title: Limit Cycle and Conserved Dynamics

abstract:
  We demonstrate that a potential coexists with limit cycle. Here the potential
determines the final distribution of population. Our demonstration consists of
three steps: We first show the existence of limit from a typical physical
sciences setting: the potential is a type of Mexican hat type, with the
strength of a magnetic field scale with the strength the potential gradient
near the limit cycle, and the friction goes to zero faster than the potential
near the limit cycle. Hence the dynamics at the limit cycle is conserved. The
diffusion matrix is nevertheless finite at the limit cycle. Secondly, we
construct the potential in the dynamics with limit cycle in a typical dynamical
systems setting. Thirdly, we argue that such a construction can be carried out
in a more general situation based on a method discovered by one of us. This
method of dealing with stochastic differential equation is in general different
from both Ito and Stratonovich calculus. Our result may be useful in many
related applications, such as in the discussion of metastability of limit cycle
and in the construction of Hopfield potential in the neural network
computation.


title:
Wave breaking and the generation of undular bores in an integrable
  shallow-water system

abstract:
  The generation of an undular bore in the vicinity of a wave-breaking point is
considered for the integrable Kaup-Boussinesq shallow water system. In the
framework of the Whitham modulation theory, an analytic solution of the
Gurevich-Pitaevskii type of problem for a generic "cubic" breaking regime is
obtained using a generalized hodograph transform, and a further reduction to a
linear Euler-Poisson equation. The motion of the undular bore edges is
investigated in detail.


title: Analytic model for a frictional shallow-water undular bore

abstract:
  We use the integrable Kaup-Boussinesq shallow water system, modified by a
small viscous term, to model the formation of an undular bore with a steady
profile. The description is made in terms of the corresponding integrable
Whitham system, also appropriately modified by friction. This is derived in
Riemann variables using a modified finite-gap integration technique for the
AKNS scheme. The Whitham system is then reduced to a simple first-order
differential equation which is integrated numerically to obtain an asymptotic
profile of the undular bore, with the local oscillatory structure described by
the periodic solution of the unperturbed Kaup-Boussinesq system. This solution
of the Whitham equations is shown to be consistent with certain jump conditions
following directly from conservation laws for the original system. A comparison
is made with the recently studied dissipationless case for the same system,
where the undular bore is unsteady.


title:
Statistical physics and stromatolite growth: new perspectives on an
  ancient dilemma

abstract:
  This paper outlines our recent attempts to model the growth and form of
microbialites from the perspective of the statistical physics of evolving
surfaces. Microbialites arise from the environmental interactions of microbial
communities (microbial mats). The mats evolve over time to form internally
laminated organosedimentary structures (stromatolites). Modern day
stromatolites exist in only a few locations, whereas ancient stromatolitic
microbialites were the only form of life for much of the Earth's history. They
existed in a wide variety of growth forms, ranging from almost perfect cones to
branched columnar structures. The coniform structures are central to the heated
debate on the oldest evidence of life. We proposed a biotic model which
considers the relationship between upward growth of a phototropic or
phototactic biofilm and mineral accretion normal to the surface. These
processes are sufficient to account for the growth and form of many ancient
stromatolities. These include domical stromatolites and coniform structures
with thickened apical zones typical of Conophyton. More angular coniform
structures, similar to the stromatolites claimed as the oldest macroscopic
evidence of life, form when the photic effects dominate over mineral accretion.


title: Scarring in a driven system with wave chaos

abstract:
  We consider acoustic wave propagation in a model of a deep ocean acoustic
waveguide with a periodic range-dependence. Formally, the wave field is
described by the Schrodinger equation with a time-dependent Hamiltonian. Using
methods borrowed from the quantum chaos theory it is shown that in the driven
system under consideration there exists a "scarring" effect similar to that
observed in autonomous quantum systems.


title:
Double waves in multi-dimensional systems of hydrodynamic type: a
  necessary condition for integrability

abstract:
  An invariant differential-geometric approach to the integrability of
(2+1)-dimensional systems of hydrodynamic type u_t+A(u)u_x+B(u)u_y=0 is
developed. It is proved that the existence of special solutions known as
`double waves' is equivalent to the diagonalizability of an arbitrary matrix of
the two-parameter family (kE+A)^{-1}(lE+B). Since the diagonalizability can be
effectively verified by differential-geometric means, this provides a simple
necessary condition for integrability.


title: Separation of variables on a non-hyperelliptic curve

abstract:
  In the paper we consider several dynamical systems that admit a separation of
variables on the algebraic curve of genus 4. The main result of the paper is an
explicit formula for the action of these systems. We find it directly from the
Hamilton-Jacobi equation. We find the action and a separation of variables for
the Kowalewski hyrostat, the Clebsch and the so(4) Schottky-Manakov spinning
tops.


title:
A new constrained mKP hierarchy and the generalized Darboux
  transformation for the mKP equation with self-consistent sources

abstract:
  The mKP equation with self-consistent sources (mKPESCS) is treated in the
framework of the constrained mKP hierarchy. We introduce a new constrained mKP
hierarchy which may be viewed as the stationary hierarchy of the mKP hierarchy
with self-consistent sources. This offers a natural way to obtain the Lax
representation for the mKPESCS. Based on the conjugate Lax pairs, we construct
the generalized Darboux transformation with arbitrary functions in time $t$ for
the mKPESCS which, in contrast with the Darboux transformation for the mKP
equation, provides a non-auto-B\"{a}cklund transformation between two mKPESCSs
with different degrees. The formula for $n$-times repeated generalized Darboux
transformation is proposed and enables us to find the rational solutions
(including the lump solutions), soliton solutions and the solutions of breather
type of the mKPESCS.


title: $q$-analogue of modified KP hierarchy and its quasi-classical limit

abstract:
  A $q$-analogue of the tau function of the modified KP hierarchy is defined by
a change of independent variables. This tau function satisfies a system of
bilinear $q$-difference equations. These bilinear equations are translated to
the language of wave functions, which turn out to satisfy a system of linear
$q$-difference equations. These linear $q$-difference equations are used to
formulate the Lax formalism and the description of quasi-classical limit. These
results can be generalized to a $q$-analogue of the Toda hierarchy. The results
on the $q$-analogue of the Toda hierarchy might have an application to the
random partition calculus in gauge theories and topological strings.


title:
Transition from Uniform Motion to Stick-Slip in the Rice-Ruina Model of
  One and Two Blocks

abstract:
  We analyze the Rice-Ruina state and rate dependent friction model. The system
consists of one or two blocks driven by springs with constant velocity on a
dry, rough surface. Our discussion is limited to the creep-like motion, when
the driving velocity is small. Two regimes of motion are observed: stick-slip
and steady sliding. The stability of the steady sliding depends on the model
parameters. Numerical and analytical results show a transition between two
regimes: the system passes directly from uniform to stick-slip motion. The
calculations are performed also for two driven blocks. Then, the transition has
the same character and it appears at the same point.


title: Long Tailed Maps as a Representation of Mixed Mode Oscillatory Systems

abstract:
  Mixed mode oscillatory (MMO) systems are known to exhibit some generic
features such as the reversal of period doubling sequences and crossover to
period adding sequences as bifurcation parameters are varied. In addition, they
exhibit a nearly one dimensional unimodal Poincare map with a longtail. We
recover these common features from a general class of two parameter family of
one dimensional maps with a unique critical point that satisfy a few general
constraints that determine the nature of the map. We derive scaling laws that
determine the parameter widths of the dominant windows of periodic orbits
sandwiched between two successive states of RL^k sequence. An example of a two
parameter map with a unique critical point is introduced to verify the
analytical results.


title:
Generalized Darboux transformations for the KP equation with
  self-consistent sources

abstract:
  The KP equation with self-consistent sources (KPESCS) is treated in the
framework of the constrained KP equation. This offers a natural way to obtain
the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we
construct the generalized binary Darboux transformation with arbitrary
functions in time $t$ for the KPESCS which, in contrast with the binary Darboux
transformation of the KP equation, provides a non-auto-B\"{a}cklund
transformation between two KPESCSs with different degrees. The formula for
N-times repeated generalized binary Darboux transformation is proposed and
enables us to find the N-soliton solution and lump solution as well as some
other solutions of the KPESCS.


title: The Solutions of the NLS Equations with Self-Consistent Sources

abstract:
  We construct the generalized Darboux transformation with arbitrary functions
in time $t$ for the AKNS equation with self-consistent sources (AKNSESCS)
which, in contrast with the Darboux transformation for the AKNS equation,
provides a non-auto-B\"{a}cklund transformation between two AKNSESCSs with
different degrees of sources. The formula for N-times repeated generalized
Darboux transformation is proposed. By reduction the generalized Darboux
transformation with arbitrary functions in time $t$ for the Nonlinear
Schr\"{o}dinger equation with self-consistent sources (NLSESCS) is obtained and
enables us to find the dark soliton, bright soliton and positon solutions for
NLS$^{+}$ESCS and NLS$^{-}$ESCS. The properties of these solution are analyzed.


title: Gurevich-Zybin system

abstract:
  We present three different linearizable extensions of the Gurevich-Zybin
system. Their general solutions are found by reciprocal transformations. In
this paper we rewrite the Gurevich-Zybin system as a Monge-Ampere equation. By
application of reciprocal transformation this equation is linearized.
Infinitely many local Hamiltonian structures, local Lagrangian representations,
local conservation laws and local commuting flows are found. Moreover, all
commuting flows can be written as Monge-Ampere equations similar to the
Gurevich-Zybin system. The Gurevich-Zybin system describes the formation of a
large scale structures in the Universe. The second harmonic wave generation is
known in nonlinear optics. In this paper we prove that the Gurevich-Zybin
system is equivalent to a degenerate case of the second harmonic generation.
Thus, the Gurevich-Zybin system is recognized as a degenerate first negative
flow of two-component Harry Dym hierarchy up to two Miura type transformations.
A reciprocal transformation between the Gurevich-Zybin system and degenerate
case of the second harmonic generation system is found. A new solution for the
second harmonic generation is presented in implicit form.


