title:
Piecewise Deterministic Quantum Dynamics and Quantum Fractals on the
  Poincare Disk

abstract:
  It is shown that piecewise deterministic dissipative quantum dynamics in a
vector space with indefinite metric can lead to well defined, positive
probabilities. The case of quantum jumps on the Poincar'e disk is studied in
details, including results of numerical simulations of quantum fractals.
  Keywords: iterated function systems, dynamical semigroups, EEQT, indefinite
metric, quantum jumps, quantum measurement, non-commuting observables,
fractals, Poincare disk


title:
Shrunk loop theorem for the topology probabilities of closed Brownian
  (or Feynman) paths on the twice punctured plane

abstract:
  The shrunk loop theorem presented here is an integral identity which
facilitates the calculation of the relative probability (or probability
amplitude) of any given topology that a free, closed Brownian or Feynman path
of a given 'duration' might have on the twice punctured plane (the plane with
two marked points). The result is expressed as a scattering series of integrals
of increasing dimensionality based on the maximally shrunk version of the path.
Physically, this applies in different contexts: (i) the topology probability of
a closed ideal polymer chain on a plane with two impassable points, (ii) the
trace of the Schroedinger Green function, and thence spectral information, in
the presence of two Aharonov-Bohm fluxes, (iii) the same with two branch points
of a Riemann surface instead of fluxes. Our theorem starts with the Stovicek
expansion for the Green function in the presence of two Aharonov-Bohm flux
lines, which itself is based on the famous Sommerfeld one puncture point
solution of 1896 (the one puncture case has much easier topology, just one
winding number). Stovicek's expansion itself can supply the results at the
expense of choosing a base point on the loop and then integrating it away. The
shrunk loop theorem eliminates this extra two dimensional integration,
distilling the topology from the geometry.


title:
Momentum Maps and Measure-valued Solutions (Peakons, Filaments and
  Sheets) for the EPDiff Equation

abstract:
  We study the dynamics of measure-valued solutions of what we call the EPDiff
equations, standing for the {\it Euler-Poincar\'e equations associated with the
diffeomorphism group (of $\mathbb{R}^n$ or an $n$-dimensional manifold $M$)}.
Our main focus will be on the case of quadratic Lagrangians; that is, on
geodesic motion on the diffeomorphism group with respect to the right invariant
Sobolev $H^1$ metric. The corresponding Euler-Poincar\'e (EP) equations are the
EPDiff equations, which coincide with the averaged template matching equations
(ATME) from computer vision and agree with the Camassa-Holm (CH) equations in
one dimension. The corresponding equations for the volume preserving
diffeomorphism group are the well-known LAE (Lagrangian averaged Euler)
equations for incompressible fluids. We first show that the EPDiff equations
are generated by a smooth vector field on the diffeomorphism group for
sufficiently smooth solutions. This is analogous to known results for
incompressible fluids--both the Euler equations and the LAE equations--and it
shows that for sufficiently smooth solutions, the equations are well-posed for
short time. In fact, numerical evidence suggests that, as time progresses,
these smooth solutions break up into singular solutions which, at least in one
dimension, exhibit soliton behavior. With regard to these non-smooth solutions,
we study measure-valued solutions that generalize to higher dimensions the
peakon solutions of the (CH) equation in one dimension. One of the main
purposes of this paper is to show that many of the properties of these
measure-valued solutions may be understood through the fact that their solution
ansatz is a momentum map. Some additional geometry is also pointed out, for
example, that this momentum map is one leg of a natural dual pair.


title: Rays, modes, wavefield structure and wavefield stability

abstract:
  Sound propagation is considered in range-independent environments and
environments consisting of a range-independent background on which a weak
range-dependent perturbation is superimposed. Recent work on propagation of
both types of environment, involving both ray- and mode-based wavefield
descriptions, have focused on the importance of $\alpha $, a ray-based
``stability parameter,'' and $\beta ,$ a mode-based ``waveguide invariant.'' It
is shown that, when $\beta $ is evaluated using asymptotic mode theory, $%
\beta =\alpha .$ Using both ray and mode concepts, known results relating to
the manner by which $\alpha $ (or $\beta $) controls both the unperturbed
wavefield structure and the stability of the perturbed wavefield are briefly
reviewed.


title: Distributed Delays Facilitate Amplitude Death of Coupled Oscillators

abstract:
  Coupled oscillators are shown to experience amplitude death for a much larger
set of parameter values when they are connected with time delays distributed
over an interval rather than concentrated at a point. Distributed delays
enlarge and merge death islands in the parameter space. Furthermore, when the
variance of the distribution is larger than a threshold the death region
becomes unbounded and amplitude death can occur for any average value of delay.
These phenomena are observed even with a small spread of delays, for different
distribution functions, and an arbitrary number of oscillators.


title: Goldstone modes in Lyapunov spectra of hard sphere systems

abstract:
  In this paper, we demonstrate how the Lyapunov exponents close to zero of a
system of many hard spheres can be described as Goldstone modes, by using a
Boltzmann type of approach. At low densities, the correct form is found for the
wave number dependence of the exponents as well as for the corresponding
eigenvectors in tangent-space. The predicted values for the Lyapunov exponents
belonging to the transverse mode are within a few percent of the values found
in recent simulations, the propagation velocity for the longitudinal mode is
within 1%, but the value for the Lyapunov exponent belonging to the
longitudinal mode deviates from the simulations by 30%. For higher densities,
the predicted values deviate more from the values calculated in the
simulations. These deviations may be due to contributions from ring collisions
and similar terms, which, even at low densities, can contribute to the leading
order.


title:
Soliton on a Cnoidal Wave Background in the coupled nonlinear
  Schr\"{o}dinger equation

abstract:
  An application of the Darboux transformation on a cnoidal wave background in
the coupled nonlinear Schr\"{o}dinger equation gives a new solution which
describes a soliton moving on a cnoidal wave. This is a generalized version of
the previously known soliton solutions of dark-bright pair. Here a dark soliton
resides on a cnoidal wave instead of on a constant background. It also exhibits
a new types of soliton solution in a self-focusing medium, which describes a
breakup of a generalized dark-bright pair into another generalized dark-bright
pair and an "oscillating" soliton. We calculate the shift of the crest of the
cnoidal wave along a soliton and the moving direction of the soliton on a
cnoidal wave.


title: Thermodynamic formalism for field driven Lorentz gases

abstract:
  We analytically determine the dynamical properties of two dimensional field
driven Lorentz gases within the thermodynamic formalism. For dilute gases
subjected to an iso-kinetic thermostat, we calculate the topological pressure
as a function of a temperature-like parameter $\ba$ up to second order in the
strength of the applied field. The Kolmogorov-Sinai entropy and the topological
entropy can be extracted from a dynamical entropy defined as a Legendre
transform of the topological pressure. Our calculations of the Kolmogorov-Sinai
entropy exactly agree with previous calculations based on a Lorentz-Boltzmann
equation approach. We give analytic results for the topological entropy and
calculate the dimension spectrum from the dynamical entropy function.


title: Perturbation Theory for the Double-Sine-Gordon Equation

abstract:
  This paper presents the perturbation theory for the double--sine--Gordon
equation. We received the system of differential equations that shows the
soliton parameters modification under perturbation's influence. In particular
case $\lambda=0$ the results of the research transform into well-known
perturbation theory for the sine--Gordon equation.


title: Construction of a natural partition of incomplete horseshoes

abstract:
  A method is presented to construct a partition of an incomplete horseshoe in
a Poincare map which is only based on unstable manifolds of outermost fixed
points and eventually their limits. Thereby this partition becomes natural from
the point of view of asymptotic scattering observations. The symbolic dynamics
derived from this partition coincides with the one derived from the
hierarchical structure of the singularities of scattering functions.


title:
Shannon information, LMC complexity and Renyi entropies: a
  straightforward approach

abstract:
  The LMC complexity, an indicator of complexity based on a probabilistic
description, is revisited. A straightforward approach allows us to establish
the time evolution of this indicator in a near-equilibrium situation and gives
us a new insight for interpreting the LMC complexity for a general non
equilibrium system. Its relationship with the Renyi entropies is also
explained. One of the advantages of this indicator is that its calculation does
not require a considerable computational effort in many cases of physical and
biological interest.

categories: nlin.AO cond-mat.stat-mech cs.IT math.IT nlin.CD physics.comp-ph q-bio.QM


title: Multiplicities of Periodic Orbit Lengths for Non-Arithmetic Models

abstract:
  Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle
groups are discussed. It is demonstrated both numerically and analytically that
at least for certain groups the mean multiplicity of periodic orbits with
exactly the same length increases exponentially with the length. The main
ingredient used is the construction of joint distribution of periodic orbits
when group matrices are transformed by field isomorphisms. The method can be
generalized to other groups for which traces of group matrices are integers of
an algebraic field of finite degree.


title:
A simple statistical explanation for the localization of energy in
  nonlinear lattices with two conserved quantities

abstract:
  The localization of energy in the discrete nonlinear Schroedinger equation is
explained with statistical methods. The partition function and the entropy of
the system are computed for low-amplitude initial conditions. Detailed
predictions for the long-time solution are derived.


title:
Intermittency is a consequence of turbulent transport in nonlinear
  systems

abstract:
  Intermittent high-amplitude structures emerge in a damped and driven discrete
nonlinear Schroedinger equation whose solutions transport both energy and
particles from sources to sinks. These coherent structures are necessary for
any solution that has statistically stationary transport properties.


title: Two twistor descriptions of the isomonodromy problem

abstract:
  The connections between Hitchin and Mason's twistor descriptions of the
isomonodromy problem are explored.


title: Quantum and Arithmetical Chaos

abstract:
  The lectures are centered around three selected topics of quantum chaos: the
Selberg trace formula, the two-point spectral correlation functions of Riemann
zeta function zeros, and of the Laplace--Beltrami operator for the modular
group. The lectures cover a wide range of quantum chaos applications and can
serve as a non-formal introduction to mathematical methods of quantum chaos.


title: Spectral properties and classical decays in quantum open systems

abstract:
  We study the relationship between the spectral properties of diffusive open
quantum maps and the classical spectrum of Ruelle-Pollicott resonances. The
leading resonances determine the asymptotic time regime for several quantities
of interest - the linear entropy, the Loschmidt echo and the correlations of
the initial state. A numerical method that allow an efficient calculation of
the leading spectrum is developed using a truncated basis adapted to the
dynamics.


