CIM Thematic Term on Mathematics and the Environment

 

School and Workshop on Dynamical Systems and Applications

 

 

List of participants

 

Photos

 

 

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, Spain)

 

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, Korea)

 

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. Balearic Islands, Spain)

 

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, Portugal)

 

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.

 

 

Shadowing in the C1 topology

Lorenzo J. Díaz

(PUC-Rio, Brazil)

 

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, Portugal)

 

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, Portugal)

 

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

(Lund Univ., Sweden)

 

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, Korea)

 

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 Bizerte, Tunisia)

 

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

Warwick Tucker

(Uppsala Univ., Sweden)

 

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, Brazil)

 

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.