CIM Thematic Term on Mathematics and the Environment
School and Workshop on
Dynamical Systems and Applications
Abstracts
Random perturbations of diffeomorphisms with dominated splitting
Vítor Araújo
(Univ. Porto, Portugal)
Abstract: We prove that the statistical
properties of random perturbations of a nonuniformly
hyperbolic diffeomorphism are described by a finite
number of stationary measures. We also give necessary and sufficient conditions
for the stochastic stability of such dynamical systems. We show that a certain
C2-open class of nonuniformly hyperbolic diffeomorphisms introduced in [Alves,
J; Bonatti, C. and Viana,
V., SRB measures for partially hyperbolic systems with mostly expanding central
direction, Invent. Math., 140 (2000), 351-398] are stochastically stable. Our
setting encompasses that of partially hyperbolic diffeomorphisms
as well. Moreover, the techniques used enable us to obtain SRB measures in this
setting through zero-noise limit measures. In addition, uniformly hyperbolic diffeomorphisms satisfy our condition and we also obtain
their stochastic stability as a corollary.
Optical chaos cryptography using semiconductor lasers
with electrooptical feedback
Yanne Chembo
(IMEDEA,
Abstract: One of the most interesting
technological application of chaos theory is cryptography. Optical chaos
cryptography basically relies on the synchronization of two semiconductor
lasers operating in the chaotic regime. Typically, an information-bearing
signal is masked within the noise-like output of a chaotic emitter, while a
synchronous receiver recognizes the chaotic component and extracts it to reveal
the originally encrypted signal. Within that frame, it can therefore be said
that encryption relies on the unpredictability of chaotic oscillations, while
decryption relies on their determinism.
In this communication, we present an optical
cryptographic system where hyperchaos is generated in
the semiconductor lasers by electrooptical feedback
loops. We first study the detrimental influence of parameter mismatch between
emitter and receiver on the quality of synchronization. Analytical techniques
are developped to study the statistical properties of
the synchronization error as a function of the different types of mismatch. We
secondly study the influence of these mismatches on the efficiency of this
optical communication scheme, and we analytically demonstrate that the
probability density function of the hyperchaotic
carrier does decisively influence the performance of the system. Our analytical
results are confronted with numerical simulatons and
experimental measures, and a good concordance is observed.
Authors: Y. Chembo Kouomou, Pere Colet,
Nicolas Gastaud, and Laurent Larger.
Rigorous numerical simulations of discrete dynamical
systems
Geon Ho Choe
(KAIST,
Abstract: Can we rigorously simulate a discrete
dynamical system on a computer? We introduce an idea for numerical simulations
on a fast computer with large amount of memory. (In the year 2004, most
personal computers satisfy such criteria.)
A discrete dynamical system in ergodic theory is based on iterations of a mapping. A loop
in computer programming corresponds to iterations of the same instruction. Here
is the big difference: There is loss of accuracy in each application of the
instruction when a dynamical system is modelled as a do loop.
A dynamical system corresponds to iterations
of T, which again corresponds to a do
loop in computer simulation. (Computer arithmetic uses finite precision,
though.)
Some theoretical models cannot be rigorously
simulated numerically except when there is no need for high precision, e.g.,
approximation of invariant measures by the Birkhoff Ergodic Theorem. Modern day technology has enabled us to do
computer experiments using almost unlimited amount of precision.
Some previously unthinkable simulations are
now done using software such as Maple on a reasonably fast PCs. The optimal
number of significant digits is given in terms of Lyapunov
exponent of T.
How to do computer experiments? Choose
sufficiently many significant digits in floating point calculations because we
iterate transformations very many times.
How many is sufficiently many? If we take too
many significant digits, then the computation would take too long.
With small number of digits, the truncation
error will ruin everything very soon. Maple allows us to do floating point
computations with practically unlimited precision. Until recently computations
with this level of accuracy was practically impossible because of small amount
of RAM and slow CPUs.
Here is the basic idea: Take D significant decimal digits in floating
point computation in defining a starting point x0. After k
iterations of T, we lose about k x entropy decimal digits from original
D digits in x0, where the
logarithmic base is 10 in the definition of entropy.
Examples to be presented are: (i) the first return time and dimension, (ii) the first
return time of irrational translations mod 1, (iii) continued fraction
algorithm in Maple, (iv) mod 2 normality.
Dynamical mechanism of anticipating synchronization in
excitable systems
Marcena Ciszak
(Univ.
Abstract: We analyze the phenomenon of
anticipating synchronization of two excitable systems with unidirectional
delayed coupling which are subject to the same external forcing. We demonstrate
for different paradigms of excitable system that, due to the coupling, the
excitability threshold for the slave system is lower than that for the master.
This allows to explain in a simple way the mechanism behind the
counterintuitive phenomenon of anticipating synchronization.