title: Frequency Locking for Combustion Synthesis in Periodic Medium

abstract:
  Solutions of a 1-D free-interface problem modeling solid combustion front
propagating in combustible mixture with periodically varying concentration of
reactant exhibit classical phenomenon of mode locking. Numerical simulation
shows a variety of locked periodic, quasi-periodic and chaotic solutions.


title:
Attraction of Spiral Waves by Localized Inhomogeneities with Small-World
  Connections in Excitable Media

abstract:
  Trapping and un-trapping of spiral tips in a two-dimensional homogeneous
excitable medium with local small-world connections is studied by numerical
simulation. In a homogeneous medium which can be simulated with a lattice of
regular neighborhood connections, the spiral wave is in the meandering regime.
When changing the topology of a small region from regular connections to
small-world connections, the tip of a spiral waves is attracted by the
small-world region, where the average path length declines with the
introduction of long distant connections. The "trapped" phenomenon also occurs
in regular lattices where the diffusion coefficient of the small region is
increased. The above results can be explained by the eikonal equation and the
relation between core radius and diffusion coefficient.


title:
Intermittent chaos in a model of financial markets with heterogeneous
  agents

abstract:
  In this paper we study the price dynamics in a simple model of financial
markets with heterogeneous agents. We concentrate on how increases in the total
number of active traders influences fluctuations of asset prices. We find that
a curious route to chaos is observed when the total number of [active traders]
increases. Particularly, we show that {\it intermittent chaos} [1] of price
fluctuations is observed as the total number of trader increases.


title:
Quasiperiodic localized oscillating solutions in the discrete nonlinear
  Schr\"odinger equation with alternating on-site potential

abstract:
  We present what we believe to be the first known example of an exact
quasiperiodic localized stable solution with spatially symmetric
large-amplitude oscillations in a non-integrable Hamiltonian lattice model. The
model is a one-dimensional discrete nonlinear Schr\"odinger equation with
alternating on-site energies, modelling e.g. an array of optical waveguides
with alternating widths. The solution bifurcates from a stationary discrete gap
soliton, and in a regime of large oscillations its intensity oscillates
periodically between having one peak at the central site, and two symmetric
peaks at the neighboring sites with a dip in the middle. Such solutions, termed
'pulsons', are found to exist in continuous families ranging arbitrarily close
both to the anticontinuous and continuous limits. Furthermore, it is shown that
they may be linearly stable also in a regime of large oscillations.


title:
Combustion Process in a Spark Ignition Engine: Dynamics and Noise Level
  Estimation

abstract:
  We analyse the experimental time series of internal pressure in a four
cylinder spark ignition engine. In our experiment, performed for different
spark advance angles, apart from usual cyclic changes of engine pressure we
observed oscillations. These oscillations are with longer time scales ranging
from one to several hundred engine cycles depending on engine working
conditions. Basing on the pressure time dependence we have calculated the heat
released per cycle. Using the time series of heat release to calculate the
correlation coarse-grained entropy we estimated the noise level for internal
combustion process. Our results show that for a smaller spark advance angle the
system is more deterministic.


title:
Cycle-to-Cycle Fluctuations of Burned Fuel Mass in Spark Ignition
  Combustion Engines

abstract:
  We examine a simple, fuel-air, model of combustion in spark ignition (si)
engine with indirect injection. In our two fluid model, variations of fuel mass
burned in cycle sequences appear due to stochastic fluctuations of a fuel feed
amount. We have shown that a small amplitude of these fluctuations affects
considerably the stability of a combustion process strongly depending on the
quality of air-fulel mixture. The largest influence was found in the limit of a
lean combustion. The possible effect of nonlinearities in the combustion
process were also discussed.


title: Geometrical optics in nonlinear media and integrable equations

abstract:
  It is shown that the geometrical optics limit of the Maxwell equations for
certain nonlinear media with slow variation along one axis and particular
dependence of dielectric constant on the frequency and fields gives rise to the
dispersionless Veselov-Novikov equation for refractive index. It is
demonstrated that the last one is amenable to the quasiclassical DBAR-dressing
method. A connection is noted between geometrical optics phenomena under
consideration and quasiconformal mappings on the plane.


title: A renormalization group theory of cultural evolution

abstract:
  We present a theory of cultural evolution based upon a renormalization group
scheme. We consider rational but cognitively limited agents who optimize their
decision making process by iteratively updating and refining the mental
representation of their natural and social environment. These representations
are built around the most important degrees of freedom of their world. Cultural
coherence among agents is defined as the overlap of mental representations and
is characterized using an adequate order parameter. As the importance of social
interactions increases or agents become more intelligent, we observe and
quantify a series of dynamic phase transitions by which cultural coherence
advances in the society. A similar phase transition may explain the so-called
"cultural explosion" in human evolution some 50,000 years ago.

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


title:
Chaotic flow and efficient mixing in a micro-channel with a polymer
  solution

abstract:
  Microscopic flows are almost universally linear, laminar and stationary
because Reynolds number, $Re$, is usually very small. That impedes mixing in
micro-fluidic devices, which sometimes limits their performance. Here we show
that truly chaotic flow can be generated in a smooth micro-channel of a uniform
width at arbitrarily low $Re$, if a small amount of flexible polymers is added
to the working liquid. The chaotic flow regime is characterized by randomly
fluctuating three-dimensional velocity field and significant growth of the flow
resistance. Although the size of the polymer molecules extended in the flow may
become comparable with the micro-channel width, the flow behavior is fully
compatible with that in a table-top channel in the regime of elastic
turbulence. The chaotic flow leads to quite efficient mixing, which is almost
diffusion independent. For macromolecules, mixing time in this microscopic flow
can be three to four orders of magnitude shorter than due to molecular
diffusion.


title:
Experimental targeting and control of spatiotemporal chaos in nonlinear
  optics

abstract:
  We demonstrate targeting and control over spatiotemporal chaos in an optical
feedback loop experiment. Different stationary target patterns are stabilized
in real-time by means of a two dimensional space extended perturbation field
driven by an interfaced computer and applied in real-space to a liquid crystal
display device inserted within a control optical loop. The flexibility of the
system in switching between different target patterns is also demonstrated.


title: Self-pulsing effect in chaotic scattering

abstract:
  We study the quantum and classical scattering of Hamiltonian systems whose
chaotic saddle is described by binary or ternary horseshoes. We are interested
in parameters of the system for which a stable island, associated with the
inner fundamental periodic orbit of the system exists and is large, but chaos
around this island is well developed. In this situation, in classical systems,
decay from the interaction region is algebraic, while in quantum systems it is
exponential due to tunneling. In both cases, the most surprising effect is a
periodic response to an incoming wave packet. The period of this self-pulsing
effect or scattering echoes coincides with the mean period, by which the
scattering trajectories rotate around the stable orbit. This period of rotation
is directly related to the development stage of the underlying horseshoe.
Therefore the predicted echoes will provide experimental access to topological
information. We numerically test these results in kicked one dimensional models
and in open billiards.


title: Mixing by polymers: experimental test of decay regime of mixing

abstract:
  By using high molecular weight fluorescent passive tracers with different
diffusion coefficients and by changing the fluid velocity we study dependence
of a characteristic mixing length on the Peclet number, $Pe$, which controls
the mixing efficiency. The mixing length is found to be related to $Pe$ by a
power law, $L_{mix}\propto Pe^{0.26\pm 0.01}$, and increases faster than
expected for an unbounded chaotic flow. Role of the boundaries in the mixing
length abnormal growth is clarified. The experimental findings are in a good
quantitative agreement with the recent theoretical predictions.


title: Flow induced ultrasound scattering: experimental studies

abstract:
  Sound scattering by a finite width beam on a single rigid body rotation
vortex flow is detected by a linear array of transducers (both smaller than a
flow cell), and analyzed using a revised scattering theory. Both the phase and
amplitude of the scattered signal are obtained on 64 elements of the detector
array and used for the analysis of velocity and vorticity fields. Due to
averaging on many pulses the signal-to-noise ratio of the phases difference in
the scattered sound signal can be amplified drastically, and the resolution of
the method in the detection of circulation, vortex radius, vorticity, and
vortex location becomes comparable with that obtained earlier by time-reversal
mirror (TRM) method (P. Roux, J. de Rosny, M. Tanter, and M. Fink, {\sl Phys.
Rev. Lett.} {\bf 79}, 3170 (1997)). The revised scattering theory includes two
crucial steps, which allow overcoming limitations of the existing theories.
First, the Huygens construction of a far field scattering signal is carried out
from a signal obtained at any intermediate plane. Second, a beam function that
describes a finite width beam is introduced, which allows using a theory
developed for an infinite width beam for the relation between a scattering
amplitude and the vorticity structure function. Structure functions of the
velocity and vorticity fields deduced from the sound scattering signal are
compared with those obtained from simultaneous particle image velocimetry (PIV)
measurements. Good quantitative agreement is found.


title: Drag Reduction by a Linear Viscosity Profile

abstract:
  Drag reduction by polymers in turbulent flows raises an apparent
contradiction: the stretching of the polymers must increase the viscosity, so
why is the drag reduced? A recent theory proposed that drag reduction in
agreement with experiments is consistent with the effective viscosity growing
linearly with the distance from the wall. With this self consistent solution
the reduction in the Reynolds stress overwhelms the increase in viscous drag.
In this Letter we show, using Direct Numerical Simulations, that a linear
viscosity profile indeed reduces the drag in agreement with the theory and in
close correspondence with direct simulations of the FENE-P model at the same
flow conditions.


title: Elastic turbulence in curvilinear flows of polymer solutions

abstract:
  Following our first report (A. Groisman and V. Steinberg, $\sl Nature$ $\bf
405$, 53 (2000)) we present an extended account of experimental observations of
elasticity induced turbulence in three different systems: a swirling flow
between two plates, a Couette-Taylor (CT) flow between two cylinders, and a
flow in a curvilinear channel (Dean flow). All three set-ups had high ratio of
width of the region available for flow to radius of curvature of the
streamlines. The experiments were carried out with dilute solutions of high
molecular weight polyacrylamide in concentrated sugar syrups. High polymer
relaxation time and solution viscosity ensured prevalence of non-linear elastic
effects over inertial non-linearity, and development of purely elastic
instabilities at low Reynolds number (Re) in all three flows. Above the elastic
instability threshold, flows in all three systems exhibit features of developed
turbulence. Those include: (i)randomly fluctuating fluid motion excited in a
broad range of spatial and temporal scales; (ii) significant increase in the
rates of momentum and mass transfer (compared to those expected for a steady
flow with a smooth velocity profile). Phenomenology, driving mechanisms, and
parameter dependence of the elastic turbulence are compared with those of the
conventional high Re hydrodynamic turbulence in Newtonian fluids.