title:
Suppression of chaos by weak resonant excitations in a nonlinear
  oscillator with a non-symmetric potential

abstract:
  We examine the Mielnikov criterion for transition to chaos in case of one
degree of freedom nonlinear oscillator with non symmetric potential. This
system subjected to an external periodic force shows homoclinic transition from
regular vibrations to chaos just before escape from the potential well.
Especially we study the effect of a second resonant excitation with different
phase on the system transition to chaos and propose a way of its control.


title:
Phase space structure of nonautonomous one-degree-of-freedom Hamiltonian
  systems with nontrivial time dependence

abstract:
  It has been recently argued that near-integrable nonautonomous
one-degree-of-freedom Hamiltonian systems are constrained by KAM theory even
when the time-dependent (nonintegrable) part of the Hamiltonian is given in the
form of a superposition of time-periodic functions with incommensurate
frequencies. Furthermore, such systems are constrained by one fewer integral of
motion than is required to render the system completely integrable. As a
consequence, the phase space in systems of this type is expected to be
partitioned into nonintersecting regular and chaotic regions. In this note we
provide numerical evidence of the existence of such a characteristic mixed
phase space structure. This is done by considering the problem of acoustic ray
dynamics in deep ocean environments, which is naturally described as a
nonautonomous one-degree-of-freedom Hamiltonian system with a multiply periodic
Hamiltonian in the independent (time-like) variable. Also, we discuss the
implications of a mixed phase space for the dynamics of that geophysical system
and another one which describes Lagrangian motion in the ocean. The latter is
also naturally described as a nonautonomous one-degree-of-freedom Hamiltonian
system with a multiply time-periodic Hamiltonian.


title: The symmetries of the Fokker - Planck equation in one dimension

abstract:
  We calculate all point symmetries of the Fokker - Planck equation in
one-dimensional Euclidean space. General expression of symmetry group action on
arbitrary solution of Fokker - Planck equation is presented. We propose new
notation for the group-theoretic analysis of PDE. The Lie prolongation formula
is derived as an example of the new notation.


title:
The Kolmogorov-Sinai entropy for dilute systems of hard particles in
  equilibrium

abstract:
  In an equilibrium system, the Kolmogorov-Sinai entropy, $h_{\mathrm{KS}}$,
equals the sum of the positive Lyapunov exponents, the exponential rates of
divergence of infinitesimal perturbations. Kinetic theory may be used to
calculate the Kolmogorov-Sinai entropy for dilute gases of many hard disks or
spheres in equilibrium at low number density $n$. The density expansion of
$h_{\mathrm{KS}}$ is $N \bar\nu A [\ln n + B + O(n)]$, where $\bar\nu$ is the
single-particle collision frequency. Previous calculations of $A$ were
succesful. Calculations of $B$, however, were unsatisfactory. In this paper, I
show how the probability distribution of the stretching factor can be
determined from a nonlinear differential equation by an iterative method. From
this the Kolmogorov-Sinai entropy follows as the average of the logarithm of
the stretching factor per unit time. I calculate approximate values of $B$ and
compare these to results from existing simulations. The agreement is good.


title:
Equatorial and related non-equilibrium states in magnetization dynamics
  of ferromagnets: Generalization of Suhl's spin-wave instabilities

abstract:
  We investigate the nonlinear dynamics underlying the evolution of a 2-D
nanoscale ferromagnetic film with uniaxial anisotropy in the presence of
perpendicular pumping. Considering the associated Landau-Lifshitz spin
evolution equation with Gilbert damping together with Maxwell equation for the
demagnetization field, we study the dynamics in terms of the stereographic
variable. We identify several new fixed points for suitable choice of external
field in a rotating frame of reference. In particular, we identify explicit
equatorial and related fixed points of the spin vector in the plane transverse
to the anisotropy axis when the pumping frequency coincides with the amplitude
of the static parallel field. We then study the linear stability of these novel
fixed points under homogeneous and spin wave perturbations and obtain a
generalized Suhl's instability criterion, giving the condition for exponential
growth of P-modes under spin wave perturbations. Two parameter phase diagrams
(in terms of amplitudes of static parallel and oscillatory perpendicular
magnetic fields) for stability are obtained, which differ qualitatively from
those for the conventional ferromagnetic resonance near thermal equilibrium and
are amenable to experimental tests.


title:
An algebraic scheme associated with the noncommutative KP hierarchy and
  some of its extensions

abstract:
  A well-known ansatz (`trace method') for soliton solutions turns the
equations of the (noncommutative) KP hierarchy, and those of certain
extensions, into families of algebraic sum identities. We develop an algebraic
formalism, in particular involving a (mixable) shuffle product, to explore
their structure. More precisely, we show that the equations of the
noncommutative KP hierarchy and its extension (xncKP) in the case of a
Moyal-deformed product, as derived in previous work, correspond to identities
in this algebra. Furthermore, the Moyal product is replaced by a more general
associative product. This leads to a new even more general extension of the
noncommutative KP hierarchy. Relations with Rota-Baxter algebras are
established.


title: Chaos and Control in Nonlinear Bloch System

abstract:
  The dynamics of two nonlinear Bloch systems is studied from the viewpoint of
bifur- cation and a particular parameter space has been explored for the
stability analysis based on stability criterion. This enables the choice of the
desired unstable periodic orbit from the numerous unstable ones present within
the attractor through the pro- cess of closed return pairs. A generalized
active control method have been discussed for two Bloch systems arising from di
erent initial conditions.


title:
Moment method, Higher order dispersion map and other effects in optical
  pulse propagation

abstract:
  Analytical and numerical procedures are applied to show that both third and
second order dispersion maps can be explicitly constructed and their mutual
effects on the optical pulse propagation are analysed. In these connection it
is also shown how the other important features such as amplification,
intra-channel Raman Scattering(IRS), fibre loss, centre frequency of the pulse
spectrum effect the propagation of pulse. Due to the presence of IRS, moment
method is adopted which is easily reduced to usual Hamiltonian technique in
absence of IRS. It is observed that the long time periodic behaviour is
stabilized by the third order dispersion map. Though the introduction of IRS
and other effects perturb the situation, one can observe that except for a very
special case it remains stable


title: Anomalous scaling of passive scalar in turbulence and in equilibrium

abstract:
  We analyze multi-point correlation functions of a tracer in an incompressible
flow at scales far exceeding the scale $L$ at which fluctuations are generated
(quasi-equilibrium domain) and compare them with the correlation functions at
scales smaller than $L$ (turbulence domain). We demonstrate that the scale
invariance can be broken in the equilibrium domain and trace this breakdown to
the statistical integrals of motion (zero modes) as has been done before for
turbulence. Employing Kraichnan model of short-correlated velocity we identify
the new type of zero modes, which break scale invariance and determine an
anomalously slow decay of correlations at large scales.

categories: nlin.CD cond-mat.stat-mech hep-th physics.flu-dyn


title: The large-scale structure of passive scalar turbulence

abstract:
  We investigate the large-scale statistics of a passive scalar transported by
a turbulent velocity field. At scales larger than the characteristic
lengthscale of scalar injection, yet smaller than the correlation length of the
velocity, the advected field displays persistent long-range correlations due to
the underlying turbulent velocity. These induce significant deviations from
equilibrium statistics for high-order scalar correlations, despite the absence
of scalar flux.


title: The symmetries of the Fokker - Planck equation in two dimensions

abstract:
  We calculate all point symmetries of the Fokker - Planck equation in
two-dimensional Euclidean space. General expression of symmetry group action on
arbitrary solution of Fokker - Planck equation is presented.


title: Aggregation of finite size particles with variable mobility

abstract:
  New model equations are derived for dynamics of self-aggregation of
finite-size particles. Differences from standard Debye-Huckel and Keller-Segel
models are: the mobility of particles depends on the configuration of their
neighbors and linear diffusion acts on locally-averaged particle density. The
evolution of collapsed states in these models reduces exactly to
finite-dimensional dynamics of interacting particle clumps. Simulations show
these collapsed (clumped) states emerge from smooth initial conditions, even in
one spatial dimension.


title: Dynamics of distributed sources

abstract:
  The dynamics of distributed sources is described by nonlinear partial
differential equations. Lagrangian analytical solutions of these (and
associated) equations are obtained and discussed in the context of Lagrangian
modeling - from the Lagrangian invariants to dynamics. Possible applications of
distributed sources and sinks to geophysical fluid dynamics and to the
cosmology are indicated.


title:
Solitary Optical pulse propagation in fused fibre coupler, Effect of
  Raman Scattering and switching

abstract:
  We analyse the motion of solitary type optical pulse in a fused optical fibre
coupler when inter channel raman scattering (IRS) is taken into account and
same type of dispersion management is used for both the core. The paper is
divided into two sections . In the first half we utilise a moment method to
study the variations of the different parameters of the Gaussian pulse, such as
chirp, frequency, energy etc. while in the second half assuming a general form
of the stationary pulse we discuss the switching phenomena due to the coupling.
It is shown that lauching a pulse in one arm of the coupler can generate pulses
in both the arms, but the patterns differs in a significant manner when the
strength of IRS gets changed


title: Counterpropagating dipole-mode vector soliton

abstract:
  We experimentally observe a counterpropagating dipole-mode vector soliton in
a photorefractive SBN:60Ce crystal. We investigate the transient formation
dynamics and show that the formation process differs significantly from the
copropagating geometry. The experimental results are compared with fully
anisotropic numerical simulations which show good qualitative agreement.


title:
An approximate threshold condition for a non-autonomous system: an
  application to a vector-borne infection

abstract:
  A non-autonomous system is proposed to model the seasonal pattern of dengue
fever.
  We found that an approximate threshold condition for infection persistence
describes all possible behavior of the system.
  As far as we know, the kind of analysis here proposed is entirely new. No
precise mathematical theorems are demonstrated but we give enough numerical
evidence to support the conclusions.


