of crucial importance to reproduce the precise parameter dependence of the
diffusion coefficent.


title: Fractal Evolution of Normalized Feedback Systems on a Lattice

abstract:
  Highly nonlinear behavior of a system of discrete sites on a lattice is
observed when a specific feedback loop is introduced into models employing
coupled map lattices, quantum cellular automata, or the real-valued analogues
of the latter. It is shown that the combination of two operations, i.e. i)
enhancement of a site's value when fulfilling a feedback condition and ii)
normalization of the system after each time step, produces relatively
short-lived spatio-temporal patterns whose mean lifetime can be considered as
emergent order parameter of the system. This mean lifetime obeys a scaling law
involving a control parameter which tunes the "fault tolerance" of the feedback
condition. Thus, within appropriate ranges of the systems variables, the
dynamical properties can be characterized by a "fractal evolution dimension"
(as opposed to a "fractal dimension").


title: Spatial solitons in an optically pumped semiconductor microresonator

abstract:
  We show experimentally and numerically the existence of stable spatial
solitons in an optically pumped semiconductor microresonator. We demonstrate
that the pump substantially reduces the light intensity necessary to sustain
the solitons and thereby reduces destabilizing thermal effects. We demonstrate
coherent switching on and off of bright solitons. We discuss differences
between pumped and unpumped below bandgap-solitons.


title: Hamiltonians for curves

abstract:
  We examine the equilibrium conditions of a curve in space when a local energy
penalty is associated with its extrinsic geometrical state characterized by its
curvature and torsion. To do this we tailor the theory of deformations to the
Frenet-Serret frame of the curve. The Euler-Lagrange equations describing
equilibrium are obtained; Noether's theorem is exploited to identify the
constants of integration of these equations as the Casimirs of the euclidean
group in three dimensions. While this system appears not to be integrable in
general, it {\it is} in various limits of interest. Let the energy density be
given as some function of the curvature and torsion, $f(\kappa,\tau)$. If $f$
is a linear function of either of its arguments but otherwise arbitrary, we
claim that the first integral associated with rotational invariance permits the
torsion $\tau$ to be expressed as the solution of an algebraic equation in
terms of the bending curvature, $\kappa$. The first integral associated with
translational invariance can then be cast as a quadrature for $\kappa$ or for
$\tau$.


title: On a Poisson reduction for Gel'fand--Zakharevich manifolds

abstract:
  We formulate and discuss a reduction theorem for Poisson pencils associated
with a class of integrable systems, defined on bi-Hamiltonian manifolds,
recently studied by Gel'fand and Zakharevich. The reduction procedure is
suggested by the bi-Hamiltonian approach to the Separation of Variables
problem.


title:
To what extent can dynamical models describe statistical features of
  turbulent flows?

abstract:
  Statistical features of "bursty" behaviour in charged and neutral fluid
turbulence, are compared to statistics of intermittent events in a GOY shell
model, and avalanches in different models of Self Organized Criticality (SOC).
It is found that inter-burst times show a power law distribution for turbulent
samples and for the shell model, a property which is shared only in a
particular case of the running sandpile model. The breakdown of self-similarity
generated by isolated events observed in the turbulent samples, is well
reproduced by the shell model, while it is absent in all SOC models considered.
On this base, we conclude that SOC models are not adequate to mimic fluid
turbulence, while the GOY shell model constitutes a better candidate to
describe the gross features of turbulence.


title:
Effects of small surface tension in Hele-Shaw multifinger dynamics: an
  analytical and numerical study

abstract:
  We study the singular effects of vanishingly small surface tension on the
dynamics of finger competition in the Saffman-Taylor problem, using the
asymptotic techniques described in [S. Tanveer, Phil. Trans. R. Soc. Lond. A
343, 155 (1993)]and [M. Siegel, and S. Tanveer, Phys. Rev. Lett. 76, 419
(1996)] as well as direct numerical computation, following the numerical scheme
of [T. Hou, J. Lowengrub, and M. Shelley,J. Comp. Phys. 114, 312 (1994)]. We
demonstrate the dramatic effects of small surface tension on the late time
evolution of two-finger configurations with respect to exact (non-singular)
zero surface tension solutions. The effect is present even when the relevant
zero surface tension solution has asymptotic behavior consistent with selection
theory.Such singular effects therefore cannot be traced back to steady state
selection theory, and imply a drastic global change in the structure of
phase-space flow. They can be interpreted in the framework of a recently
introduced dynamical solvability scenario according to which surface tension
unfolds the structually unstable flow, restoring the hyperbolicity of
multifinger fixed points.