title: Spatial patterns of desynchronization bursts in networks

abstract:
  We adapt a previous model and analysis method (the {\it master stability
function}), extensively used for studying the stability of the synchronous
state of networks of identical chaotic oscillators, to the case of oscillators
that are similar but not exactly identical. We find that bubbling induced
desynchronization bursts occur for some parameter values. These bursts have
spatial patterns, which can be predicted from the network connectivity matrix
and the unstable periodic orbits embedded in the attractor. We test the
analysis of bursts by comparison with numerical experiments. In the case that
no bursting occurs, we discuss the deviations from the exactly synchronous
state caused by the mismatch between oscillators.


title:
Invariant Submanifolds of Darboux-KP Chain and Extension of the Discete
  KP Hierarchy

abstract:
  Invariant submanifolds of the so-called Darboux-KP chain are investigated. It
is shown that restriction of dynamics on some class of invariant submanifolds
yields the extension of the discrete KP hierarchy, while the intersections
leads to Lax pairs for a broad class of differential-difference systems with
finite number of fields. Some attention is given to investigation of
self-similar reductions. It is shown that self-similar ansatzes lead to purely
discrete equations with dependence on some number of parameters together with
equations governing deformations with respect to these parameters. Some
examples are provided. In particular it is shown that well known first discrete
Painleve equation (dPI) corresponds to Volterra lattice hierarchy. It is
written down equations which naturally generalize dPI in the sense that they
have first Painleve transcedent in continuous limit.


title: On a Non-local Gas Dynamics Like Integrable Hierarchy

abstract:
  We study a new non-local hierarchy of equations of the isentropic gas
dynamics type where the pressure is a non-local function of the density. We
show that the hierarchy of equations is integrable. We construct the two
compatible Hamiltonian structures and show that the first structure has three
distinct Casimirs while the second has one. The existence of Casimirs allows us
to extend the flows to local ones. We construct an infinite series of commuting
local Hamiltonians as well as three infinite series (related to the three
Casimirs) of non-local charges. We discuss the zero curvature formulation of
the system where we obtain a simple expression for the non-local conserved
charges, which also clarifies the existence of the three series from a Lie
algebraic point of view. We point out that the non-local hierarchy of
Hunter-Zheng equations can be obtained from our non-local flows when the
dynamical variables are properly constrained.


title:
Generalized hyperbolic Ernst equations for an Einstein-Maxwell-Weyl
  field

abstract:
  A new family of nonlinear partial differential equations is presented. They
represent a generalization of the hyperbolic Ernst equations for an
Einstein-Mawxell-Weyl field in general relativity. A B\"acklund transformation
for the system of equations under consideration is given, and their direct
relation to the complete Boussinesq hierarchy of soliton equations is
illustrated.


title:
An approach to the problem of generating irreducible polynomials over
  the finite field GF(2) and its relationship with the problem of periodicity
  on the space of binary sequences

abstract:
  A method for generating irreducible polynomials of degree n over the finite
field GF(2) is proposed. The irreducible polynomials are found by solving a
system of equations that brings the information on the internal properties of
the splitting field GF(2^n) . Also, the choice of a primitive normal basis
allows us to build up a natural representation of GF(2^n) in the space of
n-binary sequences. Illustrative examples are given for the lowest orders.


title: Growth of density inhomogeneities in a flow of wave turbulence

abstract:
  We consider an advection of a passive scalar by a flow which is a
superposition of random waves. We find that such a flow can lead to an
exponential growth of the passive scalar fluctuations. We calculate the growth
rate at the fourth order in wave amplitudes and find it non-zero when either
both solenoidal and potential components are present in the flow or there are
potential waves with the same frequencies but different wavenumbers.


title: Waveguiding properties of optical vortex solitons

abstract:
  We study the properties of linear and nonlinear waveguides induced by optical
vortex solitons in both self-defocusing and self-focusing nonlinear media, for
the case of a saturable nonlinear response. We demonstrate that the
vortex-induced waveguides can guide both fundamental and first-order guided
modes which together with the vortex may form, for larger amplitudes of the
guide modes, different types of composite vortex-mode vector solitons. In the
case of the self-focusing saturable medium, we demonstrate that a
large-amplitude guided mode can stabilize the ring-like vortex structure which
usually decays due to azimuthal modulational instability.


title: Recursion operators for vacuum Einstein equations with symmetries

abstract:
  Direct and inverse recursion operator is derived for the vacuum Einstein
equations for metrics with two commuting Killing vectors that are orthogonal to
a foliation by 2-dimensional leaves.


title: Study of scalar gradient fields by geometric measure theory

abstract:
  Upper bounds of the Hausdorff volume of scalar gradient field graphs are
derived by means of geometric measure theory. The approach reproduces that
scalar gradient fields along a mean imposed scalar gradient become space
filling for sufficiently high values of Schmidt numbers Sc. The bounds are
consistent with findings from recent high-resolution numerical experiments for
1<= Sc <= 64, but too rough when compared with numerical simulations. A
Reynolds number dependence of the bounds is found due to the additional scalar
gradient stretching term in the equation of motion.


title:
Probability Density Functions of Decaying Passive Scalars in Periodic
  Domains : An Application of Sinai-Yakhot Theory

abstract:
  Employing the formalism introduced by Sinai and Yakhot [PRL, 63(18), p. 1962,
1989], we study the probability density functions (pdf's) of decaying passive
scalars in periodic domains under the influence of smooth large scale velocity
fields. The particular regime we focus on is one where the normalized scalar
pdf's attain a self-similar profile in finite time, i.e., the so called strange
or statistical eigenmode regime. In accordance with the work of Sinai and
Yakhot, the central regions of the pdf's are power laws. But the details of the
pdf profiles are dependent on the physical parameters in the problem.
Interestingly, for small Peclet numbers the pdf's {\it resemble} stretched or
pure exponential functions, whereas in the limit of large Peclet numbers, there
emerges a universal Gaussian form for the pdf. Numerical simulations are used
to verify these predictions.


title:
Topological signature of deterministic chaos in short nonstationary
  signals from an optical parametric oscillator

abstract:
  Although deterministic chaos has been predicted to occur in the triply
resonant optical parametric oscillator (TROPO) fifteen years ago, experimental
evidence of chaotic behavior in this system has been lacking so far, in marked
contrast with most nonlinear systems, where chaos has been actively tracked and
found. This situation is probably linked to the high sensitivity of the TROPO
to perturbations, which adversely affects stationary operation at high power.
We report the experimental observation in this system of a burst of irregular
behavior of duration 80 microseconds. Although the system is highly
nonstationary over this time interval, a topological analysis allows us to
extract a clearcut signature of deterministic chaos from a time series segment
of only 9 base cycles (3 microseconds). This result suggests that
nonstationarity is not necessarily an obstacle to the characterization of
chaos.


title: A phase-field description of surface-tension-driven instability

abstract:
  In this paper we report on 2D numerical simulations concerning linear and
nonlinear evolution of surface-tension-driven instability in two-fluid systems
heated from below using classical and phase-field models. In the phase-field
formalism, one introduces an order parameter called phase-field function to
characterize thermodynamically the phases. All the system parameters are
assumed to vary continuously from one fluid bulk to another (as linear
functions of the phase-field). The Navier-Stokes equation (with some extra
terms) and the heat equation are written only once for the whole system. The
evolution of the phase-field is described by the Cahn-Hilliard equation. In the
sharp-interface limit the results found by the phase-field formalism recover
the results given by the classical formulation.


title:
An observation of a nascent fractal pattern in MD simulation for a
  fragmentation of an fcc lattice

abstract:
  To seek for a possible origin of fractal pattern in nature, we perform a
molecular dynamics simulation for a fragmentation of an infinite fcc lattice.
The fragmentation is induced by the initial condition of the model that the
lattice particles have the Hubble-type radial expansion velocities. As time
proceeds, the average density decreases and density fluctuation develops. By
using the box counting method, it is found that the frequency-size plot of the
density follows instantaneously a universal power-law for each Hubble constant
up to the size of a cross-over. This cross-over size corresponds to the maximum
size of fluctuation and is found to obey a dynamical scaling law as a function
of time. This instantaneous generation of a nascent fractal is purely of
dynamical origin and it shows us a new formation mechanism of a fractal
patterns different from the traditional criticality concept.


title:
Anomalous scaling and universality in hydrodynamic systems with
  power-law forcing

abstract:
  The problem of the interplay between normal and anomalous scaling in
turbulent systems stirred by a random forcing with a power law spectrum is
addressed. We consider both linear and nonlinear systems. As for the linear
case, we study passive scalars advected by a 2d velocity field in the inverse
cascade regime. For the nonlinear case, we review a recent investigation of 3d
Navier-Stokes turbulence, and we present new quantitative results for shell
models of turbulence. We show that to get firm statements is necessary to reach
considerably high resolutions due to the presence of unavoidable subleading
terms affecting all correlation functions. All findings support universality of
anomalous scaling for the small scale fluctuations.


title: Semiclassical Foundation of Universality in Quantum Chaos

abstract:
  We sketch the semiclassical core of a proof of the so-called
Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical
chaos has a quantum energy spectrum with universal fluctuations on the scale of
the mean level spacing. We show how in the semiclassical limit all system
specific properties fade away, leaving only ergodicity, hyperbolicity, and
combinatorics as agents determining the contributions of pairs of classical
periodic orbits to the quantum spectral form factor. The small-time form factor
is thus reproduced semiclassically. Bridges between classical orbits and (the
non-linear sigma model of) quantum field theory are built by revealing the
contributing orbit pairs as topologically equivalent to Feynman diagrams.