Ergodicity of semifocusing planar
billiards
Gianluigi Del Magno
(IST,
Abstract (joint work with R. Markarian): Bunimovich and Donnay introduced, independently, a class of hyperbolic
planar billiards bounded by special focusing arcs (called absolute focusing)
and, possibly, straight lines. Such a class includes the celebrated stadium and
its generalizations constructed by Bunimovich, Wojtkowski and Markarian. In this
talk, we show that Bunimovich-Donnay billiards are ergodic. Our approach is based on Sinai’s ideas. The main
ingredient of the proof is an adapted version of the local ergodic
theorem originally proved for Hamiltonian systems by Liverani
and Wojtkowski.
Lorenzo J. Díaz
(
Abstract: We prove that among the nonhyperbolic robustly transitive C1-diffeomorphisms
those that do not satisfy the shadowing property are generic (i.e., they form a
residual subset). We explain how the non-shadowing property is related to the
creation of heterodimensional cycles.
Spectral intervals approximation via SVD-based
techniques: theoretical and numerical aspects
Cinzia Elia
(Univ. Studi di Bari, Italy)
Abstract: Lyapunov
exponents and exponential dicothomy spectrum are
common tools to study stability properties of dynamical systems. We approximate
numerically these quantities and their associated directions, for linear and
non linear systems of ordinary differential equations. In order to do so, we
use a smooth singular value decomposition of the fundamental matrix solution of
the system itself or of its linearization along a particular trajectory.
Numerical examples are given.
Dimension theory beyond uniformly hyperbolic dynamics
Katrin Gelfert
(IST,
Abstract: From the point of view of dimension
there is an almost complete understanding of uniformly hyperbolic dynamical
systems. Beyond that theory still needs to be advanced. In this talk we pursue
some new perspectives. We present results on the fractal dimension of invariant
sets. Based on the thermodynamic
formalism we derive upper dimension bounds in terms of the topological pressure
of certain functions which relate to the growth rate of volumes. For some
systems which satisfy our assumptions we point out (partially) volume expanding
systems. We remark that we are not
demanding the system to be uniformly hyperbolic. Our approach conduces to new
results even in the case of self-affine sets.
Topological entropy, homological growth and zeta
functions on graphs
Roman Hric
(IST,
Abstract: We establish a precise relation
between the topological entropy, the entropy arising from the homological
growth, and the exponential growth rate of the number of periodic points of negative
type for piecewise monotone graph maps showing that the first one is the
maximum of the latter two. This nontrivially extends a result of Milnor and
Thurston on piecewise monotone interval maps. For this purpose we generalize
the concept of Milnor-Thurston zeta function introducing negative zeta function
and involving Lefschetz zeta function. The talk is
based on new results from a joint work with Joao Ferreira Alves
and Jose Sousa Ramos.
Hamiltonian long-wave models for free surfaces and
interfaces
Henrik Kalisch
(
Abstract: The equations for the motion of a
system consisting of an ideal fluid which has a free surface, a free interface
or both, can be viewed as a Hamiltonian system with infinitely many degrees of
freedom. This formulation is used to develop an algorithm for the derivation of
long-wave model equations for the evolution of the free surface and interface.
Some of these models are well-known, such as the Benjamin-Ono, intermediate
long-wave and Korteweg-deVries equations. However,
our method yields equations of arbitrary order, and is not restricted to
small-amplitude waves, so that we can also obtain the shallow-water equations
and higher-order corrections.
The waiting time for some interval transformations
Dong Han Kim
(KIAS,
Abstract: We discuss the asymptotic behaviour
of the waiting time for some transformation on the interval including
irrational rotations. Like the first return time formula, logarithm of the
waiting time to a ball divided by logarithm of the radius of the ball goes to 1
as the radius goes to 0 for many transformations. Also I'd like to consider the
relation between waiting time formula and the dynamical Borel-Cantelli
lemma.
Time-one maps of identity isotopies on surfaces
Patrice Le Calvez
(Univ. Paris-Nord, France)
Abstract: We will state some
results about homeomorphisms of surfaces which are time one map of an isotopy starting from the identity. A natural example is a diffeomorphism associated to a time dependant vector field,
periodic in time. We will see that certain theorems about time one map of flows
induced by a time independent vector field may be extended to that situation.
This includes for example results about existence of periodic orbits for
Hamiltonian homeomorphisms, properties about linking numbers of periodic
orbits...
Multidimensional renormalisation and KAM
theory
João Lopes-Dias
(IST, Lisboa)
Abstract: We construct a
multidimensional renomalisation scheme using the
geodesic multidimensional continued fractions which apply to the problem of
stability of quasiperiodic motion. As an example, we
prove several KAM-type results for Hamiltonians close to integrable
and vector fields on the d-torus.