title: Generalized Benney Lattice and the Heavenly Equation

abstract:
  We generalize the Benney lattice and show that the new system of equations
can be reduced to a generalized Chaplygin gas as well as the heavenly equation.
We construct two infinite sets of conserved charges and show that one of the
sets can be obtained from the Lax function. The conserved densities are related
to Legendre polynomials and we present closed form expressions for the
generating functions for these densities which also determines the Riemann
invariants of the problem. We prove that the system is bi-Hamiltonian and that
the conserved charges are in involution with respect to either of the
Hamiltonian structures. We show that the associated generalized elastic medium
equations are bi-Hamiltonian as well. We also bring out various other
interesting features of our model.


title: Partition functions for Matrix Models and Isomonodromic Tau functions

abstract:
  For one-matrix models with polynomial potentials, the explicit relationship
between the partition function and the isomonodromic tau function for the 2x2
polynomial differential systems satisfied by the associated orthogonal
polynomials is derived.


title: Scar functions in the Bunimovich Stadium billiard

abstract:
  In the context of the semiclassical theory of short periodic orbits, scar
functions play a crucial role. These wavefunctions live in the neighbourhood of
the trajectories, resembling the hyperbolic structure of the phase space in
their immediate vicinity. This property makes them extremely suitable for
investigating chaotic eigenfunctions. On the other hand, for all practical
purposes reductions to Poincare sections become essential. Here we give a
detailed explanation of resonances and scar functions construction in the
Bunimovich stadium billiard and the corresponding reduction to the boundary.
Moreover, we develop a method that takes into account the departure of the
unstable and stable manifolds from the linear regime. This new feature extends
the validity of the expressions.


title: Algebraic decay in hierarchical graphs

abstract:
  We study the algebraic decay of the survival probability in open hierarchical
graphs. We present a model of a persistent random walk on a hierarchical graph
and study the spectral properties of the Frobenius-Perron operator. Using a
perturbative scheme, we derive the exponent of the classical algebraic decay in
terms of two parameters of the model. One parameter defines the geometrical
relation between the length scales on the graph, and the other relates to the
probabilities for the random walker to go from one level of the hierarchy to
another. The scattering resonances of the corresponding hierarchical quantum
graphs are also studied. The width distribution shows the scaling behavior
$P(\Gamma) \sim 1/\Gamma$.


title:
Coupled Replicator Equations for the Dynamics of Learning in Multiagent
  Systems

abstract:
  Starting with a group of reinforcement-learning agents we derive coupled
replicator equations that describe the dynamics of collective learning in
multiagent systems. We show that, although agents model their environment in a
self-interested way without sharing knowledge, a game dynamics emerges
naturally through environment-mediated interactions. An application to
rock-scissors-paper game interactions shows that the collective learning
dynamics exhibits a diversity of competitive and cooperative behaviors. These
include quasiperiodicity, stable limit cycles, intermittency, and deterministic
chaos--behaviors that should be expected in heterogeneous multiagent systems
described by the general replicator equations we derive.


title: Higher-order solitons in the N-wave system

abstract:
  The soliton dressing matrices for the higher-order zeros of the
Riemann-Hilbert problem for the $N$-wave system are considered. For the
elementary higher-order zero, i.e. whose algebraic multiplicity is arbitrary
but the geometric multiplicity is 1, the general soliton dressing matrix is
derived. The theory is applied to the study of higher-order soliton solutions
in the three-wave interaction model. The simplest higher-order soliton solution
is presented. In the generic case, this solution describes the breakup of a
higher-order pumping wave into two higher-order elementary waves, and the
reverse process. In non-generic cases, this solution could describe ($i$) the
merger of a pumping sech wave and an elementary sech wave into two elementary
waves (one sech and the other one higher-order); ($ii)$ the breakup of a
higher-order pumping wave into two elementary sech waves and one pumping sech
wave; and the reverse processes. This solution could also reproduce fundamental
soliton solutions as a special case.