title:
Stages of Relaxation of Patterns and the Role of Stochasticity on the
  Final Stage

abstract:
  The disorder function formalism [Gunaratne et.al., Phys. Rev. E, {\bf 57},
5146 (1998)]^M is used to show that pattern relaxation in an experiment on a
vibrated layer of brass beads^M occurs in three distinct stages. During stage
I, all lengthscales associated with ^M moments of the disorder grow at a single
universal rate, given by $L(t) \sim t^{0.5}$. In stage II, pattern evolution is
non-universal and includes a range of growth indices. Relaxation in the final
stage is characterized by a single, non-universal index. We use analysis of
patterns from the Swift-Hohenberg equation to argue that mechanisms that
underlie the observed pattern evolution are linear spatio-temporal dynamics
(stage I), non-linear saturation (stage II), and stochasticity (stage III)


title:
Resonant and Non-Resonant Modulated Amplitude Waves for Binary
  Bose-Einstein Condensates in Optical Lattices

abstract:
  We consider a system of two Gross-Pitaevskii (GP) equations, in the presence
of an optical-lattice (OL) potential, coupled by both nonlinear and linear
terms. This system describes a Bose-Einstein condensate (BEC) composed of two
different spin states of the same atomic species, which interact linearly
through a resonant electromagnetic field. In the absence of the OL, we find
plane-wave solutions and examine their stability. In the presence of the OL, we
derive a system of amplitude equations for spatially modulated states which are
coupled to the periodic potential through the lowest-order subharmonic
resonance. We determine this averaged system's equilibria, which represent
spatially periodic solutions, and subsequently examine the stability of the
corresponding solutions with direct simulations of the coupled GP equations. We
find that symmetric (equal-amplitude) and asymmetric (unequal-amplitude)
dual-mode resonant states are, respectively, stable and unstable. The unstable
states generate periodic oscillations between the two condensate components,
which is possible only because of the linear coupling between them. We also
find four-mode states, but they are always unstable. Finally, we briefly
consider ternary (three-component) condensates.

categories: nlin.CD cond-mat.soft nlin.PS physics.atom-ph


title: Consistent nonlinear dynamics: identifying model inadequacy

abstract:
  Empirical modelling often aims for the simplest model consistent with the
data. A new technique is presented which quantifies the consistency of the
model dynamics as a function of location in state space. As is well-known,
traditional statistics of nonlinear models like root-mean-square (RMS) forecast
error can prove misleading. Testing consistency is shown to overcome some of
the deficiencies of RMS error, both within the perfect model scenario and when
applied to data from several physical systems using previously published
models. In particular, testing for consistent nonlinear dynamics provides
insight towards (i) identifying when a delay reconstruction fails to be an
embedding, (ii) allowing state dependent model selection and (iii) optimising
local neighbourhood size. It also provides a more relevant (state dependent)
threshold for identifying false nearest neighbours.


title: Classical R-matrices for generalized so(p,q) tops

abstract:
  An integrable deformation of the known integrable model of two interacting
p-dimensional and q-dimensional spherical tops is considered. After reduction
this system gives rise to the generalized Lagrange and the Kowalevski tops. The
corresponding Lax matrices and classical r-matrices are calculated.


title: A case for biotic morphogenesis of coniform stromatolites

abstract:
  Mathematical models have recently been used to cast doubt on the biotic
origin of stromatolites. Here by contrast we propose a biotic model for
stromatolite morphogenesis which considers the relationship between upward
growth of a phototropic or phototactic biofilm ($v$) and mineral accretion
normal to the surface ($\lambda$). These processes are sufficient to account
for the growth and form of many ancient stromatolities. Domical stromatolites
form when $v$ is less than or comparable to $\lambda$. Coniform structures with
thickened apical zones, typical of Conophyton, form when $v >> \lambda$. More
angular coniform structures, similar to the stromatolites claimed as the oldest
macroscopic evidence of life, form when $v >>> \lambda$.


title:
Scattering and Trapping of Nonlinear Schroedinger Solitons in External
  Potentials

abstract:
  Soliton motion in some external potentials is studied using the nonlinear
Schr\"odinger equation. Solitons are scattered by a potential wall. Solitons
propagate almost freely or are trapped in a periodic potential. The critical
kinetic energy for reflection and trapping is evaluated approximately with a
variational method.


title: Multidimensional integrable boundary problems

abstract:
  A method of looking for boundary conditions consistent with the integrability
property of multidimensional Kadomtsev-Petviashvili (KP) type equations is
discussed. The method is based on involutions of the Lax pair taken at the
border plane. New classes of boundary conditions for the KP and Hirota
equations are proposed consistent with the Lax pair. The boundary problem on
the stripe 0<y<1 for the KP equation is discussed, its exact solutions are
found. Ward's problem on discrete versions of the generalized Toda chains is
briefly discussed.


title:
A note on fermionic flows of the N=(1|1) supersymmetric Toda lattice
  hierarchy

abstract:
  We extend the Sato equations of the N=(1|1) supersymmetric Toda lattice
hierarchy by two new infinite series of fermionic flows and demonstrate that
the algebra of the flows of the extended hierarchy is the Borel subalgebra of
the N=(2|2) loop superalgebra.


title: Quasi-Exact Solvability and the direct approach to invariant subspaces

abstract:
  We propose a more direct approach to constructing differential operators that
preserve polynomial subspaces than the one based on considering elements of the
enveloping algebra of sl(2). This approach is used here to construct new
exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line
which are not Lie-algebraic. It is also applied to generate potentials with
multiple algebraic sectors. We discuss two illustrative examples of these two
applications: an interesting generalization of the Lam\'e potential which
posses four algebraic sectors, and a quasi-exactly solvable deformation of the
Morse potential which is not Lie-algebraic.


title:
Resonance tongues and patterns in periodically forced reaction-diffusion
  systems

abstract:
  Various resonant and near-resonant patterns form in a light-sensitive
Belousov-Zhabotinsky (BZ) reaction in response to a spatially-homogeneous
time-periodic perturbation with light. The regions (tongues) in the forcing
frequency and forcing amplitude parameter plane where resonant patterns form
are identified through analysis of the temporal response of the patterns.
Resonant and near-resonant responses are distinguished. The unforced BZ
reaction shows both spatially-uniform oscillations and rotating spiral waves,
while the forced system shows patterns such as standing-wave labyrinths and
rotating spiral waves. The patterns depend on the amplitude and frequency of
the perturbation, and also on whether the system responds to the forcing near
the uniform oscillation frequency or the spiral wave frequency. Numerical
simulations of a forced FitzHugh-Nagumo reaction-diffusion model show both
resonant and near-resonant patterns similar to the BZ chemical system.


title:
Soliton Lattice and Single Soliton Solutions of the Associated Lam\'e
  and Lam\'e Potentials

abstract:
  We obtain the exact nontopological soliton lattice solutions of the
Associated Lam\'e equation in different parameter regimes and compute the
corresponding energy for each of these solutions. We show that in specific
limits these solutions give rise to nontopological (pulse-like) single
solitons, as well as to different types of topological (kink-like) single
soliton solutions of the Associated Lam\'e equation. Following Manton, we also
compute, as an illustration, the asymptotic interaction energy between these
soliton solutions in one particular case. Finally, in specific limits, we
deduce the soliton lattices, as well as the topological single soliton
solutions of the Lam\'e equation, and also the sine-Gordon soliton solution.


title:
Total and partial amplitude death in networks of diffusively coupled
  oscillators

abstract:
  Networks of weakly nonlinear oscillators are considered with diffusive and
time-delayed coupling. Averaging theory is used to determine parameter ranges
for which the network experiences amplitude death, whereby oscillations are
quenched and the equilibrium solution has a large domain of attraction. The
amplitude death is shown to be a common phenomenon, which can be observed
regardless of the precise nature of the nonlinearities and under very general
coupling conditions. In addition, when the network consists of dissimilar
oscillators, there exist parameter values for which only parts of the network
are suppressed. Sufficient conditions are derived for total and partial
amplitude death in arbitrary network topologies with general nonlinearities,
coupling coefficients, and connection delays.


title: Chaos synchronization in long-range coupled map lattices

abstract:
  We investigate the synchronization phenomenon in coupled chaotic map lattices
where the couplings decay with distance following a power-law. Depending on the
lattice size, the coupling strength and the range of the interactions, complete
chaos synchronization may be attained. The synchronization domain in parameter
space can be analytically delimited by means of the condition of negativity of
the largest transversal Lyapunov exponent. Here we analyze in detail the role
of all the system parameters in the ability of the lattice to achieve complete
synchronization, testing analytical results with the outcomes of numerical
experiments.


title: Numerical Techniques for the Study of Long-Time Correlations

abstract:
  In the study of long-time correlations extremely long orbits must be
calculated. This may be accomplished much more reliably using fixed-point
arithmetic. Use of this arithmetic on the Cray-1 computer is illustrated.


title: Fluctuation Formula in the Nose-Hoover thermostated Lorentz gas

abstract:
  In this paper we examine numerically the Gallavotti-Cohen fluctuation formula
for phase-space contraction rate and entropy production rate fluctuations in
the Nos\'e-Hoover thermostated periodic Lorentz gas. Our results indicate that
while the phase-space contraction rate fluctuations violate the fluctuation
formula near equilibrium states, the entropy production rate fluctuations obey
this formula near and far from equilibrium states as well.


title: Quantum decay of an open chaotic system: a semiclassical approach

abstract:
  We study the quantum probability to survive in an open chaotic system in the
framework of the van Vleck-Gutzwiller propagator and present the first such
calculation that accounts for quantum interference effects. Specifically we
calculate quantum deviations from the classical decay after the break time for
both broken and preserved time-reversal symmetry. The source of these
corrections is identified in interfering pairs of correlated classical
trajectories. In our approach the quantized chaotic system is modelled by a
quatum graph.


title: Dimension dependent energy thresholds for discrete breathers

abstract:
  Discrete breathers are time-periodic, spatially localized solutions of the
equations of motion for a system of classical degrees of freedom interacting on
a lattice. We study the existence of energy thresholds for discrete breathers,
i.e., the question whether, in a certain system, discrete breathers of
arbitrarily low energy exist, or a threshold has to be overcome in order to
excite a discrete breather. Breather energies are found to have a positive
lower bound if the lattice dimension d is greater than or equal to a certain
critical value d_c, whereas no energy threshold is observed for d<d_c. The
critical dimension d_c is system dependent and can be computed explicitly,
taking on values between zero and infinity. Three classes of Hamiltonian
systems are distinguished, being characterized by different mechanisms
effecting the existence (or non-existence) of an energy threshold.