title:
Multiple Attractor in Newton -Leipnik System, Peak to Peak dynamics and
  Chaos Control

abstract:
  The chaotic properties of Newton-Leipnik system are discussed from the view
point of strange attractors. Previously, two strange attractors of this system
were illustrated which occured from two different initial conditions under the
same parameter condition. It is found that above system also exhibits multiple
attractors under different parameter values but same initial condition and we
have shown the existence of three other strange attractors with varying
dimensionality under different parametric conditions. The properties of these
attractors are then analyzed on the basis of Lyapunov exponents, power spectra,
recurrence analysis and peak-to-peak dynamics. The peak-to-peak dynamics relies
on the low dimensionality of the chaotic attractor and allows to approximately
model the system. Peak-to-peak plot along with return-time plot are then
effectively used to solve the optimal control problem of the system which
reverts the system to a periodic situation.


title: Synchronization in networks of general, weakly nonlinear oscillators

abstract:
  We present a general approach to the study of synchrony in networks of weakly
nonlinear systems described by singularly perturbed equations of the type
$x''+x+\epsilon f(x,x')=0$. By performing a perturbative calculation based on
normal form theory we analytically obtain an $\O(\epsilon)$ approximation to
the Floquet multipliers that determine the stability of the synchronous
solution. The technique allows us to prove and generalize recent results
obtained using heuristic approaches, as well as reveal the structure of the
approximating equations. We illustrate the results in several examples, and
discuss extensions to the analysis of stability of multisynchronous states in
networks with complex architectures.


title:
Electrostatic acoustic modes in a self-gravitating complex plasma with
  variable charge impurities

abstract:
  A linear theory of dust acoustic (DA) and dust ion-acoustic (DIA) waves in a
self-gravitating collisional dusty plasma contaminated by variable-charge
impurities is presented for a self-consistent closed system. The ion-drag force
arising from the ion orbital motion as well as momentum transfer from all the
ions and self-gravitation are taken into consideration. The physical processes,
viz., dust-charge relaxation, ionization, recombination and collisional
dissipations are also taken into account self-consistently. The generalized
dispersion relation describing the coupling of the acoustic modes with the
dust-charge relaxation mode (CRM) and damping of the waves is derived and
analyzed numerically.


title: The symmetries of the Fokker - Planck equation in three dimensions

abstract:
  We calculate all point symmetries of the Fokker - Planck equation in
three-dimensional Euclidean space. General expression of symmetry group action
on arbitrary solution of Fokker - Planck equation is presented.


title: Return-Map Cryptanalysis Revisited

abstract:
  As a powerful cryptanalysis tool, the method of return-map attacks can be
used to extract secret messages masked by chaos in secure communication
schemes. Recently, a simple defensive mechanism was presented to enhance the
security of chaotic parameter modulation schemes against return-map attacks.
Two techniques are combined in the proposed defensive mechanism: multistep
parameter modulation and alternative driving of two different transmitter
variables. This paper re-studies the security of this proposed defensive
mechanism against return-map attacks, and points out that the security was much
over-estimated in the original publication for both ciphertext-only attack and
known/chosen-plaintext attacks. It is found that a deterministic relationship
exists between the shape of the return map and the modulated parameter, and
that such a relationship can be used to dramatically enhance return-map attacks
thereby making them quite easy to break the defensive mechanism.


title:
Logarithmically modified scaling of temperature structure functions in
  thermal convection

abstract:
  Using experimental data on thermal convection, obtained at a Rayleigh number
of 1.5 $\times 10^{11}$, it is shown that the temperature structure functions
$<\Delta T_{r}^p>$, where $\Delta T_r$ is the absolute value of the temperature
increment over a distance $r$, can be well represented in an intermediate range
of scales by $r^{\zeta_p} \phi (r)^{p}$, where the $\zeta_p$ are the scaling
exponents appropriate to the passive scalar problem in hydrodynamic turbulence
and the function $\phi (r) = 1-a(\ln r/r_h)^2$. Measurements are made in the
midplane of the apparatus near the sidewall, but outside the boundary layer.


title:
Equivalence of Many-Gluon Green Functions in Duffin-Kemmer-Petieu and
  Klein-Gordon-Fock Statistical Quantum Field Theories

abstract:
  We prove the equivalence of many-gluon Green functions in
Duffin-Kemmer-Petieu and Klein-Gordon-Fock statistical quantum field theories.
The proof is based on the functional integral formulation for the statistical
generating functional in a finite-temperature quantum field theory. As an
illustration, we calculate one-loop polarization operators in both theories and
show that their expressions indeed coincide.


title: Effect of Noise on the Standard Mapping

abstract:
  The effect of a small amount of noise on the standard mapping is considered.
Whenever the standard mapping possesses accelerator modes (where the action
increases approximately linearly with time), the diffusion coefficient contains
a term proportional to the reciprocal of the variance of the noise term. At
large values of the stochasticity parameter, the accelerator modes exhibit a
universal behavior. As a result the dependence of the diffusion coefficient on
the stochasticity parameter also shows some universal behavior.


title: Two-state, Reversible, Universal Cellular Automata in Three Dimensions

abstract:
  A novel two-state, Reversible Cellular Automata (RCA) is described. This
three-dimensional RCA is shown to be capable of universal computation.
Additionally, evidence is offered that this RCA Is capable of universal
construction.


title: Long-Time Correlations in the Stochastic Regime

abstract:
  The phase space for Hamiltonians of two degrees of freedom is usually divided
into stochastic and integrable components. Even when well into the stochastic
regime, integrable orbits may surround small stable regions or islands. The
effect of these islands on the correlation function for the stochastic
trajectories is examined. Depending on the value of the parameter describing
the rotation number for the elliptic fixed point at the center of the island,
the long-time correlation function may decay as t^-5 or exponentially, but more
commonly it decays much more slowly (roughly as t^-1). As a consequence these
small islands may have a profound effect on the properties of the stochastic
orbits. In particular, there is evidence that the evolution of a distribution
of particles is no longer governed by a diffusion equation.


title: Long-Time Correlations in Stochastic Systems

abstract:
  In recent years, there has been considerable interest in understanding the
motion in Hamiltonian systems when phase space is divided into stochastic and
integrable regions. This paper studies one aspect of this problem, namely, the
motion of trajectories in the stochastic sea when there is a small island
present. The results show that the particle can be stuck close to the island
for very long times. For the standard mapping, where accelerator modes are
possible, it appears that the mean squared displacement of particles in the
stochastic sea may increase faster than linearly with time indicating
non-diffusive behavior.


title: Polymer dynamics in chaotic flows with strong shear component

abstract:
  We consider the internal dynamics of the polymer molecule which is injected
in the chaotic flow with strong mean shear component. The flow geometry
corresponds to the recent experiments on the elastic turbulence (Groisman,
Steinberg 2000). The passive polymer in such flows experiences aperiodic
tumbling. We present a detailed study of the statistical properties of such
polymer dynamics. First we obtain the stationary probability distribution
function of the polymer orientation. Secondly we find the distribution of the
time periods between consequent events of tumbling, and finally we find the
tails of the polymer size distribution.


title: Optical Vortices and Vortex Solitons

abstract:
  Optical vortices are phase singularities nested in electromagnetic waves that
constitute a fascinating source of phenomena in the physics of light and
display deep similarities to their close relatives, quantized vortices in
superfluids and Bose-Einstein condensates. We present a brief overview of the
major advances in the study of optical vortices in different types of nonlinear
media, with emphasis on the properties of {\em vortex solitons}. Self-focusing
nonlinearity leads, in general, to the azimuthal instability of a
vortex-carrying beam, but it can also support novel types of stable or
meta-stable self-trapped beams carrying nonzero angular momentum, such as
ring-like solitons, necklace beams, and soliton clusters. We describe vortex
solitons created by multi-component beams, by parametrically coupled beams in
quadratic nonlinear media, and in partially incoherent light, as well as
discrete vortex solitons in periodic photonic lattices.


title:
Additive Equivalence in Turbulent Drag Reduction by Flexible and Rodlike
  Polymers

abstract:
  We address the "Additive Equivalence" discovered by Virk and coworkers: drag
reduction affected by flexible and rigid rodlike polymers added to turbulent
wall-bounded flows is limited from above by a very similar Maximum Drag
Reduction (MDR) asymptote. Considering the equations of motion of rodlike
polymers in wall-bounded turbulent ensembles, we show that although the
microscopic mechanism of attaining the MDR is very different, the macroscopic
theory is isomorphic, rationalizing the interesting experimental observations.


title:
A 2-Component Generalization of the Camassa-Holm Equation and Its
  Solutions

abstract:
  An explicit reciprocal transformation between a 2-component generalization of
the Camassa-Holm equation, called the 2-CH system, and the first negative flow
of the AKNS hierarchy is established, this transformation enables one to obtain
solutions of the 2-CH system from those of the first negative flow of the AKNS
hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH
system are presented.


title:
Reconfigurable directional couplers and junctions optically induced by
  nondiffracting Bessel beams

abstract:
  We put forward the concept of reconfigurable structures optically induced by
mutually incoherent nondiffracting Bessel beams in Kerr-type nonlinear media.
We address collinear couplers and X-junctions, and show how the switching
properties of such structures can be tuned by varying the intensity of the
Bessel beams.


title: A new self-synchronizing stream cipher

abstract:
  A new self-synchronizing stream cipher (SSSC) is proposed based on one-way
and nearest neighbor coupled integer maps. Some ideas of spatiotemporal chaos
synchronization and chaotic cryptography are applied in this new SSSC system.
Several principles of constructing optimal SSSC are discussed, and the methods
realizing these principles are specified. This SSSC is compared with several
SSSC systems in security by applying chosen-ciphertext attacks. It is shown
that our new system can provide SSSC with high security and fairly fast
performance.


title:
Analytical and Numerical Investigation of the Phase-Locked Loop with
  Time Delay

abstract:
  We derive the normal form for the delay-induced Hopf bifurcation in the
first-order phase-locked loop with time delay by the multiple scaling method.
The resulting periodic orbit is confirmed by numerical simulations. Further
detailed numerical investigations demonstrate exemplarily that this system
reveals a rich dynamical behavior. With phase portraits, Fourier analysis and
Lyapunov spectra it is possible to analyze the scaling properties of the
control parameter in the period-doubling scenario, both qualitatively and
quantitatively. Within the numerical accuracy there is evidence that the
scaling constant of the time-delayed phase-locked loop coincides with the
Feigenbaum constant $\delta \approx 4.669$ in one-dimensional discrete systems.