title:
Generalized dimensions of Feigenbaum's attractor from
  renormalization-group functional equations

abstract:
  A method is suggested for the computation of the generalized dimensions of
fractal attractors at the period-doubling transition to chaos. The approach is
based on an eigenvalue problem formulated in terms of functional equations,
with a coefficient expressed in terms of Feigenbaum's universal fixed-point
function. The accuracy of the results is determined only by precision of the
representation of the universal function.


title: Floquet Stability Analysis of Ott-Grebogi-Yorke and Difference Control

abstract:
  Stabilization of instable periodic orbits of nonlinear dynamical systems has
been a widely explored field theoretically and in applications. The techniques
can be grouped in time-continuous control schemes based on Pyragas, and the two
Poincar\'e-based chaos control schemes, Ott-Gebogi-Yorke (OGY) and difference
control. Here a new stability analysis of these two Poincar\'e-based chaos
control schemes is given by means of Floquet theory. This approach allows to
calculate exactly the stability restrictions occuring for small measurement
delays and for an impulse length shorter than the length of the orbit. This is
of practical experimental relevance; to avoid a selection of the relative
impulse length by trial and error, it is advised to investigate whether the
used control scheme itself shows systematic limitations on the choice of the
impulse length. To investigate this point, a Floquet analysis is performed. For
OGY control the influence of the impulse length is marginal. As an unexpected
result, difference control fails when the impulse length is taken longer than a
maximal value that is approximately one half of the orbit length for small
Ljapunov numbers and decreases with the Ljapunov number.


title:
Numerical aspects of eigenvalue and eigenfunction computations for
  chaotic quantum systems

abstract:
  We give an introduction to some of the numerical aspects in quantum chaos.
The classical dynamics of two--dimensional area--preserving maps on the torus
is illustrated using the standard map and a perturbed cat map. The quantization
of area--preserving maps given by their generating function is discussed and
for the computation of the eigenvalues a computer program in Python is
presented. We illustrate the eigenvalue distribution for two types of perturbed
cat maps, one leading to COE and the other to CUE statistics. For the
eigenfunctions of quantum maps we study the distribution of the eigenvectors
and compare them with the corresponding random matrix distributions. The Husimi
representation allows for a direct comparison of the localization of the
eigenstates in phase space with the corresponding classical structures.
Examples for a perturbed cat map and the standard map with different parameters
are shown. Billiard systems and the corresponding quantum billiards are another
important class of systems (which are also relevant to applications, for
example in mesoscopic physics). We provide a detailed exposition of the
boundary integral method, which is one important method to determine the
eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several
methods to determine the eigenvalues from the Fredholm equation and illustrate
them for the stadium billiard. The occurrence of spurious solutions is
discussed in detail and illustrated for the circular billiard, the stadium
billiard, and the annular sector billiard. We emphasize the role of the normal
derivative function to compute the normalization of eigenfunctions, momentum
representations or autocorrelation functions in a very efficient and direct
way. Some examples for these quantities are given and discussed.


title: Boolean Dynamics with Random Couplings

abstract:
  This paper reviews a class of generic dissipative dynamical systems called
N-K models. In these models, the dynamics of N elements, defined as Boolean
variables, develop step by step, clocked by a discrete time variable. Each of
the N Boolean elements at a given time is given a value which depends upon K
elements in the previous time step.
  We review the work of many authors on the behavior of the models, looking
particularly at the structure and lengths of their cycles, the sizes of their
basins of attraction, and the flow of information through the systems. In the
limit of infinite N, there is a phase transition between a chaotic and an
ordered phase, with a critical phase in between.
  We argue that the behavior of this system depends significantly on the
topology of the network connections. If the elements are placed upon a lattice
with dimension d, the system shows correlations related to the standard
percolation or directed percolation phase transition on such a lattice. On the
other hand, a very different behavior is seen in the Kauffman net in which all
spins are equally likely to be coupled to a given spin. In this situation,
coupling loops are mostly suppressed, and the behavior of the system is much
more like that of a mean field theory.
  We also describe possible applications of the models to, for example, genetic
networks, cell differentiation, evolution, democracy in social systems and
neural networks.