title: The case of the missing neutrality

abstract:
  The concept of neutral evolutionary networks being a significant factor in
evolutionary dynamics was first proposed by Huynen {\em et al.} about 7 years
ago. In one sense, the principle is easy to state -- because most mutations to
an organism are deleterious, one would expect that neutral mutations that don't
affect the phenotype will have disproportionately greater representation
amongst successor organisms than one would expect if each mutation was equally
likely. So it was with great surprise that I noted neutral mutations being very
rare in a visualisation of phylogenetic trees generated in {\em Tierra}, since
I already knew that there was a significant amount of neutrality in the Tierra
genotype-phenotype map. The paper reports on an investigation into this
mystery.


title: Corrections of the NIST Statistical Test Suite for Randomness

abstract:
  It is well known that the NIST statistical test suite was used for the
evaluation of AES candidate algorithms. We have found that the test setting of
Discrete Fourier Transform test and Lempel-Ziv test of this test suite are
wrong. We give four corrections of mistakes in the test settings. This suggests
that re-evaluation of the test results should be needed.


title:
Dynamics of positive- and negative-mass solitons in optical lattices and
  inverted traps

abstract:
  We study the dynamics of one-dimensional solitons in the attractive and
repulsive Bose-Einstein condensates (BECs) loaded into an optical lattice (OL),
which is combined with an external parabolic potential. First, we demonstrate
analytically that, in the repulsive BEC, where the soliton is of the gap type,
its effective mass is \emph{negative}. This gives rise to a prediction for the
experiment: such a soliton cannot be not held by the usual parabolic trap, but
it can be captured (performing harmonic oscillations) by an anti-trapping
inverted parabolic potential. We also study the motion of the soliton a in long
system, concluding that, in the cases of both the positive and negative mass,
it moves freely, provided that its amplitude is below a certain critical value;
above it, the soliton's velocity decreases due to the interaction with the OL.
At a late stage, the damped motion becomes chaotic. We also investigate the
evolution of a two-soliton pulse in the attractive model. The pulse generates a
persistent breather, if its amplitude is not too large; otherwise, fusion into
a single fundamental soliton takes place. Collisions between two solitons
captured in the parabolic trap or anti-trap are considered too. Depending on
their amplitudes and phase difference, the solitons either perform stable
oscillations, colliding indefinitely many times, or merge into a single
soliton. Effects reported in this work for BECs can also be formulated for
optical solitons in nonlinear photonic crystals. In particular, the capture of
the negative-mass soliton in the anti-trap implies that a bright optical
soliton in a self-defocusing medium with a periodic structure of the refractive
index may be stable in an anti-waveguide.


title:
Alternative pathways of dewetting for a thin two-layer film of soft
  matter

abstract:
  We consider two stacked ultra-thin layers of different liquids on a solid
substrate. Using long-wave theory, we derive coupled evolution equations for
the free liquid-liquid and liquid-gas interfaces. Linear and non-linear
analyses show that depending on the long-range van-der-Waals forces and the
ratio of the layer thicknesses, the system follows different pathways of
dewetting. The instability may be driven by varicose or zigzag modes and leads
to film rupture either at the liquid-gas interface or at the substrate.


title:
Intermittency and the passive nature of the magnitude of the magnetic
  field

abstract:
  It is shown that the statistical properties of the magnitude of the magnetic
field in turbulent electrically conducting media resemble, in the inertial
range, those of passive scalars in fully developed three-dimensional fluid
turbulence. This conclusion, suggested by the data from Advanced Composition
Explorer, is supported by a brief analysis of the appropriate
magnetohydrodynamic equations.


title:
"Clusterization" and intermittency of temperature fluctuations in
  turbulent convection

abstract:
  Temperature time traces are obtained in turbulent thermal convection at high
Rayleigh numbers. Measurements are made in the midplane of the apparatus, near
the sidewall but outside the boundary layer. A telegraph approximation for
temperature traces is generated by setting the fluctuation amplitude to 1 or 0
depending on whether or not it exceeds the mean value. Unlike the standard
diagnostics of intermittency, the telegraph approximation allows one to
distinguish the tendency of events to cluster ("clusterization") from their
large-scale variability in amplitude. A qualitative conclusion is that
amplitude intermittency might mitigate clusterization effects.


title:
Complex Behaviour in a Discrete Coupled Logistic Model for the Symbiotic
  Interaction of Two Species

abstract:
  A symmetrical cubic discrete coupled logistic equation is proposed to model
the symbiotic interaction of two isolated species. The coupling depends on the
population size of both species and on a positive constant $\lambda$, named the
mutual benefit. Different dynamical regimes are obtained when the mutual
benefit is modified. For small $\lambda$, the species become extinct. For
increasing $\lambda$, the system stabilizes in a synchronized state or
oscillates in a 2 periodic orbit. For the greatest permitted values of
$\lambda$, the dynamics evolves into a quasiperiodic, into a chaotic scenario
or into extinction. The basins for these regimes are visualized as coloured
figures on the plane. These patterns suffer different change as consequence of
basins' bifurcations. The use of the critical curves let us to determine the
influence of the zones with different number of first rank preimages in those
bifurcation mechanisms.


title:
Two approaches for Helmholtz equation: generalized Darboux Trasformation
  and the method of d-bar problem

abstract:
  Two approaches to solution of the two-dimensional Helmholtz equation with a
"wave number" are proposed. The results can be applied both in numerical areas
of physics and in the theory of nonlinear equations. The first approach is
based on the requirement of the covariance of equation under the generalized
Darboux transformation (Moutard transformation). It allows to construct a new
solution of equation, using a given initial solution of the equation.
Simultaneously we obtain the "dressing" relation for the "wave number". The
simplest examples of the approach are considered in detail. In the second
approach the Green-Cauchy formula (the $\bar \partial$-method) is applied to
reduce the solution of the equation to the solution of a system of singular
integral equations.


title: Calculating adiabatic evolution of the perturbed DNLS/MNLS solitons

abstract:
  A symbolic computation technique is developed to calculate adiabatic
evolution equations for parameters of the perturbed DNLS/MNLS solitons obtained
by the recently developed direct perturbation theory [X.-J. Chen and J. Yang,
Phys. Rev. E {\bf 65}, 066608(2002)]. Effects of the intrapulse Raman
scattering, third-order group velocity dispersion, and narrow-banded filters
with nonlinear gain on the MNLS solitons are studied as examples.


title:
Evidence of universality for the May-Wigner stability theorem for random
  networks with local dynamics

abstract:
  We consider a random network of nonlinear maps exhibiting a wide range of
local dynamics, with the links having normally distributed interaction
strengths. The stability of such a system is examined in terms of the
asymptotic fraction of nodes that persist in a non-zero state. Scaling results
show that the probability of survival in the steady state agrees remarkably
well with the May-Wigner stability criterion derived from linear stability
arguments. This suggests universality of the complexity-stability relation for
random networks with respect to arbitrary global dynamics of the system.


title: Coarse-Grained Picture for Controlling Complex Quantum Systems

abstract:
  We propose a coarse-grained picture to control ``complex'' quantum dynamics,
i.e., multi-level-multi-level transition with a random interaction. Assuming
that optimally controlled dynamics can be described as a Rabi-like oscillation
between an initial and final state, we derive an analytic optimal field as a
solution to optimal control theory. For random matrix systems, we numerically
confirm that the analytic optimal field steers an initial state to a target
state which both contains many eigenstates.


title: A Note on the Painleve Property of Coupled KdV Equations

abstract:
  We prove that one system of coupled KdV equations, claimed by Hirota, Hu, and
Tang to pass the Painleve test for integrability, actually fails the test at
the highest resonance of the generic branch and therefore must be
non-integrable.


title: Coarse-Grained Picture for Controlling Quantum Chaos

abstract:
  We propose a coarse-grained picture to analyze control problems for quantum
chaos systems. Using optimal control theory, we first show that almost perfect
control is achieved for random matrix systems and a quantum kicked rotor.
Second, under the assumption that the controlled dynamics is well described by
a Rabi-type oscillaion between unperturbed states, we derive an analytic
expression for the optimal field. Finally we numerically confirm that the
analytic field can steer an initial state to a target state in random matrix
systems.


title: Updating Schemes in Random Boolean Networks: Do They Really Matter?

abstract:
  In this paper we try to end the debate concerning the suitability of
different updating schemes in random Boolean networks (RBNs). We quantify for
the first time loose attractors in asyncrhonous RBNs, which allows us to
analyze the complexity reduction related to different updating schemes. We also
report that all updating schemes yield very similar critical stability values,
meaning that the "edge of chaos" does not depend much on the updating scheme.
After discussion, we conclude that synchonous RBNs are justifiable theoretical
models of biological networks.

categories: nlin.AO cond-mat.other cs.CC nlin.CG q-bio.MN q-bio.OT q-bio.QM


title:
Combining the ApEn statistic with surrogate data analysis for the
  detection of nonlinear dynamics in time series

abstract:
  We tested the natural combination of surrogate data analysis with the ApEn
regularity statistic developed by Pincus [Proc. Natl. Acad. Sci. USA 88 (1991)
2297] by applying it to some popular models of nonlinear dynamics and publicly
available experimental time series. We found that this easily implemented
combination provided a useful method for discriminating signals governed by
nonlinear dynamics from those governed by linear dynamics and noise. An
apparently novel physical interpretation of ApEn also is supplied.


title: Quantization scheme for modular q-difference equations

abstract:
  Modular pairs of some second order q-difference equations are considered.
These equations may be interpreted as a quantum mechanics of a sort of
hyperelliptic pendulum. It is shown the quantization of a spectrum may be
provided by the condition of the analyticity of the wave function. Baxter's t-Q
equations for the quantum relativistic Toda chain in the ``strong coupling
regime'' are related to the system considered, and the quantization condition
for Q-operator is also considered.


title: Multiscale SOC in turbulent convection

abstract:
  Using data obtained in a laboratory thermal convection experiment at high
Rayleigh numbers, it is shown that the multiscaling properties of the observed
mean wind reversals are quantitatively consistent with analogous multiscaling
properties of the Bak-Tang-Wiesenfeld prototype model of self-organized
criticality in two dimensions.