title: On complex adaptive systems and terrorism

abstract:
  Complex adaptive systems (CAS) are ubiquitous in nature. They are basic in
social sciences. An overview of CAS is given with emphasize on the occurrence
of bad side effects to seemingly wise decisions. Hence application to terrorism
is given. Some conclusions on how to deal with this phenomena are proposed.


title: Spatial vector solitons in nonlinear photonic crystal fibers

abstract:
  We study spatial vector solitons in a photonic crystal fiber (PCF) made of a
material with the focusing Kerr nonlinearity. We show that such two-component
localized nonlinear waves consist of two mutually trapped components confined
by the PCF linear and the self-induced nonlinear refractive indices, and they
bifurcate from the corresponding scalar solitons. We demonstrate that, in a
sharp contrast with an entirely homogeneous nonlinear Kerr medium where both
scalar and vector spatial solitons are unstable and may collapse, the periodic
structure of PCF can stabilize the otherwise unstable two-dimensional spatial
optical solitons. We apply the matrix criterion for stability of these
two-parameter solitons, and verify it by direct numerical simulations.


title: Partial and complete linearization of PDEs based on conservation laws

abstract:
  A method based on infinite parameter conservation laws is described to factor
linear differential operators out of nonlinear partial differential equations
(PDEs) or out of differential consequences of nonlinear PDEs. This includes a
complete linearization to an equivalent linear PDE (-system) if that is
possible. Infinite parameter conservation laws can be computed, for example,
with the computer algebra package ConLaw.


title: Universal Scaling in Saddle-Node Bifurcation Cascades (I)

abstract:
  A saddle-node bifurcation cascade is studied in the logistic equation, whose
bifurcation points follow an expression formally identical to the one given by
Feigenbaum for period doubling cascade. The Feigenbaum equation is generalized
because it rules several objects, which do not have to be orbits. The outcome
is that an attractor of attractors appears, and information about the birth,
death and scaling of windows is obtained.


title:
Universal Scaling in Saddle-Node Bifurcation Cascades (II) Intermittency
  Cascade

abstract:
  The presence of saddle-node bifurcation cascade in the logistic equation is
associated with an intermittency cascade; in a similar way as a saddle-node
bifurcation is associated with an intermittency. We merge the concepts of
bifurcation cascade and intermittency. The mathematical tools necessary for
this process will describe the structure of the Myrberg-Feigenbaum point.


title:
Continuous and Discrete Homotopy Operators with Applications in
  Integrability Testing

abstract:
  We introduce calculus-based formulas for the continuous Euler and homotopy
operators. The 1D continuous homotopy operator automates integration by parts
on the jet space. Its 3D generalization allows one to invert the total
divergence operator. As a practical application, we show how the operators can
be used to symbolically compute local conservation laws of nonlinear systems of
partial differential equations in multi-dimensions.
  By analogy to the continuous case, we also present concrete formulas for the
discrete Euler and homotopy operators. Essentially, the discrete homotopy
operator carries out summation by parts. We use it to algorithmically invert
the forward difference operator. We apply the discrete operator to compute
fluxes of differential-difference equations in (1+1) dimensions.
  Our calculus-based approach allows for a straightforward implementation of
the operators in major computer algebra system, such as Mathematica and Maple.
The symbolic algorithms for integration and summation by parts are illustrated
with elementary examples. The algorithms to compute conservation laws are
illustrated with nonlinear PDEs and their discretizations arising in fluid
dynamics and mathematical physics.


title: On continual classes of evolution equations

abstract:
  We reproduce our old result on the complete class of local evolution
equations admitting the original Miura transformation, and then consider this
continual class of evolution equations from the standpoint of zero-curvature
representations.


title:
Reconfigurable soliton networks optically-induced by arrays of
  nondiffracting Bessel beams

abstract:
  We address the propagation of solitons in reconfigurable two-dimensional
networks induced optically by arrays of nondiffracting Bessel beams in
Kerr-type nonlinear media. We show that broad soliton beams can move across
networks having different topologies almost without radiation losses, opening
new prospects for all-optical soliton manipulation. We also discuss various
switching scenarios for solitons launched into multi-core directional couplers
optically-induced by suitable arrays of Bessel beams.


title:
Characterization of chaotic dynamics in the vocalization of Cervus
  elaphus corsicanus

abstract:
  Chaos, oscillations, instabilities, intermittency represent only some
nonlinear examples apparent in natural world. These phenomena appear in any
field of study, and advances in complex and nonlinear dynamic techniques bring
about opportunities to better understand animal signals.
  In this work we suggest an analysis method based on the characterization of
the vocal fold dynamics by means of the nonlinear time series analysis, and by
the computations of the parameters typical of chaotic oscillations: Attractor
reconstruction, Spectrum of Lyapunov Exponents and Maximum Lyapunov Exponent
was used to reconstruct the dynamic of the vocal folds. Identifying a sort of
of vocal fingerprint can be useful in biodiversity monitoring and understanding
the health status of a given animal.
  This method was applied to the vocalization of the Cervus elaphus corsicanus,
the Sardinian Red Deer.


title: Acceleration and vortex filaments in turbulence

abstract:
  We report recent results from a high resolution numerical study of fluid
particles transported by a fully developed turbulent flow. Single particle
trajectories were followed for a time range spanning more than three decades,
from less than a tenth of the Kolmogorov time-scale up to one large-eddy
turnover time. We present some results concerning acceleration statistics and
the statistics of trapping by vortex filaments.


title: Reconnection of Stable/Unstable Manifolds of the Harper Map

abstract:
  The Harper map is one of the simplest chaotic systems exhibiting reconnection
of invariant manifolds. The method of asymptotics beyond all orders (ABAO) is
used to construct stable/unstable manifolds of the Harper map. When the
parameter changes to the reconnection threshold, the stable/unstable manifolds
are shown to acquire new oscillatory portion corresponding to the heteroclinic
tangle after the reconnection.


title: Motion representation of one-dimensional cellular automaton rules

abstract:
  Generalizing the motion representation we introduced for number-conserving
rules, we give a systematic way to construct a generalized motion
representation valid for non-conservative rules using the expression of the
current, which appears in the discrete version of the continuity equation,
completed by the discrete analogue of the source term. This new representation
is general, but not unique, and can be used to represent, in a more visual way,
any one-dimensional cellular automaton rule. A few illustrative examples are
presented.


title:
On deformations of standard R-matrices for integrable
  infinite-dimensional systems

abstract:
  Simple deformations, with a parameter $\epsilon$, of classical $R$-matrices
which follow from decomposition of appropriate Lie algebras, are considered. As
a result nonstandard Lax representations for some well known integrable systems
are presented as well as new integrable evolution equations are constructed.


title:
Extended Prelle-Singer Method and Integrability/Solvability of a Class
  of Nonlinear $n$th Order Ordinary Differential Equations

abstract:
  We discuss a method of solving $n^{th}$ order scalar ordinary differential
equations by extending the ideas based on the Prelle-Singer (PS) procedure for
second order ordinary differential equations. We also introduce a novel way of
generating additional integrals of motion from a single integral. We illustrate
the theory for both second and third order equations with suitable examples.
Further, we extend the method to two coupled second order equations and apply
the theory to two-dimensional Kepler problem and deduce the constants of motion
including Runge-Lenz integral.


title:
Desynchronized wave patterns in synchronized chaotic regions of coupled
  map lattices

abstract:
  We analyze the size limits of coupled map lattices with diffusive coupling at
the crossover of low-dimensional to high-dimensional chaos. We investigate the
existence of standing-wave-type periodic patterns, within the low-dimensional
limit, in addition to the stable synchronous chaotic states depending upon the
initial conditions. Further, we bring out a controlling mechanism to explain
the emergence of standing-wave patterns in the coupled map lattices. Finally,
we give an analytic expression in terms of the unstable periodic orbits of the
isolated map to represent the standing-wave patterns.


title: Laws of Energy Gradient for Instabilities

abstract:
  Transition to turbulence is due to the instability of a laminar flow subject
to a disturbance. This complicated problem can be explained using a new
proposed energy gradient theory in our previous study. This theory is extended
to the instability of fluid material systems in this study. The instability of
fluid material systems may lead to the evolution of natural environments and
the occurrence of catastrophic events in the world. To better describe these
phenomena and to understand the physical mechanism behind them are very
important. In order to more generally describe the instability of fluid
material systems, laws of energy gradient are summarized for static and motion
systems, respectively. These laws could be applicable to various flow problems
and material systems. Examples are shown that many events in the world could be
explained using these laws.


title:
Investigation of Turbulent transition in plane Couette flows Using
  Energy Gradient Method

abstract:
  The energy gradient method has been proposed with the aim of better
understanding the mechanism of flow transition from laminar flow to turbulent
flow. In this method, it is demonstrated that the transition to turbulence
depends on the relative magnitudes of the transverse gradient of the total
mechanical energy which amplifies the disturbance and the energy loss from
viscous friction which damps the disturbance, for given imposed disturbance.
For a given flow geometry and fluid properties, when the maximum of the
function K (a function standing for the ratio of the gradient of total
mechanical energy in the transverse direction to the rate of energy loss due to
viscous friction in the streamwise direction) in the flow field is larger than
a certain critical value, it is expected that instability would occur for some
initial disturbances. In this paper, using the energy gradient analysis, the
equation for calculating the energy gradient function K for plane Couette flow
is derived. The result indicates that K reaches the maximum at the moving
walls. Thus, the fluid layer near the moving wall is the most dangerous
position to generate initial oscillation at sufficient high Re for given same
level of normalized perturbation in the domain. The critical value of K at
turbulent transition, which is observed from experiments, is about 370 for
plane Couette flow when two walls move in opposite directions (anti-symmetry).
This value is about the same as that for plane Poiseuille flow and pipe
Poiseuille flow (385-389). Therefore, it is concluded that the critical value
of K at turbulent transition is about 370-389 for wall-bounded parallel shear
flows which include both pressure (symmetrical case) and shear driven flows
(anti-symmetrical case).