title:
Alternative dispersionless limit of N=2 supersymmetric KdV-type
  hierarchies

abstract:
  We present a systematic procedure for obtaining the dispersionless limit of a
class of N=1 supersymmetric systems starting from the Lax description of their
dispersive counterparts. This is achieved by starting with an N=2
supersymmetric system and scaling the fields in an alternative manner so as to
maintain N=1 supersymmetry. We illustrate our method by working out explicitly
the examples of the dispersionless supersymmetric two boson hierarchy and the
dispersionless supersymmetric Boussinesq hierarchy.


title: Eigenstates Ignoring Regular and Chaotic Phase-Space Structures

abstract:
  We report the failure of the semiclassical eigenfunction hypothesis if
regular classical transport coexists with chaotic dynamics. All eigenstates,
instead of being restricted to either a regular island or the chaotic sea,
ignore these classical phase-space structures. We argue that this is true even
in the semiclassical limit for extended systems with transporting regular
islands such as the standard map with accelerator modes.


title: Scattering of a solitary pulse on a local defect or breather

abstract:
  A model is introduced to describe guided propagation of a linear or nonlinear
pulse which encounters a localized nonlinear defect, that may be either static
or breather-like one (the model with the static defect applies to an optical
pulse in a long fiber link with an inserted additional section of a nonlinear
fiber). In the case when the host waveguide is linear, the pulse has a Gaussian
shape. In that case, an immediate result of its interaction with the nonlinear
defect can be found in an exact analytical form, amounting to transformation of
the incoming Gaussian into an infinite array of overlapping Gaussian pulses.
Further evolution of the array in the linear host medium is found numerically
by means of the Fourier transform. An important ingredient of the linear medium
is the third-order dispersion, that eventually splits the array into individual
pulses. If the host medium is nonlinear, the input pulse is taken as a
fundamental soliton. The soliton is found to be much more resistant to the
action of the nonlinear defect than the Gaussian pulse in the linear host
medium. In this case, the third-order-dispersion splits the soliton proper and
wavepackets generated by the action of the defect.


title:
On the Algebraic Bethe Ansatz for XXX spin chain: creation operators
  "beyond the equator"

abstract:
  Considering the XXX spin-1/2 chain in the framework of the Algebraic Bethe
Ansatz (ABA) we make the following short comment: the product of the creation
operators corresponding to the recently found solution of the Bethe equations
"on the wrong side of the equator" (hep-th/9808153) is just zero (not only its
action on the pseudovacuum).


title:
Spatial solitons in a medium composed of self-focusing and
  self-defocusing layers

abstract:
  We introduce a model combining Kerr nonlinearity with a periodically changing
sign ("nonlinearity management") and a Bragg grating (BG). The main result,
obtained by means of systematic simulations, is presented in the form of a
soliton's stability diagram on the parameter plane of the model; the diagram
turns out to be a universal one, as it practically does not depend on the
soliton's power. Moreover, simulations of the nonlinear Schroedinger (NLS)
model subjected to the same "nonlinearity management" demonstrate that the same
diagram determines the stability of the NLS solitons, unless they are very
narrow. The stability region of very narrow NLS solitons is much smaller, and
soliton splitting is readily observed in that case. The universal diagram shows
that a minimum non-zero average value of the Kerr coefficient is necessary for
the existence of stable solitons. Interactions between identical solitons with
an initial phase difference between them are simulated too in the BG model,
resulting in generation of stable moving solitons. A strong spontaneous
symmetry breaking is observed in the case when in-phase solitons pass through
each other due to attraction between them.