title:
Non-Markov stochastic dynamics of real epidemic process of respiratory
  infections

abstract:
  The study of social networks and especially of the stochastic dynamics of the
diseases spread in human population has recently attracted considerable
attention in statistical physics. In this work we present a new statistical
method of analyzing the spread of epidemic processes of grippe and acute
respiratory track infections (ARTI) by means of the theory of discrete
non-Markov stochastic processes. We use the results of our last theory (Phys.
Rev. E 65, 046107 (2002)) to study statistical memory effects, long - range
correlation and discreteness in real data series, describing the epidemic
dynamics of human ARTI infections and grippe. We have carried out the
comparative analysis of the data of the two infections (grippe and ARTI) in one
of the industrial districts of Kazan, one of the largest cities of Russia. The
experimental data are analyzed by the power spectra of the initial time
correlation function and the memory functions of junior orders, the phase
portraits of the four first dynamic variables, the three first points of the
statistical non-Markov parameter and the locally averaged kinetic and
relaxation parameters. The received results give an opportunity to provide
strict quantitative description of the regular and stochastic components in
epidemic dynamics of social networks taking into account their time
discreteness and effects of statistical memory. They also allow to reveal the
degree of randomness and predictability of the real epidemic process in the
specific social network.


title:
Stringent Numerical Test of the Poisson Distribution for Finite Quantum
  Integrable Hamiltonians

abstract:
  Using a new class of exactly solvable models based on the pairing
interaction, we show that it is possible to construct integrable Hamiltonians
with a Wigner distribution of nearest neighbor level spacings. However, these
Hamiltonians involve many-body interactions and the addition of a small
integrable perturbation very quickly leads the system to a Poisson
distribution. Besides this exceptional cases, we show that the accumulated
distribution of an ensemble of random integrable two-body pairing hamiltonians
is in perfect agreement with the Poisson limit. These numerical results for
quantum integrable Hamiltonians provide a further empirical confirmation to the
work of the Berry and Tabor in the semiclassical limit.


title: Analysis of Data Clusters Obtained by Self-Organizing Methods

abstract:
  The self-organizing methods were used for the investigation of financial
market. As an example we consider data time-series of Dow Jones index for the
years 2002-2003 (R. Mantegna, cond-mat/9802256). In order to reveal new
structures in stock market behavior of the companies drawing up Dow Jones index
we apply SOM (Self-Organizing Maps) and GMDH (Group Method of Data Handling)
algorithms. Using SOM techniques we obtain SOM-maps that establish a new
relationship in market structure. Analysis of the obtained clusters was made by
GMDH.


title: Overcoming Obstacles to Integrability I: Perturbed Diffusive Systems

abstract:
  Obstacles to integrability sometimes hamper the standard Normal Form analysis
of perturbed integrable evolution equations. One is then forced to account for
them by the Normal Form, which is the dynamical equation obeyed by the
zero-order term. This spoils the integrability of the normal form, and the
simple wave-nature of the zero-order approximation. To avoid both undesired
results, one must require that the obstacles be accounted for by the
higher-order terms in the expansion of the solution (the Near-Identity
Transformation). We show that this goal can be achieved if the higher-order
terms are allowed to depend explicitly on the independent variables,t and x (an
option that is not considered in the standard NF analysis). In addition, a
particular algorithm for the construction of the NIT leads to a canonical form
for the obstacles; they are expressed in terms of the symmetries of the
unperturbed equation. The canonical form ensures the explicit vanishing of the
obstacles in the case of a single-wave zero-order solution. In the case of a
multiple-wave solution, the effect of the canonical obstacles on the
higher-order terms in the NIT is confined to the region of interaction among
the waves. This often facilitates the derivation of closed-form expressions for
the asymptotic effect of the obstacles. These ideas are demonstrated for the
cases of the perturbed Burgers and heat diffusion equations.


title: Compatible Poisson Structures of Toda Type Discrete Hierarchy

abstract:
  An algebra isomorphism between algebras of matrices and difference operators
is used to investigate the discrete integrable hierarchy. We find local and
non-local families of R-matrix solutions to the modified Yang-Baxter equation.
The three R-theoretic Poisson structures and the Suris quadratic bracket are
derived. The resulting family of bi-Poisson structures include a seminal
discrete bi-Poisson structure of Kupershmidt at a special value.


title: Encircling an Exceptional Point

abstract:
  We calculate analytically the geometric phases that the eigenvectors of a
parametric dissipative two-state system described by a complex symmetric
Hamiltonian pick up when an exceptional point (EP) is encircled. An EP is a
parameter setting where the two eigenvalues and the corresponding eigenvectors
of the Hamiltonian coalesce. We show that it can be encircled on a path along
which the eigenvectors remain approximately real and discuss a microwave cavity
experiment, where such an encircling of an EP was realized. Since the
wavefunctions remain approximately real, they could be reconstructed from the
nodal lines of the recorded spatial intensity distributions of the electric
fields inside the resonator. We measured the geometric phases that occur when
an EP is encircled four times and thus confirmed that for our system an EP is a
branch point of fourth order.


title:
An Experimental Study of Robustness to Asynchronism for Elementary
  Cellular Automata

abstract:
  Cellular Automata (CA) are a class of discrete dynamical systems that have
been widely used to model complex systems in which the dynamics is specified at
local cell-scale. Classically, CA are run on a regular lattice and with perfect
synchronicity. However, these two assumptions have little chance to truthfully
represent what happens at the microscopic scale for physical, biological or
social systems. One may thus wonder whether CA do keep their behavior when
submitted to small perturbations of synchronicity.
  This work focuses on the study of one-dimensional (1D) asynchronous CA with
two states and nearest-neighbors. We define what we mean by ``the behavior of
CA is robust to asynchronism'' using a statistical approach with macroscopic
parameters. and we present an experimental protocol aimed at finding which are
the robust 1D elementary CA. To conclude, we examine how the results exposed
can be used as a guideline for the research of suitable models according to
robustness criteria.


title: Superscars

abstract:
  Wave functions of plane polygonal billiards are investigated. It is
demonstrated that they have clear structures (superscars) related with families
of classical periodic orbits which do not disappear at large energy.


title:
Long-range dispersion and spatial diffusion of fault waves in the
  Burridge-Knopoff earthquake model

abstract:
  The Burridge-Knopoff model of earthquakes has recently gained increased
interest for the consistency of the predicted energy released by sismic faults,
with the Gutenberg-Richter scaling law. The present work suggests an
improvement of this model to account for long-range dispersions and large
spatial diffusion of sismic faults. An enhancement of the threshold speed of
shock waves driven by translated fault fronts is pointed out and shown to
result from the interactions between components of the system situated far
aways them and others. Due to the enhanced threshold speed, size of the sismic
fault gets increased but a control effect can still be gained from tunable
dispersion extent irrespective of the total length of the system. To the
viewpoint of the Burridge-Knopoff block-lattice model, this last consideration
introduces the possibility of sizable but finite interactions among infinitely
aligned massive blocks. Implications on the fault wave propagation are examined
by numerical simulations of the improved nonlinear partial differential
equation.


title:
Nonlinear optical responses and gap-soliton lattices in layered
  dielectric materials

abstract:
  A new approach to the nonlinear reponse of a multiplayer structure consisting
of alternating dielectric materials with intensity-dependent dielectric
constants is considered in both analytical and numerical viewpoints. The
nonlinear feedback of the structure gives rise to an artificial potential
acting as a mean-field feedback on each single dielectric material due to other
layers. For superlattices resulting from a regular periodic arrangement of an
infinite number of single dielectric layers, it is shown that an analytical
expression of this mean-field feedback can be obtained and waveshapes of the
associate gap solitons are derived. The case of a multilayer with dispersive
interlayer interactions is considered numerically and leads to strongly
localized gap solitons. Implicatons of the artificial soliton structure as for
possible theoretical modelling of soliton-compression phenomena are discussed.


title:
Quasi-continuum approximation to the Nonlinear Schr\"odinger equation
  with Long-range dispersions

abstract:
  The long-wavelength, weak-dispersion limit of the discrete nonlinear
Schr\"odinger equation with long-range dispersion is analytically considered.
This continuum approximation is carried out irrespective of the dispersion
range and hence can be assumed exact in the weak dispersion regime. For
nonlinear Schr\"odinger equations showing finite dispersion extents, the
long-range parameter is still a relevant control parameter allowing to tune the
dispersion from short-range to long-range regimes with respect to the
dispersion extent. The long-range Kac-Baker potential becomes unappropriate in
this context owing to an "edge anomaly" consisting of vanishing maximum
dispersion frequency and group velocity(and in turn soliton width) in the
"Debye" limit. An improved Kac-Baker potential is then considered which gives
rise to a non-zero maximum frequency, and allows for soliton excitations with
finite widths in the nonlinear Schr\"odinger system subjected to the long-range
but finite-extent dispersion.


title: On the differential invariants of a family of diffusion equations

abstract:
  The equivalence transformation algebra $L_{\cal E}$ for the class of
equations $u_t -u_{xx}=f(u, u_x) $ is obtained. After getting the differential
invariants with respect to $L_{\cal E}$, some results which allow to linearize
a subclass of equations are showed. Equations like the standard deterministic
KPZ equation fall in this subclass.


title: Signatures of homoclinic motion in quantum chaos

abstract:
  Homoclinic motion plays a key role in the organization of classical chaos in
Hamiltonian systems. In this Letter, we show that it also imprints a clear
signature in the corresponding quantum spectra. By numerically studying the
fluctuations of the widths of wavefunctions localized along periodic orbits we
reveal the existence of an oscillatory behavior, that is explained solely in
terms of the primary homoclinic motion. Furthermore, our results indicate that
it survives the semiclassical limit.


title: How can we think the complex?

abstract:
  This chapter does not deal with specific tools and techniques for managing
complex systems, but proposes some basic concepts that help us to think and
speak about complexity. We review classical thinking and its intrinsic
drawbacks when dealing with complexity. We then show how complexity forces us
to build models with indeterminacy and unpredictability. However, we can still
deal with the problems created in this way by being adaptive, and profiting
from a complex system's capability for selforganization, and the distributed
intelligence this may produce.