title: Energy Gradient Theory of Hydrodynamic Instability

abstract:
  A new universal theory for flow instability and turbulent transition is
proposed in this study. Flow instability and turbulence transition have been
challenging subjects for fluid dynamics for a century. The critical condition
of turbulent transition from theory and experiments differs largely from each
other for Poiseuille flows. In this paper, a new mechanism of flow instability
and turbulence transition is presented for parallel shear flows and the energy
gradient theory of hydrodynamic instability is proposed. It is stated that the
total energy gradient in the transverse direction and that in the streamwise
direction of the main flow dominate the disturbance amplification or decay. A
new dimensionless parameter K for characterizing flow instability is proposed
for wall bounded shear flows, which is expressed as the ratio of the energy
gradients in the two directions. It is thought that flow instability should
first occur at the position of Kmax which may be the most dangerous position.
This speculation is confirmed by Nishioka et al's experimental data. Comparison
with experimental data for plane Poiseuille flow and pipe Poiseuille flow
indicates that the proposed idea is really valid. It is found that the
turbulence transition takes place at a critical value of Kmax of about 385 for
both plane Poiseuille flow and pipe Poiseuille flow, below which no turbulence
will occur regardless the disturbance. More studies show that the theory is
also valid for plane Couette flows and Taylor-Couette flows between concentric
rotating cylinders.


title:
Discrete soliton collisions in a waveguide array with saturable
  nonlinearity

abstract:
  We study the symmetric collisions of two mobile breathers/solitons in a model
for coupled wave guides with a saturable nonlinearity. The saturability allows
the existence of breathers with high power. Three main regimes are observed:
breather fusion, breather reflection and breather creation. The last regime
seems to be exclusive of systems with a saturable nonlinearity, and has been
previously observed in continuous models. In some cases a ``symmetry breaking''
can be observed, which we show to be an numerical artifact.


title: Construction of Hypergeometric Solutions to the q-Painlev\'e Equations

abstract:
  Hypergeometric solutions to the q-Painlev\'e equations are constructed by
direct linearization of disrcrete Riccati equations. The decoupling factors are
explicitly determined so that the linear systems give rise to q-hypergeometric
equations.


title:
Coupled nonlinear Schroedinger field equations for electromagnetic wave
  propagation in nonlinear left-handed materials

abstract:
  For an isotropic and homogeneous nonlinear left-handed materials, for which
the effective medium approximation is valid, Maxwell's equations for electric
and magnetic fields lead naturally, within the slowly varying envelope
approximation, to a system of coupled nonlinear Schroedinger equations. This
system is equivalent to the well-known Manakov model that under certain
conditions, is completely integrable and admits bright and dark soliton
solutions. It is demonstrated that left- and right-handed (normal) nonlinear
mediums may have compound bright and dark soliton solutions, respectively.
These results are also supported by numerical calculations.


title:
The Fermi-Pasta-Ulam "numerical experiment": history and pedagogical
  perspectives

abstract:
  The Fermi-Pasta-Ulam (FPU) pioneering numerical experiment played a major
role in the history of computer simulation because it introduced this concept
for the first time. Moreover, it raised a puzzling question which was answered
more than 10 years later. After an introduction to this problem, we briefly
review its history and then suggest some simple numerical experiments, with a
provided Matlab code, to study various aspects of the ``FPU'' problem.


title:
Lagrangian statistics of particle pairs in homogeneous isotropic
  turbulence

abstract:
  We present a detailed investigation of the particle pair separation process
in homogeneous isotropic turbulence. We use data from direct numerical
simulations up to Taylor's Reynolds number 280 following the evolution of about
two million passive tracers advected by the flow over a time span of about
three decades. We present data for both the separation distance and the
relative velocity statistics. Statistics are measured along the particle pair
trajectories both as a function of time and as a function of their separation,
i.e. at fixed scales. We compare and contrast both sets of statistics in order
to gain an insight into the mechanisms governing the separation process. We
find very high levels of intermittency in the early stages, that is, for travel
times up to order ten Kolmogorov time scales. The fixed scale statistics allow
us to quantify anomalous corrections to Richardson diffusion in the inertial
range of scales for those pairs that separate rapidly. It also allows a
quantitative analysis of intermittency corrections for the relative velocity
statistics.


title: Local Average and Scaling Property of $1/f^{\alpha}$ Random Fields

abstract:
  We use the techniques developed in [1] to study the local average of random
fields with spectral density $1/f^{\alpha}$. We study their scaling properties
and show that the self-similarity of $1/f$ random fields is preserved under the
local average. We study how the original random fields can be recovered from
the locally averaged ones. We also study the derivative of the locally averaged
random fields as a way to get the spectral density. Finally, we propose the
generalization of local average by means of an arbitrary response function.


title: Discrete peakons

abstract:
  We demonstrate for the first time the possibility for explicit construction
in a discrete Hamiltonian model of an exact solution of the form $\exp(-|n|)$,
i.e., a discrete peakon. These discrete analogs of the well-known, continuum
peakons of the Camassa-Holm equation [Phys. Rev. Lett. {\bf 71}, 1661 (1993)]
are found in a model different from their continuum siblings. Namely, we
observe discrete peakons in Klein-Gordon-type and nonlinear Schr\"odinger-type
chains with long-range interactions. The interesting linear stability
differences between these two chains are examined numerically and illustrated
analytically. Additionally, inter-site centered peakons are also obtained in
explicit form and their stability is studied. We also prove the global
well-posedness for the discrete Klein-Gordon equation, show the instability of
the peakon solution, and the possibility of a formation of a breathing peakon.


title: Foundations of Swarm Intelligence: From Principles to Practice

abstract:
  Swarm Intelligence (SI) is a relatively new paradigm being applied in a host
of research settings to improve the management and control of large numbers of
interacting entities such as communication, computer and sensor networks,
satellite constellations and more. Attempts to take advantage of this paradigm
and mimic the behavior of insect swarms however often lead to many different
implementations of SI. The rather vague notions of what constitutes
self-organized behavior lead to rather ad hoc approaches that make it difficult
to ascertain just what SI is, assess its true potential and more fully take
advantage of it. This article provides a set of general principles for SI
research and development. A precise definition of self-organized behavior is
described and provides the basis for a more axiomatic and logical approach to
research and development as opposed to the more prevalent ad hoc approach in
using SI concepts. The concept of Pareto optimality is utilized to capture the
notions of efficiency and adaptability. A new concept, Scale Invariant Pareto
Optimality is described and entails symmetry relationships and scale invariance
where Pareto optimality is preserved under changes in system states. This
provides a mathematical way to describe efficient tradeoffs of efficiency
between different scales and further, mathematically captures the notion of the
graceful degradation of performance so often sought in complex systems.


title: Continuous Symmetries of Difference Equations

abstract:
  Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.

categories: nlin.SI hep-ph hep-th math-ph math.GR math.MP math.SP


title: Nonlinear dual-core photonic crystal fiber couplers

abstract:
  We study nonlinear modes of dual-core photonic crystal fiber couplers made of
a material with the focusing Kerr nonlinearity. We find numerically the
profiles of symmetric, antisymmetric, and asymmetric nonlinear modes, and
analyze all-optical switching based on instability of the symmetric mode. We
also describe elliptic spatial solitons controlled by the waveguide boundaries.


title: Elementary Darboux transformations and factorization

abstract:
  A general theorem on factorization of matrices with polynomial entries is
proven and it is used to reduce polynomial Darboux matrices to linear ones.
Some new examples of linear Darboux matrices are discussed.


title: Spectral fluctuations and 1/f noise in the order-chaos transition regime

abstract:
  Level fluctuations in quantum system have been used to characterize quantum
chaos using random matrix models. Recently time series methods were used to
relate level fluctuations to the classical dynamics in the regular and chaotic
limit. In this we show that the spectrum of the system undergoing order to
chaos transition displays a characteristic $f^{-\gamma}$ noise and $\gamma$ is
correlated with the classical chaos in the system. We demonstrate this using a
smooth potential and a time-dependent system modeled by Gaussian and circular
ensembles respectively of random matrix theory. We show the effect of short
periodic orbits on these fluctuation measures.


title:
Formation of Multifractal Population Patterns from Reproductive Growth
  and Local Resettlement

abstract:
  We consider the general character of the spatial distribution of a population
that grows through reproduction and subsequent local resettlement of new
population members. We present several simple one and two-dimensional point
placement models to illustrate possible generic behavior of these
distributions. We show, numerically and analytically, that these models all
lead to multifractal spatial distributions of population. Additionally, we make
qualitative links between our models and the example of the Earth at Night
image, showing the Earth's nighttime man-made light as seen from space. The
Earth at Night data suffer from saturation of the sensing photodetectors at
high brightness (`clipping'), and we account for how this influences the
determined dimension spectrum of the light intensity distribution.


title:
Emergence of multiple time scales in coupled oscillators with plastic
  frequencies

abstract:
  A coupled-phase oscillator model where each oscillator has an angular
velocity that varies due to the interaction with other oscillators is studied.
This model is proposed to deepen the understanding of the relationship between
the coexistence and the plasticity of time scales in complex systems. It is
found that initial conditions close to a one cluster states self-organize into
multiple clusters with different angular velocities. Namely, hierarchization of
the time scales emerges through the multi-clustering process in phases.
Analyses for the clusters solution, the stability of the solution and the
mechanism determining the time scales are reported.


title:
Variational and Potential Formulation for Stochastic Partial
  Differential Equations

abstract:
  There is recent interest in finding a potential formulation for Stochastic
Partial Differential Equations (SPDEs). The rationale behind this idea lies in
obtaining all the dynamical information of the system under study from one
single expression. In this Letter we formally provide a general Lagrangian
formalism for SPDEs using the Hojman et al. method. We show that it is possible
to write the corresponding effective potential starting from an s-equivalent
Lagrangean, and that this potential is able to reproduce all the dynamics of
the system, once a special differential operator has been applied. This
procedure can be used to study the complete time evolution and spatial
inhomogeneities of the system under consideration, and is also suitable for the
statistical mechanics description of the problem. Keywords: stochastic partial
differential equations, variational formulation, effective potential. PACS:
45.20.Jj; 02.50.-r; 02.50.Ey.