title: Relaxation and Noise in Chaotic Systems

abstract:
  For a class of idealized chaotic systems (hyperbolic systems) correlations
decay exponentially in time. This result is asymptotic and rigorous. The decay
rate is related to the Ruelle-Pollicott resonances. Nearly all chaotic model
systems, that are studied by physicists, are not hyperbolic. For many such
systems it is known that exponential decay takes place for a long time. It may
not be asymptotic, but it may persist for a very long time, longer than any
time of experimental relevance. In this review a heuristic method for
calculation of this exponential decay of correlations in time is presented. It
can be applied to model systems, where there are no rigorous results concerning
this exponential decay. It was tested for several realistic systems (kicked
rotor and kicked top) in addition to idealized systems (baker map and perturbed
cat map). The method consists of truncation of the evolution operator
(Frobenius-Perron operator), and performing all calculations with the resulting
finite dimensional matrix. This finite dimensional approximation can be
considered as coarse graining, and is equivalent to the effect of noise. The
exponential decay rate of the chaotic system is obtained when the
dimensionality of the approximate evolution operator is taken to infinity,
resulting in infinitely fine resolution, that is equivalent to vanishing noise.
The corresponding Ruelle-Pollicott resonances can be calculated for many
systems that are beyond the validity of the Ruelle-Pollicott theorem.


title: Stretching and Curvature of Material Lines in Chaotic Flows

abstract:
  As a streak of dye is advected by a chaotic flow, it stretches and folds and
becomes indistinguishable from a one-dimensional idealized material line. The
variation along a material line of the total stretching experienced by fluid
elements is examined, and it is found that it can be decomposed into an overall
time-dependent factor, constant along the line, and a smooth time-independent
deviation. The stretching is related by a power law to the curvature of the
line near sharp bends. This is confirmed numerically and motivated by a simple
model. A conservation law for Lyapunov exponents explains deviations from a
power-law.


title: The space of initial conditions for linearisable mappings

abstract:
  We apply the algebraic-geometric techniques developed for the study of
mappings which have the singularity confinement property to mappings which are
integrable through linearisation. The main difference with respect to the
previous studies is that the linearisable mappings have generically unconfined
singularities. Despite this fact we are able to provide a complete description
of the dynamics of these mappings and derive rigorously their growth
properties.


title: The semiclassical limit of chaotic eigenfunctions

abstract:
  A generic chaotic eigenfunction has a non-universal contribution consisting
of scars of short periodic orbits. This contribution, which can not be
explained in terms of random universal waves, survives the semiclassical limit
(when $\hbar$ goes to zero). In this limit, the sum of scarred intensities is a
simple function of $\eta\equiv \sqrt{\pi /2} (f-1) h^{-1}_T (\sum
\lambda_i^2)^{1/2} $, with $f$ the degrees of freedom, $h_T$ the topological
entropy and $\{\lambda_i\}$ the set of positive Lyapunov exponents. Moreover,
the fluctuations of this representation go to zero as $1/|\ln \hbar|$. For this
reasson, we will be able to provide a detailed description of a generic chaotic
eigenfunction in the semiclassical limit.


title: A Study Of A New Class Of Discrete Nonlinear Schroedinger Equations

abstract:
  A new class of 1D discrete nonlinear Schr${\ddot{\rm{o}}}$dinger Hamiltonians
with tunable nonlinerities is introduced, which includes the integrable
Ablowitz-Ladik system as a limit. A new subset of equations, which are derived
from these Hamiltonians using a generalized definition of Poisson brackets, and
collectively refered to as the N-AL equation, is studied. The symmetry
properties of the equation are discussed. These equations are shown to possess
propagating localized solutions, having the continuous translational symmetry
of the one-soliton solution of the Ablowitz-Ladik nonlinear
Schr${\ddot{\rm{o}}}$dinger equation. The N-AL systems are shown to be suitable
to study the combined effect of the dynamical imbalance of nonlinearity and
dispersion and the Peierls-Nabarro potential, arising from the lattice
discreteness, on the propagating solitary wave like profiles. A perturbative
analysis shows that the N-AL systems can have discrete breather solutions, due
to the presence of saddle center bifurcations in phase portraits. The
unstaggered localized states are shown to have positive effective mass. On the
other hand, large width but small amplitude staggered localized states have
negative effective mass. The collison dynamics of two colliding solitary wave
profiles are studied numerically. Notwithstanding colliding solitary wave
profiles are seen to exhibit nontrivial nonsolitonic interactions, certain
universal features are observed in the collison dynamics. Future scopes of this
work and possible applications of the N-AL systems are discussed.