title: Multifractal concentrations of inertial particles in smooth random flows

abstract:
  Collisionless suspensions of inertial particles (finite-size impurities) are
studied in 2D and 3D spatially smooth flows. Tools borrowed from the study of
random dynamical systems are used to identify and to characterise in full
generality the mechanisms leading to the formation of strong inhomogeneities in
the particle concentration.
  Phenomenological arguments are used to show that in 2D, heavy particles form
dynamical fractal clusters when their Stokes number (non-dimensional viscous
friction time) is below some critical value. Numerical simulations provide
strong evidence for this threshold in both 2D and 3D and for particles not only
heavier but also lighter than the carrier fluid. In 2D, light particles are
found to cluster at discrete (time-dependent) positions and velocities in some
range of the dynamical parameters (the Stokes number and the mass density ratio
between fluid and particles). This regime is absent in 3D, where evidence is
that the Hausdorff dimension of clusters in phase space (position-velocity)
remains always above two.
  After relaxation of transients, the phase-space density of particles becomes
a singular random measure with non-trivial multiscaling properties. Theoretical
results about the projection of fractal sets are used to relate the
distribution in phase space to the distribution of the particle positions.
Multifractality in phase space implies also multiscaling of the spatial
distribution of the mass of particles. Two-dimensional simulations, using
simple random flows and heavy particles, allow the accurate determination of
the scaling exponents: anomalous deviations from self-similar scaling are
already observed for Stokes numbers as small as $10^{-4}$.


title:
Macroscopic detection of the strong stochasticity threshold in
  Fermi-Pasta-Ulam chains of oscillators

abstract:
  The largest Lyapunov exponent of a system composed by a heavy impurity
embedded in a chain of anharmonic nearest-neighbor Fermi-Pasta-Ulam oscillators
is numerically computed for various values of the impurity mass $M$. A
crossover between weak and strong chaos is obtained at the same value
$\epsilon_{_T}$ of the energy density $\epsilon$ (energy per degree of freedom)
for all the considered values of the impurity mass $M$. The threshold $\epsi
lon_{_T}$ coincides with the value of the energy density $\epsilon$ at which a
change of scaling of the relaxation time of the momentum autocorrelation
function of the impurity ocurrs and that was obtained in a previous work ~[M.
Romero-Bastida and E. Braun, Phys. Rev. E {\bf65}, 036228 (2002)]. The complete
Lyapunov spectrum does not depend significantly on the impurity mass $M$. These
results suggest that the impurity does not contribute significantly to the
dynamical instability (chaos) of the chain and can be considered as a probe for
the dynamics of the system to which the impurity is coupled. Finally, it is
shown that the Kolmogorov-Sinai entropy of the chain has a crossover from weak
to strong chaos at the same value of the energy density that the crossover
value $\epsilon_{_T}$ of largest Lyapunov exponent. Implications of this result
are discussed.


title: On Separation of Variables for Homogeneous SL(r) Gaudin Systems

abstract:
  By means of a recently introduced bihamiltonian structure for the homogeneous
Gaudin models, we find a new set of Separation Coordinates for the sl(r) case.


title: Saturation of Turbulent Drag Reduction in Dilute Polymer Solutions

abstract:
  Drag reduction by polymers in turbulent wall-bounded flows exhibits universal
and non-universal aspects. The universal maximal mean velocity profile was
explained in a recent theory. The saturation of this profile and the crossover
back to the Newtonian plug are non-universal, depending on Reynolds number Re,
concentration of polymer $c_p$ and the degree of polymerization $N_p$. We
explain the mechanism of saturation stemming from the finiteness of
extensibility of the polymers, predict its dependence on $c_p$ and $N$ in the
limit of small $c_p$ and large Re, and present the excellent comparison of our
predictions to experiments on drag reduction by DNA.


title: Theoretical derivation of 1/f noise in quantum chaos

abstract:
  It was recently conjectured that 1/f noise is a fundamental characteristic of
spectral fluctuations in chaotic quantum systems. This conjecture is based on
the behavior of the power spectrum of the excitation energy fluctuations, which
is different for chaotic and integrable systems. Using random matrix theory we
derive theoretical expressions that explain the power spectrum behavior at all
frequencies. These expressions reproduce to a good approximation the power laws
of type 1/f (1/f^2) characteristics of chaotic (integrable) systems, observed
in almost the whole frequency domain. Although we use random matrix theory to
derive these results, they are also valid for semiclassical systems.


title: Universal spectral statistics in quantum graphs

abstract:
  We prove that the spectrum of an individual chaotic quantum graph shows
universal spectral correlations, as predicted by random--matrix theory. The
stability of these correlations with regard to non--universal corrections is
analyzed in terms of the linear operator governing the classical dynamics on
the graph.


title: Patterns that Grow

abstract:
  Chemotactic bacteria have been observed to congregate into highly regular
patterns. When the bacteria are placed in the center of a dish, a wave of
bacteria can travel outward, leaving a regular pattern of spots or stripes in
its wake. Although chemotaxis and excretion of an attractant can readily cause
a pattern forming instability from a uniform state, they are not capable of
generating patterns starting from a single spot. These patterns are apparently
formed with the help of bacterial growth and depletion of nutrients in the
growth medium.


title: Dispersion of tracer particles in a compressible flow

abstract:
  The turbulent diffusion of Lagrangian tracer particles has been studied in a
flow on the surface of a large tank of water and in computer simulations. The
effect of flow compressibility is captured in images of particle fields. The
velocity field of floating particles has a divergence, whose probability
density function shows exponential tails. Also studied is the motion of pairs
and triplets of particles. The mean square separation <Delta(t)^2> is fitted to
the scaling form <Delta(t)^2> ~ t^alpha, and in contrast with the
Richardson-Kolmogorov prediction, an extended range with a reduced scaling
exponent of alpha=1.65 pm 0.1 is found. Clustering is also manifest in strongly
deformed triangles spanned within triplets of tracers.


title: Lagrangian statistics in fully developed turbulence

abstract:
  The statistical properties of fluid particles transported by a fully
developed turbulent flow are investigated by means of high resolution direct
numerical simulations. Single trajectory statistics is investigated in a time
range spanning more than three decades, from less than a tenth of the
Kolmogorov timescale up to one large-eddy turnover time. Acceleration and
velocity statistics show a neat quantitative agreement with recent experimental
results. Trapping effects in vortex filaments give rise to enhanced small-scale
intermittency on Lagrangian observables.


title: Another integrable case in the Lorenz model

abstract:
  A scaling invariance in the Lorenz model allows one to consider the usually
discarded case sigma=0. We integrate it with the third Painlev\'e function.


title: Measurement of Mean Flows in Faraday Waves

abstract:
  We measure the velocities of the mean flows which are driven by curved rolls
in a pattern formation system. Curved rolls in Faraday waves are generated in
experimental cells consisting of channels with varying widths. The mean flow
magnitudes are found to scale linearly with roll curvatures and squares of wave
amplitudes, agreeing with prediction from the analysis of phase dynamics
expansion. The effects of the mean flows on reducing roll curvatures are also
seen.


title:
Non-regular (non-St\"ackel) R-separation and general modulated soliton
  of wave equation

abstract:
  We present a class of orthogonal non-regular in a sense of Kalnins and Miller
(hence non-St\"ackel) coordinates which are R-separable in 3-dim. Helmholtz
equation. One family of parametric surfaces consists of parallel Dupin
cyclides, the other two consist of circular cones. This coordinate system is
used to simplify derivation of Friedlander's formulae for a general "simple
progressive solution of the wave equation" (modulated soliton of wave equation)
in $E^3$ and to correct some errors of his paper. The extended version of the
paper will be published soon.


title: Homogeneous shear turbulence

abstract:
  The addition of suitable volume forces to the Navier-Stokes equation allows
to simulate flows in the presence of a homogeneous shear. Because of the
explicit form of the driving the flows are accessible to rigorous mathematical
treatment and to accurate quantitative modelling in their global properties.
The statistics of the fluctuations provide insight into generic behaviour of
non-equilibrium systems and into the presistence of anisotropies at small
scales. Correlation functions can be used to identify dominant large scale
dynamical processes that are relevant for most of the momentum transport across
the shear. The numerical studies of homogeneous shear flows complement
analytical and experimental investigations and contribute to bridging the gap
between ideal homogeneous isotropic turbulence and the more realistic
heterogeneous turbulence.


title:
Long-Time Behaviour and Self-Similarity in a Coagulation Equation with
  Input of Monomers

abstract:
  For a coagulation equation with Becker-Doring type interactions and
time-independent monomer input we study the detailed long-time behaviour of
nonnegative solutions and prove the convergence to a self-similar function.


title:
The self-adjoint 5-point and 7-point difference operators, the
  associated Dirichlet problems, Darboux transformations and Lelieuvre formulas

abstract:
  We present some basic properties of two distinguished discretizations of
elliptic operators: the self-adjoint 5-point and 7-point schemes on a two
dimensional lattice. We first show that they allow to solve Dirichlet boundary
value problems; then we present their Darboux transformations. Finally we
construct their Lelieuvre formulas and we show that, at the level of the normal
vector and in full analogy with their continuous counterparts, the self-adjoint
5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice
whose elementary quadrilaterals are planar), while the self-adjoint 7-point
scheme characterizes a generic 2D lattice.


title:
Plume motion and large-scale circulation in a cylindrical
  Rayleigh-B\'enard cell

abstract:
  We used the time correlation of shadowgraph images to determine the angle
$\Theta$ of the horizontal component of the plume velocity above (below) the
center of the bottom (top) plate of a cylindrical Rayleigh-B\'enard cell of
aspect ratio $\Gamma \equiv D/L = 1$ ($D$ is the diameter and $L \simeq 87$ mm
the height) in the Rayleigh-number range $7\times 10^7 \leq R \leq 3\times
10^{9}$ for a Prandtl number $\sigma = 6$. We expect that $\Theta$ gives the
direction of the large-scale circulation. It oscillates time-periodically. Near
the top and bottom plates $\Theta(t)$ has the same frequency but is
anti-correlated.


title:
Modulational instability, solitons and beam propagation in spatially
  nonlocal nonlinear media

abstract:
  We present an overview of recent advances in the understanding of optical
beams in nonlinear media with a spatially nonlocal nonlinear response. We
discuss the impact of nonlocality on the modulational instability of plane
waves, the collapse of finite-size beams, and the formation and interaction of
spatial solitons.