title: Scalar Decay in Chaotic Mixing

abstract:
  I review the local theory of mixing, which focuses on infinitesimal blobs of
scalar being advected and stretched by a random velocity field. An advantage of
this theory is that it provides elegant analytical results. A disadvantage is
that it is highly idealised. Nevertheless, it provides insight into the
mechanism of chaotic mixing and the effect of random fluctuations on the rate
of decay of the concentration field of a passive scalar.


title: Route to Chaos in Three-Dimensional Maps of Logistic Type

abstract:
  A route to chaos is studied in 3-dimensional maps of logistic type.
Mechanisms of period doubling for invariant closed curves (ICC) are found for
specific 3-dimensional maps. These bifurcations cannot be observed for ICC in
the 2-dimensional case. When the parameter of the system is modified, localized
oscillations occur on the ICC that give rise to weakly chaotic rings, then to
chaotic attractors, which finally disappear by contact bifurcations. These maps
can be considered as models for the symbiotic interaction of three species.


title:
Efficient detection of periodic orbits in chaotic systems by stabilising
  transformations

abstract:
  An algorithm for detecting periodic orbits in chaotic systems [Phys. Rev. E,
60 (1999), pp.~6172--6175], which combines the set of stabilising
transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78
(1997), pp.~4733--4736] with a modified semi-implicit Euler iterative scheme
and seeding with periodic orbits of neighbouring periods, has been shown to be
highly efficient when applied to low-dimensional systems. The difficulty in
applying the algorithm to higher-dimensional systems is mainly due to the fact
that the number of the stabilising transformations grows extremely fast with
increasing system dimension. Here we analyse the properties of stabilising
transformations and propose an alternative approach for constructing a smaller
set of transformations. The performance of the new approach is illustrated on
the four-dimentional kicked double rotor map and the six-dimensional system of
three coupled Henon maps.


title:
Relating pain intensity of newborns to onset of nonlinear phenomena in
  cry recordings

abstract:
  The cries of several full term newborns, recorded during blood sampling, were
analyzed. Spectrograms showed the appearance of irregular patterns related to
the pain assessed using the method of the DAN scale. In particular, the
appearance of Noise concentration Patterns (NP) in spectrograms was related to
the increase of the pain suffered by the newborns. In this scenario, pain
constitutes a bifurcation parameter for the vocal folds dynamic, inducing a
Ruelle-Takens-Newhouse chaotic transition.


title:
Birhythmicity, Synchronization, and Turbulence in an Oscillatory System
  with Nonlocal Inertial Coupling

abstract:
  We consider a model where a population of diffusively coupled limit-cycle
oscillators, described by the complex Ginzburg-Landau equation, interacts
nonlocally via an inertial field. For sufficiently high intensity of nonlocal
inertial coupling, the system exhibits birhythmicity with two oscillation modes
at largely different frequencies. Stability of uniform oscillations in the
birhythmic region is analyzed by means of the phase dynamics approximation.
Numerical simulations show that, depending on its parameters, the system has
irregular intermittent regimes with local bursts of synchronization or
desynchronization.


title:
An equation-free approach to coupled oscillator dynamics: the Kuramoto
  model example

abstract:
  We present an equation-free multi-scale approach to the computational study
of the collective dynamics of the Kuramoto model [{\it Chemical Oscillations,
Waves, and Turbulence}, Springer-Verlag (1984)], a prototype model for coupled
oscillator populations. Our study takes place in a reduced phase space of
coarse-grained "observables" of the system: the first few moments of the
oscillator phase angle distribution. We circumvent the derivation of explicit
dynamical equations (approximately) governing the evolution of these
coarse-grained macroscopic variables; instead we use the {\it equation-free
framework} [Kevrekidis {\it et al.}, {\it Comm. Math. Sci.} {\bf 1}(4), 715
(2003)] to computationally solve these equations without obtaining them in
closed form. In this approach, the numerical tasks for the conceptually
existing but unavailable coarse-grained equations are implemented through short
bursts of appropriately initialized simulations of the "fine-scale", detailed
coupled oscillator model. Coarse projective integration and coarse fixed point
computations are illustrated.


title: Painleve tests, singularity structure and integrability

abstract:
  After a brief introduction to the Painlev\'{e} property for ordinary
differential equations, we present a concise review of the various methods of
singularity analysis which are commonly referred to as Painlev\'{e} tests. The
tests are applied to several different examples, and we discuss the connection
between singularity structure and integrability for ordinary and partial
differential equations.


title: Self-tuning to the Hopf bifurcation in fluctuating systems

abstract:
  The problem of self-tuning a system to the Hopf bifurcation in the presence
of noise and periodic external forcing is discussed. We find that the response
of the system has a non-monotonic dependence on the noise-strength, and
displays an amplified response which is more pronounced for weaker signals. The
observed effect is to be distinguished from stochastic resonance. For the
feedback we have studied, the unforced self-tuned Hopf oscillator in the
presence of fluctuations exhibits sharp peaks in its spectrum. The implications
of our general results are briefly discussed in the context of sound detection
by the inner ear.

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


title:
Absolute and relative choreographies in the problem of point vortices
  moving on a plane

abstract:
  We obtained new periodic solutions in the problems of three and four point
vortices moving on a plane. In the case of three vortices, the system is
reduced to a Hamiltonian system with one degree of freedom, and it is
integrable. In the case of four vortices, the order is reduced to two degrees
of freedom, and the system is not integrable. We present relative and absolute
choreographies of three and four vortices of the same intensity which are
periodic motions of vortices in some rotating and fixed frame of reference,
where all the vortices move along the same closed curve. Similar choreographies
have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the
n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the
choreographies that appear in vortex dynamics have a number of distinct
features.


title: Tsunami wave runup on coasts of narrow bays

abstract:
  The runup of tsunami waves on the coasts of the barrow bays, channels and
straits is studied in the framework of the nonlinear shallow water theory.
Using the narrowness of the water channel, the one-dimensional equations are
applied; they include the variable cross-section of channel. It is shown that
the analytical solutions can be obtained with use of the hodograph (Legendre)
transformation similar to the wave runup on the plane beach. As a result, the
linear wave equation is derived and all physical variables (water displacement,
fluid velocity, coordinate and time) can be determined. The dynamics of the
moving shoreline (boundary of the flooding zone) is investigated in details. It
is shown that all analytical formulas for the moving shoreline can be obtained
explicitly. Two examples of the incident wave shapes are analysed: sine wave
and parabolic pulse. The last example demonstrates that even for approaching of
the crest only, the flooding can appear very quickly; then water will recede
relatively slowly, and then again quickly return to the initial state.


title: Experimental Synchronization of Spatiotemporal Chaos in Nonlinear Optics

abstract:
  We demonstrate that a unidirectional coupling between a pattern forming
system and its replica induces complete synchronization of the slave to the
master system onto a spatiotemporal chaotic state.


title: Local evolution equations for non-Markovian processes

abstract:
  A Fokker-Planck equation approach for the treatment of non-Markovian
stochastic processes is proposed. The approach is based on the introduction of
fictitious trajectories sharing with the real ones their local structure and
initial conditions. Different statistical quantities are generated by different
construction rules for the trajectories, which coincide only in the Markovian
case. The merits and limitations of the approach are discussed and applications
to transport in ratchets and to anomalous diffusion are illustated.


title:
Harmonic oscillator states as oscillating localized structures near
  Hopf-Turing instability boundary

abstract:
  A set of coupled complex Ginzburg-landau type amplitude equations which
operates near a Hopf-Turing instability boundary is analytically investigated
to show localized oscillatory patterns. The spatial structure of those patterns
are the same as quantum mechanical harmonic oscillator stationary states and
can have even or odd symmetry depending on the order of the state. It has been
seen that the underlying Turing state plays a major role in the selection of
the order of such solutions.


title: 1D spirals: is multi stability essential?

abstract:
  The origin of 1D spirals or antisymmetric 1D pulses is explaind so far on the
basis of multistability of spatially inhomogeneous and temporally oscillatory
phases and so called nonvariational effects. Thus, coupled amplitude equations
which are valid near a co-dimention 2 point and provides with the necessary
multistable environment are commonly in use in the numerical calculations to
generate such structures. In the present work we analytically show that a
complex Ginzburg-Landau type amplitude equation which is valid in the Hopf
region of phase space near an instability threshold admits solutions like
antisymmetric pulses traveling in alternate directions from a core. The pulses
can have well defined spatial profile like Hermite polynomial of order unity on
the side of the direction of its motion. Thus, a pulse moving in the positive
direction of x-axis from near a core at origin will be a peak while that moving
in the negative direction is a trough and a temporal oscillation of such
structures would give the impression of alternative peak and troughs are moving
appart being generated from a common core.


title: A Weakly Nonlinear Analysis of Impulsively-Forced Faraday Waves

abstract:
  Parametrically-excited surface waves, forced by a periodic sequence of
delta-function impulses, are considered within the framework of the
Zhang-Vi\~nals model (J. Fluid Mech. 1997). The exact impulsive-forcing
results, in the linear and weakly nonlinear regimes, are compared with
numerical results for sinusoidal and multifrequency forcing. We find
surprisingly good agreement between impulsive forcing results and those
obtained using a two-term truncated Fourier series representation of the
impulsive forcing function. As noted previously by Bechhoefer and Johnson (Am.
J. Phys. 1996), in the case of two equally-spaced impulses per period there are
only subharmonic modes of instability. The familiar situation of alternating
subharmonic and harmonic resonance tongues emerges for unequally-spaced
impulses. We extend the linear analysis for two impulses per period to the
weakly nonlinear regime for one-dimensional waves. Specifically, we derive an
analytic expression for the cubic Landau coefficient in the bifurcation
equation as a function of the dimensionless fluid parameters and spacing
between the two impulses. As the capillary parameter is varied, one finds a
parameter region of wave amplitude suppression, which is due to a familiar 1:2
spatio-temporal resonance between the subharmonic mode of instability and a
damped harmonic mode. This resonance occurs for impulsive forcing even when
harmonic resonance tongues are absent from the neutral stability curve. The
strength of this resonance feature can be tuned by varying the spacing between
the impulses. This finding is interpreted in terms of a recent symmetry-based
analysis of multifrequency forced Faraday waves by Porter, Topaz and Silber
(Phys. Rev. Lett. 2004, Phys. Rev. E 2004).