title: Turbulent Pair Diffusion

abstract:
  Kinematic Simulations of turbulent pair diffusion in planar turbulence with a
-5/3 energy spectrum reproduce the results of the laboratory measurements of
Jullien Phys. Rev. Lett. 82, 2872 (1999), in particular the stretched
exponential form of the PDF of pair separations and their correlation
functions. The root mean square separation is found to be strongly dependent on
initial conditions for very long stretches of times. This dependence is
consistent with the topological picture of turbulent pair diffusion where pairs
initially close enough travel together for long stretches of time and separate
violently when they meet straining regions around hyperbolic points. A new
argument based on the divergence of accelerations is given to support this
picture.


title: Phase Structure of Resource Allocation Games

abstract:
  We consider a class of games that are generalizations of the minority game,
in that the demand and supply of the resource are specified independently. This
allows us to study systems in which agents compete under different demand
loads. Among other features, we find the existence of a robust phase change
with a coexistence region as the demand load is varied, separating regions with
nearly balanced supply and demand from regions of scarce or abundant resources.
The coexistence region exists when the amount of information used by the agents
to make their choices is greater than a critical value, which is related to the
point at which there is a phase transition in the standardd minority game.


title: Travelling fronts in an array of coupled symmetric bistable units

abstract:
  A symmetry breaking mechanism is shown to occur in an array composed of
symmetric bistable Lorenz units coupled through a nearest neighbour scheme.
When the coupling is increased, we observe the route: standing --> oscillating
--> travelling front. In some circumstances, this route can be described in
terms of a gluing of two cycles on the plane. In this case, the asymptotic
behaviour of the velocity of the front is found straightforward. However, it
may also happen that the gluing bifurcation involves a saddle-focus fixed
point. If so, front dynamics may become quite complex displaying several
oscillating and propagating regimes, including chaotic (Shil'nikov-type) front
propagation. Finally, we compare our results with different couplings and with
other discrete bistable media.


title: Stable Quantum Resonances in Atom Optics

abstract:
  A theory for stabilization of quantum resonances by a mechanism similar to
one leading to classical resonances in nonlinear systems is presented. It
explains recent surprising experimental results, obtained for cold Cesium atoms
when driven in the presence of gravity, and leads to further predictions. The
theory makes use of invariance properties of the system, that are similar to
those of solids, allowing for separation into independent kicked rotor
problems. The analysis relies on a fictitious classical limit where the small
parameter is {\em not} Planck's constant, but rather the detuning from the
frequency that is resonant in absence of gravity.


title: Generalized synchronization of chaos in non-invertible maps

abstract:
  The properties of functional relation between a non-invertible chaotic drive
and a response map in the regime of generalized synchronization of chaos are
studied. It is shown that despite a very fuzzy image of the relation between
the current states of the maps, the functional relation becomes apparent when a
sufficient interval of driving trajectory is taken into account. This paper
develops a theoretical framework of such functional relation and illustrates
the main theoretical conclusions using numerical simulations.


title: Random Coupling of Chaotic Maps leads to Spatiotemporal Synchronisation

abstract:
  We investigate the spatiotemporal dynamics of a network of coupled chaotic
maps, with varying degrees of randomness in coupling connections. While
strictly nearest neighbour coupling never allows spatiotemporal synchronization
in our system, randomly rewiring some of those connections stabilises entire
networks at $x^*$, where $x^*$ is the strongly unstable fixed point solution of
the local chaotic map. In fact, the smallest degree of randomness in spatial
connections opens up a window of stability for the synchronised fixed point in
coupling parameter space. Further, the coupling $\epsilon_{bifr}$ at which the
onset of spatiotemporal synchronisation occurs, scales with the fraction of
rewired sites p as a power law, for 0.1 < p < 1. We also show that the
regularising effect of random connections can be understood from stability
analysis of the probabilistic evolution equation for the system, and
approximate analytical expressions for the range and $\epsilon_{bifr}$ are
obtained.