title:
Experimental realization of a highly secure chaos communication under
  strong channel noise

abstract:
  A one-way coupled spatiotemporally chaotic map lattice is used to contruct
cryptosystem. With the combinatorial applications of both chaotic computations
and conventional algebraic operations, our system has optimal cryptographic
properties much better than the separative applications of known chaotic and
conventional methods. We have realized experiments to pratice duplex voice
secure communications in realistic Wired Public Switched Telephone Network by
applying our chaotic system and the system of Advanced Encryption Standard
(AES), respectively, for cryptography. Our system can work stably against
strong channel noise when AES fails to work.


title:
$p$-adic discrete dynamical systems and their applications in physics
  and cognitive sciences

abstract:
  This review is devoted to dynamical systems in fields of $p$-adic numbers:
origin of $p$-adic dynamics in $p$-adic theoretical physics (string theory,
quantum mechanics and field theory, spin glasses), continuous dynamical systems
and discrete dynamical systems. The main attention is paid to discrete
dynamical systems - iterations of maps in the field of $p$-adic numbers (or
their algebraic extensions): conjugate maps, ergodicity, random dynamical
systems, behaviour of cycles, holomorphic dynamics. dynamical systems in finite
fields. We also discuss applications of $p$-adic discrete dynamical systems to
cognitive sciences and psychology.


title: On Markovian behaviour of $p$-adic random dynamical systems

abstract:
  We study Markovian and non-Markovian behaviour of stochastic processes
generated by $p$-adic random dynamical systems. Given a family of $p$-adic
monomial random mappings generating a random dynamical system. Under which
conditions do the orbits under such a random dynamical system form Markov
chains? It is necessary that the mappings are Markov dependent. We show,
however, that this is in general not sufficient. In fact, in many cases we have
to require that the mappings are independent. Moreover we investigate some
geometric and algebraic properties for $p-$adic monomial mappings as well as
for the $p-$adic power function which are essential to the formation of
attractors. $p$-adic random dynamical systems can be useful in so called
$p$-adic quantum phytsics as well as in some cognitive models.


title:
Reduced Singular Solutions of EPDiff Equations on Manifolds with
  Symmetry

abstract:
  The EPDiff equation governs geodesic flow on the diffeomorphisms with respect
to a chosen metric, which is typically a Sobolev norm on the tangent space of
vector fields. EPDiff admits a remarkable ansatz for its singular solutions,
called ``diffeons,'' whose momenta are supported on embedded subspaces of the
ambient space. Diffeons are true solitons for some choices of the norm. The
diffeon solution ansatz is a momentum map. Consequently. the diffeons evolve
according to canonical Hamiltonian equations. We examine diffeon solutions on
Einstein spaces that are "mostly" symmetric, i.e., whose quotient by a subgroup
of the isometry group is 1-dimensional. An example is the two-sphere, whose
isometry group $\SO{3}$ contains $S^1$. In this situation, the singular
diffeons (called ``Puckons'') are supported on latitudes (``girdles'') of the
sphere. For this $S^1$ symmetry of the two-sphere, the canonical Hamiltonian
dynamics for Puckons reduces from integral partial differential equations to a
dynamical system of ordinary differential equations for their colatitudes.
Explicit examples are computed numerically for the motion and interaction of
the Puckons on the sphere with respect to the $H^1$ norm. We analyse this case
and several other 2-dimensional examples. From consideration of these
2-dimensional spaces, we outline the theory for reduction of diffeons on a
general manifold possessing a metric equivalent to the warped product of the
line with the bi-invariant metric of a Lie group.


title: Antispiral waves are sources in oscillatory reaction-diffusion media

abstract:
  Spiral and antispiral waves are studied numerically in two examples of
oscillatory reaction-diffusion media and analytically in the corresponding
complex Ginzburg-Landau equation (CGLE). We argue that both these structures
are sources of waves in oscillatory media, which are distinguished only by the
sign of the phase velocity of the emitted waves. Using known analytical results
in the CGLE, we obtain a criterion for the CGLE coefficients that predicts
whether antispirals or spirals will occur in the corresponding
reaction-diffusion systems. We apply this criterion to the FitzHugh-Nagumo and
Brusselator models by deriving the CGLE near the Hopf bifurcations of the
respective equations. Numerical simulations of the full reaction-diffusion
equations confirm the validity of our simple criterion near the onset of
oscillations. They also reveal that antispirals often occur near the onset and
turn into spirals further away from it. The transition from antispirals to
spirals is characterized by a divergence in the wavelength. A tentative
interpretaion of recent experimental observations of antispiral waves in the
Belousov-Zhabotinsky reaction in a microemulsion is given.


title:
Spontaneous Emergence of Complex Optimal Networks through Evolutionary
  Adaptation

abstract:
  An important feature of many complex systems, both natural and artificial, is
the structure and organization of their interaction networks with interesting
properties. Here we present a theory of self-organization by evolutionary
adaptation in which we show how the structure and organization of a network is
related to the survival, or in general the performance, objectives of the
system. We propose that a complex system optimizes its network structure in
order to maximize its overall survival fitness which is composed of short-term
and long-term survival components. These in turn depend on three critical
measures of the network, namely, efficiency, robustness and cost, and the
environmental selection pressure. Using a graph theoretical case study, we show
that when efficiency is paramount the "Star" topology emerges and when
robustness is important the "Circle" topology is found. When efficiency and
robustness requirements are both important to varying degrees, other classes of
networks such as the "Hub" emerge. Our assumptions and results are consistent
with observations across a wide variety of applications.

categories: nlin.AO cond-mat.stat-mech cs.MA nlin.CG q-bio.QM


title: Lie Symmetries and Exact Solutions of First Order Difference Schemes

abstract:
  We show that any first order ordinary differential equation with a known Lie
point symmetry group can be discretized into a difference scheme with the same
symmetry group. In general, the lattices are not regular ones, but must be
adapted to the symmetries considered. The invariant difference schemes can be
so chosen that their solutions coincide exactly with those of the original
differential equation.


title:
Controlling chaotic transport in a Hamiltonian model of interest to
  magnetized plasmas

abstract:
  We present a technique to control chaos in Hamiltonian systems which are
close to integrable. By adding a small and simple control term to the
perturbation, the system becomes more regular than the original one. We apply
this technique to a model that reproduces turbulent ExB drift and show
numerically that the control is able to drastically reduce chaotic transport.


title: Integrable KP and KdV cellular automata out of a hyperelliptic curve

abstract:
  The goal of this paper is to present a solution of the cellular automaton
associated with the discrete KdV equation using algebro-geometric solution of
the discrete KP equation over a finite field out of a hyperelliptic curve.


title: Nonlinear states and nonlinear tunneling in a potential well

abstract:
  A nonlinear Schroedinger model in a square well and managed nonlinearity is
shown to possess nonlinear states as continuous extensions of the linear
levels. The solutions are remarkably stable up to a threshold amplitude where a
soliton is emitted and propagates outside the well. The analytic expression of
the threshold is given in terms of the well size for each level. This process
of nonlinear tunneling results from an instability of the evanescent wave
inside the walls and can find experimental realization in a proposed nonlinear
fiber Bragg gratings resonator.


title:
Chaotic oscillations in nonlinear system of interacting oscillators with
  the interaction of the fourth order

abstract:
  Dynamics of two anharmonic oscillators with interaction of the fourth order
has been investigated. The conditions at realization of which system is
integrable are established. The exact analytical solution of the nonlinear
equations in the case of adiabatic isolation of a system of oscillators has
been obtained


title: Supersymmetry and algebraic deformations

abstract:
  We describe a class of algebraically solvable SUSY models by considering the
deformation of invariant polynomial flags by means of the Darboux
transformation. The algebraic deformations corresponding to the addition of a
bound state to a shape-invariant potential are particularly interesting. The
polynomial flags in question are indexed by a deformation parameter m=1,2,...
and lead to new algebraically solvable models. We illustrate these ideas by
considering deformations of the hyperbolic P\"oschl-Teller potential.


title:
The Boussinesq integrable system. Compatible lattice and continuum
  structures

abstract:
  We consider the discrete Boussinesq integrable system and the compatible set
of differential difference, and partial differential equations. The latter not
only encode the complete hierarchy of the Boussisesq equation, but also
incorporate the hyperbolic Ernst equations for an Einstein-Maxwell-Weyl field
in general relativity. We demonstrate a specific symmetry reduction of the
partial differential equations, to a six-parameter, second order coupled system
of ordinary differential equations, which is conjectured to be of Garnier type.


title: Anisotropic fluctuations in turbulent sheared flows

abstract:
  An experimental analysis of small-scales anisotropic turbulent fluctuations
has been performed in two different flows. We analyzed anisotropic properties
of an homogeneous shear flows and of a turbulent boundary layer by means of two
cross-wire probes to obtain multi-point multi-component measurements. Data are
analyzed at changing inter-probe separation without the use of Taylor
hypothesis. The results are consistent with the ``exponent-only'' scenario for
universality, i.e. all experimental data can be fit by fixing the same set of
anisotropic scaling exponents at changing only prefactors, for different shear
intensities and boundary conditions.


title:
Second harmonic generation: Goursat problem on the semi-strip and
  explicit solutions

abstract:
  A rigorous and complete solution of the initial-boundary-value (Goursat)
problem for second harmonic generation (and its matrix analog) on the
semi-strip is given in terms of the Weyl functions. A wide class of the
explicit solutions and their Weyl functions is obtained also.


title:
Affine Lie group approach to a derivative nonlinear Schr\"odinger
  equatoin and its similarity reduction

abstract:
  The generalized Drinfel'd-Sokolov hierarchies studied by de
Groot-Hollowood-Miramontes are extended from the viewpoint of Sato-Wilson
dressing method. In the A_1^(1) case, we obtain the hierarchy that include the
derivative nonlinear Schr\"odinger equation. We give two types of affine Weyl
group symmetry of the hierarchy based on the Gauss decomposition of the A_1^(1)
affine Lie group. The fourth Painlev\'e equation and their Weyl group symmetry
are obtained as a similarity reduction. We also clarify the connection between
these systems and monodromy preserving deformations.