title:
Construction of a secure cryptosystem based on spatiotemporal chaos and
  its application in public channel cryptography

abstract:
  By combining the one-way coupled chaotic map lattice system with a
bit-reverse operation, we construct a new cryptosystem which is extremely
sensitive to the system parameters even for low-dimensional systems. The
security of this new algorithm is investigated and mechanism of the sensitivity
is analyzed. We further apply this cryptosystem to the public channel
cryptography, based on "Merkle's puzzles", by employing it both as
pseudo-random-number (PN) generators and symmetric encryptor. With the
properties of spatiotemporal chaos, the new scheme is rich with new features
and shows some advantages in comparison with the conventional ones.


title: Two-body problem on a sphere. Reduction, stochasticity, periodic orbits

abstract:
  We consider the problem of two interacting particles on a sphere. The
potential of the interaction depends on the distance between the particles. The
case of Newtonian-type potentials is studied in most detail. We reduce this
system to a system with two degrees of freedom and give a number of remarkable
periodic orbits. We also discuss integrability and stochastization of the
motion.


title: Imaginary Fields

abstract:
  Nonlinear dynamical modeling of interaction between automatic and conscious
processes in the human brain is considered in terms of the quaternion fields.
The interaction is due, particularly, to the nonlinear firing rate of neurons.
Possible connection of consciousness with the general field theory is
indicated. A new type of symmetry between dynamics of real and imaginary fields
is pointed out.


title: Motion of a circular cylinder and n point vortices in a perfect fluid

abstract:
  The paper studies the system of a rigid body interacting dynamically with
point vortices in a perfect fluid. For arbitrary value of vortex strengths and
circulation around the cylinder the system is shown to be Hamiltonian (the
corresponding Poisson bracket structure is rather complicated). We also reduced
the number of degrees of freedom of the system by two using the reduction by
symmetry technique and performed a thorough qualitative analysis of the
integrable system of a cylinder interacting with one vortex.


title: Euler-Poisson Equations and Integrable Cases

abstract:
  In this paper we propose a new approach to the study of integrable cases
based on intensive computer methods' application. We make a new investigation
of Kovalevskaya and Goryachev-Chaplygin cases of Euler-Poisson equations and
obtain many new results in rigid body dynamics in absolute space. Also we
present the visualization of some special particular solutions.


title:
Integrability of the problem of motion of a cylinder and a point vortex
  in a perfect fluid

abstract:
  Motion of a cylinder dynamically interacting with n point vortices in a
perfect fluid is considered. A nonliniear Poisson structure and two integrals
of motion are found. The equations of motion a priori are not Hamiltonian. For
n=1, the problem is integrable.


title:
Critical bending point in the Lyapunov localization spectra of
  many-particle systems

abstract:
  The localization spectra of Lyapunov vectors in many-particle systems at low
density exhibit a characteristic bending behavior. It is shown that this
behavior is due to a restriction on the maximum number of the most localized
Lyapunov vectors determined by the system configuration and mutual
orthogonality. For a quasi-one-dimensional system this leads to a predicted
bending point at n_c \approx 0.432 N for an N particle system. Numerical
evidence is presented that confirms this predicted bending point as a function
of the number of particles N.


title: Controlling soliton explosions

abstract:
  We investigate the dynamics of solitons in generalized Klein-Gordon equations
in the presence of nonlinear damping and spatiotemporal perturbations. We will
present different mechanisms for soliton explosions. We show (both analytically
and numerically) that some space-dependent perturbations or nonlinear damping
can make the soliton internal mode unstable leading to soliton explosion. We
will show that, in some cases, while some conditions are satisfied, the soliton
explodes becoming a permanent, extremely complex, spatiotemporal dynamics. We
believe these mechanisms can explain some of the phenomena that recently have
been reported to occur in excitable media. We present a method for controlling
soliton explosions.


title: Self-pulsing dynamics of ultrasound in a magnetoacoustic resonator

abstract:
  A theoretical model of parametric magnetostrictive generator of ultrasound is
considered, taking into account magnetic and magnetoacoustic nonlinearities.
The stability and temporal dynamics of the system is analized with standard
techniques revealing that, for a given set of parameters, the model presents a
homoclinic or saddle--loop bifurcation, which predicts that the ultrasound is
emitted in the form of pulses or spikes with arbitrarily low frequency.


title: Complexiton solutions to integrable equations

abstract:
  Complexiton solutions (or complexitons for short) are exact solutions newly
introduced to integrable equations. Starting with the solution classification
for a linear differential equation, the Korteweg-de Vries equation and the Toda
lattice equation are considered as examples to exhibit complexiton structures
of nonlinear integrable equations. The crucial step in the solution process is
to apply the Wronskian and Casoratian techniques for Hirota's bilinear
equations. Correspondence between complexitons of the Korteweg-de Vries
equation and complexitons of the Toda lattice equation is provided.


title: Very fine structures in scalar mixing

abstract:
  We explore very fine scales of scalar dissipation in turbulent mixing, below
Kolmogorov and around Batchelor scales, by performing direct numerical
simulations at much finer grid resolution than is usually adopted in the past.
We consider the resolution in terms of a local, fluctuating Batchelor scale and
study the effects on the tails of the probability density function and
multifractal properties of the scalar dissipation. The origin and importance of
these very fine-scale fluctuations are discussed. One conclusion is that they
are unlikely to be related to the most intense dissipation events.


title: Second order additive invariants in elementary cellular automata

abstract:
  We investigate second order additive invariants in elementary cellular
automata rules. Fundamental diagrams of rules which possess additive invariants
are either linear or exhibit singularities similar to singularities of rules
with first-order invariant. Only rules which have exactly one invariants
exhibit singularities. At the singularity, the current decays to its
equilibrium value as a power law $t^{\alpha}$, and the value of the exponent
$\alpha$ obtained from numerical simulations is very close to -1/2. This is in
agreements with values previously reported for number-conserving rules, and
leads to a conjecture that regardless of the order of the invariant, exponent
$\alpha$ seems to have a universal value of 1/2.


title:
Separation of variables for a lattice integrable system and the inverse
  problem

abstract:
  We investigate the relation between the local variables of a discrete
integrable lattice system and the corresponding separation variables, derived
from the associated spectral curve. In particular, we have shown how the
inverse transformation from the separation variables to the discrete lattice
variables may be factorised as a sequence of canonical transformations,
following the procedure outlined by Kuznetsov.


title: Dynamics of rolling disk

abstract:
  In the paper we present the qualitative analysis of rolling motion without
slipping of a homogeneous round disk on a horisontal plane. The problem was
studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its
integrability. The behavior of the point of contact on a plane is investigated
and conditions under which its trajectory is finit are obtained. The
bifurcation diagrams are constructed.


title: On the History of the Development of the Nonholonomic Dynamics

abstract:
  The main directions in the development of the nonholonomic dynamics are
briefly considered in this paper. The first direction is connected with the
general formalizm of the equations of dynamics that differs from the Lagrangian
and Hamiltonian methods of the equations of motion's construction. The second
direction, substantially more important for dynamics, includes investigations
concerning the analysis of the specific nonholonomic problems. We also point
out rather promising direction in development of nonholonomic systems that is
connected with intensive use of the modern computer-aided methods.


title: Localized patterns and hole solutions in one-dimension extended sytem

abstract:
  The existence, stability properties, and bifurcation diagrams of localized
patterns and hole solutions in one-dimensional extended systems is studied from
the point of view of front interactions. An adequate envelope equation is
derived from a prototype model that exhibits these particle-type solutions.
This equation allow us to obtain an analytical expression for the front
interaction, which is in good agreement with numerical simulations.


title: Nonlinear dynamics and chaos in parametric sound generation

abstract:
  A theoretical analysis of the subharmonic generation process in an acoustical
resonator (interferometer) with plane walls is performed. It is shown that,
when both the pumping wave and the generated subharmonic are detuned with
respect to the resonator modes, the fields can display complex temporal
behaviour such as self-pulsing and chaos. A discussion about the acoustical
parameters required for the experimental observation of the phenomenon is
given.


title: Gap Solitons in Periodic Discrete Nonlinear Schroedinger Equations

abstract:
  It is shown that the periodic DNLS, with cubic nonlinearity, possesses gap
solutions, i. e. standing waves, with the frequency in a spectral gap, that are
exponentially localized in spatial variable. The proof is based on the linking
theorem in combination with periodic approximations.


title: Filling Gaps in Chaotic Time Series

abstract:
  We propose a method for filling arbitrarily wide gaps in deterministic time
series. Crucial to the method is the ability to apply Takens' theorem in order
to reconstruct the dynamics underlying the time series. We introduce a
functional to evaluate how compatible is a filling sequence of data with the
reconstructed dynamics. An algorithm for minimizing the functional with a
reasonable computational effort is then discussed.


title: Mode Selection in the Spontaneous Motion of an Alcohol Droplet

abstract:
  An alcohol (pentanol) droplet exhibits spontaneous agitation on an aqueous
solution, driven by a solutal Marangoni effect. We found that the droplet's
mode of motion is controlled by its volume. A droplet with a volume of less
than $0.1 \mu\rm{l}$ shows irregular translational motion, whereas
intermediate-sized droplets of $0.1-200 \mu\rm{l}$ show vectorial motion. When
the volume is above $300 \mu\rm{l}$, the droplet splits into smaller drops.
These experimental results regarding mode selection are interpreted in terms of
the wave number selection depending on the droplet volume.


title: "Locally homogeneous turbulence" Is it an inconsistent framework?