Interval exchange maps, renormalisation and
continued fractions
Stefano Marmi
(Scuola Normale Superiore di Pisa, Italy)
Abstract (joint work with
Pierre Moussa and Jean-Christophe
Yoccoz): Interval exchange maps are characterized by
combinatorial and metric data. The analysis of first return times on an
interval (renormalisation) leads to several generalisations of the classical
continued fraction algorithm (Rauzy, Veech, Zorich). A further
acceleration of these schemes can be used to characterise a class of interval
exchange maps of “Roth type” for which the cohomological
equation can be solved.
Dynamic of a class of linear abelian groups on Rn
Habib Marzougui
(Faculty of Sciences of
Abstract (joint with A. Ayadi):
In this work, we characterize the dynamic of orbits of some Abelian
subgroups G of GL(n,R).
We show that there exists a dense open set U in Rn with all its orbits are
minimal of the same type and where the complementary Rn -
U is a union of finite G-invariant vectorial subspaces of Rn. As a consequence, G has
height at most n.
There are singular-hyperbolic flows without
spectral decomposition
Maria José
Pacifico
(UFRJ, Brazil)
Abstract: A flow Xt is singular-hyperbolic if its singularities
are hyperbolic and the non-wandering set Ω(Xt) is: (1) the closure of its closed
orbits, and (2) partially hyperbolic with volume expanding central direction,
either for Xt
or X-t. We show that every
closed 3-manifold supports a C∞ singular-hyperbolic flow Xt
whose Ω(Xt) has dense closed orbits and is not a union of disjoint homoclinic
classes and singularities. In another words, the flow Xt does not satisfy
the spectral decomposition theorem [2]. This shows, in particular, that the
result in [Theorem A, 1] that implies that each element of a residual subset of
such flows satisfies the spectral decomposition theorem is in some sense
optimal. The construction of such examples involves subtle geometric
modifications of simpler flows on 3-manifolds. This corresponds to a joint work
with S. Bautista and C. A. Morales.
References:
[1] C. Morales and M. J. Pacifico.
On the dynamics of singular hyperbolic systems. Preprint, 2003.
[2] S. Smale. Differential
dynamical systems. Bull. AMS, 73, 1967, 747-817.
Neuronal coding from a dynamical system
viewpoint
Khashayar Pakdaman
(Univ. Paris 6-7, France)
Abstract: Neurons are one of
the key ingredients in information transmission and processing in nervous
systems. Neurons encode incoming information intro trains of stereotyped
electrical discharges whose shape and amplitude depends little on the
stimulation. It is the timing and the number of such discharges that is
stimulus dependent. One could compare these discharges to the dots in the Morse
code. One the central issues in neurosciences has been to determine the neuronal
code, that is the relation between the incoming signal and the outgoing
discharge train. While the details of the "code" remain to be eludicated, much understanding has been gained on it
through approaches inspired by dynamical system (DS) theory. This presentation is concerned with these. It
will review some of the classical experimental and theoretical investigations
of neuronal coding from the DS standpoint, and will end up on recent
developments.
Validated numerics and the art of dividing by zero
(
Abstract: We will discuss a
modern approach to numerical computations, based on interval analysis. Although
the theory has been well known since the mid 60's, it is not until recently
that fast and efficient implementations of interval algorithms have appeared.
Today there exist many good interval packages for e.g. Maple, Matlab, C++, and Fortran, much due to the fact that there
now is one globally accepted standard for floating point computations.
Interval analysis is the
mathematical foundation of so called auto-validating algorithms. By computing
with intervals instead of single points, important properties of the real line
can be captured and used in the algorithms. This leads to very robust methods,
well suited for ill-conditioned problems.
Auto-validating algorithms
produce mathematically correct results, incorporating not only the computer's
internal representation of the floating point numbers and its rounding
procedures, but also all discretisation errors of the
underlying numerical method. Thus the result comes equipped with guaranteed
error bounds. With today's computing speeds, this appears to be the only
reasonable way to keep track of error propagation.
There are many situations in
which the interval algorithm returns a guaranteed result faster than the
floating point version delivers an "approximation" (which can be very
wrong indeed). Throughout the talk, several such applications will be
presented, e.g. root-finding, implicit curve generation, and chaos theory.
On the boundary of instability
Marcelo Viana
(IMPA,
Abstract: As part of a startegy to reach a global understanding of dynamics, one
would like to have a better understanding of the boundary of {structurally stable
dynamical systems}, especially of the phenomena that may cause a dynamical
system to cease to be stable. Two classical mechanisms are known, both related
to periodic orbits: loss of hyperbolicity on some
periodic orbit, or loss of transversality between a
pair of stable and unstable manifolds. It has been conjectured that,
generically, that is all there is. This conjecture remains wide open. But recents show that its natural formulation in probabilistic
terms is actually false: there is a "positive probability" subset of
the boundary of stability formed by Kupka-Smale maps,
i.e. such that all their periodic points are hyperbolic and all their stable
and unstable manifolds are transverse.