abstract:
  In his first 1941 paper Kolmogorov assumed that the velocity has increments
which are homogeneous and independent of the velocity at a suitable reference
point. This assumption of local homogeneity is consistent with the nonlinear
dynamics only in an asymptotic sense when the reference point is far away. The
inconsistency is illustrated numerically using the Burgers equation.
Kolmogorov's derivation of the four-fifths law for the third-order structure
function and its anisotropic generalization are actually valid only for
homogeneous turbulence, but a local version due to Duchon and Robert still
holds. A Kolomogorov--Landau approach is proposed to handle the effect of
fluctuations in the large-scale velocity on small-scale statistical properties;
it is is only a mild extension of the 1941 theory and does not incorporate
intermittency effects.


title: Integrable Quasiclassical Deformations of Cubic Curves

abstract:
  A general scheme for determining and studying hydrodynamic type systems
describing integrable deformations of algebraic curves is applied to cubic
curves. Lagrange resolvents of the theory of cubic equations are used to derive
and characterize these deformations.


title:
Complex sine-Gordon-2: a new algorithm for multivortex solutions on the
  plane

abstract:
  We present a new vorticity-raising transformation for the second integrable
complexification of the sine-Gordon equation on the plane. The new
transformation is a product of four Schlesinger maps of the Painlev\'{e}-V to
itself, and allows a more efficient construction of the $n$-vortex solution
than the previously reported transformation comprising a product of $2n$ maps.


title: PI-controlled bioreactor as a generalized Lienard system

abstract:
  It is shown that periodic orbits can occur in Cholette's bioreactor model
working under the influence of a PI-controller. We find a diffeomorphic
coordinate transformation that turns this controlled enzymatic reaction system
into a generalized Lienard form. Furthermore, we give sufficient conditions for
the existence and uniqueness of limit cycles in the new coordinates. We also
perform numerical simulations illustrating the possibility of the existence of
a local center (period annulus). A result with possible practical applications
is that the oscillation frequency is a function of the integral control gain
parameter


title:
Drinfeld Twists and Symmetric Bethe Vectors of Supersymmetric Fermion
  Models

abstract:
  We construct the Drinfeld twists (factorizing $F$-matrices) of the
$gl(m|n)$-invariant fermion model. Completely symmetric representation of the
pseudo-particle creation operators of the model are obtained in the basis
provided by the $F$-matrix (the $F$-basis). We resolve the hierarchy of the
nested Bethe vectors in the $F$-basis for the $gl(m|n)$ supersymmetric model.

categories: nlin.SI cond-mat.str-el hep-th math-ph math.MP


title: Integrable 1D Toda cellular automata

abstract:
  First, we recall the algebro-geometric method of construction of finite field
valued solutions of the discrete KP equation and next we perform a reduction of
the dKP equation to the discrete 1D Toda equation. This gives a method of
construction of solutions of the discrete 1D Toda equation taking values in a
finite field.


title: Scaling properties in the production range of shear dominated flows

abstract:
  Recent developments in turbulence are focused on the effect of large scale
anisotropy on the small scale statistics of velocity increments. According to
Kolmogorov, isotropy is recovered in the large Reynolds number limit as the
scale is reduced and, in the so-called inertial range, universal features
-namely the scaling exponents of structure functions - emerge clearly. However
this picture is violated in a number of cases, typically in the high shear
region of wall bounded flows. The common opinion ascribes this effect to the
contamination of the inertial range by the larger anisotropic scales, i.e. the
residual anisotropy is assumed as a weak perturbation of an otherwise isotropic
dynamics. In this case, given the rotational invariance of the Navier-Stokes
equations, the isotropic component of the structure functions keeps the same
exponents of isotropic turbulence. This kind of reasoning fails when the
anisotropic effects are strong as in the production range of shear dominated
flows. This regime is analyzed here by means of both numerical and experimental
data for a homogeneous shear flow. A well defined scaling behavior is found to
exist, with exponents which differ substantially from those of classical
isotropic turbulence. Contrary to what predicted by the perturbation approach,
such a deep alteration concerns the isotropic sector itself. The general
validity of these results is discussed in the context of turbulence near solid
walls, where more appropriate closure models for the coarse grained
Navier-Stokes equations would be advisable.


title:
A phase diagram for spatio-temporal intermittency in the sine circle map
  lattice

abstract:
  We study the phase diagram of the sine circle map lattice with random initial
conditions and identify the various types of dynamical behaviour which appear
here. We focus on the regions which show spatio-temporal intermittency and
characterise the accompanying scaling behaviour. Directed percolation exponents
are seen at some points in the parameter space in the neighbourhood of
bifurcation boundaries. We discuss this behaviour as well as other types of
behaviour seen in the parameter space in the context of the phase diagram.


title:
Exact solitary wave solutions for a discrete $\lambda \phi^4$ field
  theory in 1+1 dimensions

abstract:
  We have found exact, periodic, time-dependent solitary wave solutions of a
discrete $\phi^4$ field theory model. For finite lattices, depending on whether
one is considering a repulsive or attractive case, the solutions are either
Jacobi elliptic functions $\sn(x,m)$ (which reduce to the kink function
$\tanh(x)$ for $m\to 1$), or they are $\dn(x,m)$ and $\cn(x,m)$ (which reduce
to the pulse function $\sech(x)$ for $m\to 1$). We have studied the stability
of these solutions numerically, and we find that our solutions are linearly
stable in most cases. We show that this model is a Hamiltonian system, and that
the effective Peierls-Nabarro barrier due to discreteness is zero not only for
the two localized modes but even for all three periodic solutions. We also
present results of numerical simulations of scattering of kink--anti-kink and
pulse--anti-pulse solitary wave solutions.


title:
Sufficient Conditions for Fast Switching Synchronization in Time Varying
  Network Topologies

abstract:
  In previous work, empirical evidence indicated that a time-varying network
could propagate sufficient information to allow synchronization of the
sometimes coupled oscillators, despite an instantaneously disconnected
topology. We prove here that if the network of oscillators synchronizes for the
static time-average of the topology, then the network will synchronize with the
time-varying topology if the time-average is achieved sufficiently fast. Fast
switching, fast on the time-scale of the coupled oscillators, overcomes the
descychnronizing decoherence suggested by disconnected instantaneous networks.
This result agrees in spirit with that of where empirical evidence suggested
that a moving averaged graph Laplacian could be used in the master-stability
function analysis. A new fast switching stability criterion here-in gives
sufficiency of a fast-switching network leading to synchronization. Although
this sufficient condition appears to be very conservative, it provides new
insights about the requirements for synchronization when the network topology
is time-varying. In particular, it can be shown that networks of oscillators
can synchronize even if at every point in time the frozen-time network topology
is insufficiently connected to achieve synchronization.


title: Vector-soliton collision dynamics in nonlinear optical fibers

abstract:
  We consider the interactions of two identical, orthogonally polarized vector
solitons in a nonlinear optical fiber with two polarization directions,
described by a coupled pair of nonlinear Schroedinger equations. We study a
low-dimensional model system of Hamiltonian ODE derived by Ueda and Kath and
also studied by Tan and Yang. We derive a further simplified model which has
similar dynamics but is more amenable to analysis. Sufficiently fast solitons
move by each other without much interaction, but below a critical velocity the
solitons may be captured. In certain bands of initial velocities the solitons
are initially captured, but separate after passing each other twice, a
phenomenon known as the two-bounce or two-pass resonance. We derive an analytic
formula for the critical velocity. Using matched asymptotic expansions for
separatrix crossing, we determine the location of these "resonance windows."
Numerical simulations of the ODE models show they compare quite well with the
asymptotic theory.


title:
Social Cognitive Maps, Swarm Perception and Distributed Search on
  Dynamic Landscapes

abstract:
  Swarm Intelligence (SI) is the property of a system whereby the collective
behaviors of (unsophisticated) entities interacting locally with their
environment cause coherent functional global patterns to emerge. SI provides a
basis with which it is possible to explore collective (or distributed) problem
solving without centralized control or the provision of a global model. To
tackle the formation of a coherent social collective intelligence from
individual behaviors, we discuss several concepts related to self-organization,
stigmergy and social foraging in animals. Then, in a more abstract level we
suggest and stress the role played not only by the environmental media as a
driving force for societal learning, as well as by positive and negative
feedbacks produced by the many interactions among agents. Finally, presenting a
simple model based on the above features, we will address the collective
adaptation of a social community to a cultural (environmental, contextual) or
media informational dynamical landscape, represented here - for the purpose of
different experiments - by several three-dimensional mathematical functions
that suddenly change over time. Results indicate that the collective
intelligence is able to cope and quickly adapt to unforeseen situations even
when over the same cooperative foraging period, the community is requested to
deal with two different and contradictory purposes. KEYWORDS: Swarm
Intelligence and Perception, Social Cognitive Maps, Social Foraging,
Self-Organization, Distributed Search and Optimization


title: Stickiness in mushroom billiards

abstract:
  We investigate dynamical properties of chaotic trajectories in mushroom
billiards. These billiards present a well-defined simple border between a
single regular region and a single chaotic component. We find that the
stickiness of chaotic trajectories near the border of the regular region occurs
through an infinite number of marginally unstable periodic orbits. These orbits
have zero measure, thus not affecting the ergodicity of the chaotic region.
Notwithstanding, they govern the main dynamical properties of the system. In
particular, we show that the marginally unstable periodic orbits explain the
periodicity and the power-law behavior with exponent $\gamma=2$ observed in the
distribution of recurrence times.


title:
A kinetic approach to Bose-Einstein condensates: self-phase modulation
  and Bogoliubov oscillations

abstract:
  A kinetic approach to the Bose-Einstein condensates (BECs) is proposed, based
on the Wigner-Moyal equation (WME). In the quasi-classical limit, the WME
reduces to the particle number conservation equation. Two examples of
application are: i) a self-phase modulation of a BE condensate beam where we
show that part of the beam is decelerated and eventually stops as a result of
the gradient of the effective self-potential; ii) the derivation of a kinetic
dispersion relation for sound waves in the BECs, including a collisionless
Landau damping.