Can we trust in numerical computations of chaotic solutions of dynamical systems ?

From Laser Dynamics to Topology of Chaos, (celebrating the 70th birthday of Prof Robert Gilmore, Rouen, June 28-30, 2011), Ed. Ch. Letellier.Since the famous paper of E. Lorenz in 1963 numerical computations using computers play a central role in order to display and analyze solutions of nonlinear dynamical systems. By these means new structures have been emphasized like hyperbolic and/or strange attractors. However theoretical proofs of their existence are very di¢ cult and limited to very special linear cases. Computer aided proofs are also complex and require special interval arithmetic analysis. Nevertheless, numerous researchers in several fields linked to chaotic dynamical systems are confident in the numerical solutions they found using popular software and publish without checking carefully the reliability of their results. In the simple case of discrete dynamical systems (e.g. Hénon map) there are concerns about the nature of what a computer find out : long unstable pseudo-orbits or strange attractors? The shadowing property and its generalizations which ensure that pseudo-orbits of a homeomorphism can be traceable by actual orbits even if rounding errors are not inevitable are not of great help in order to validate the numerical results. Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more difficult to handle in this scope and researchers have to be very cautious to back up theory with numerical computations. We present a survey of the topic based on these, only few, but well studied models

Application of observer-based chaotic synchronization and identifiability to original CSK model for secure information transmission

International audienceThe modified Lozi system is analyzed as chaotic PRNG and synchronized via observers. The objective of the study is to investigate chaotic-based encryption method that preserves CSK model advantages, but improves the security level. The CSK model have been discussed to message encryption because it implies better resistance against noise, but there are many evidences of the model weaknesses. The investigation provides the original CSK model analyses of secure message transmission over the communication channel by examining identifiability and observability; switched regimes detection; sensitivity to initial conditions and session key; NIST tests of the encrypted signal; correlation between wrong decrypted messages; system ergodicity. The proposed model has a significant effect on the security level of the transmitted signal that successfully passed chaotic and randomness tests. The results suggest that the original CSK model can be used for information security applications

Statistical and spectral analysis of a new weakly coupled maps system

The complexity of chaotic maps, exhibiting deterministic, but also stochastic features makes them very promising candidates in the rush for new and secure communication technologies. However, most of the chaotic systems proposed in the litterature turned out to be unsuitable for chaotic encryption, since they don't satisfy the required spectral and statistical properties for closeness to random signals. Unlike these papers, we propose to implement the new ultra weakly coupled maps system firstly introduced by Lozi. We demonstrate that this function is highly performant, and beats most of the classically used random number generators. Next, in the context of a master-slave synchronization, an observer is required to recover the chaotic signal (and thus, the original message). Therefore, two different piece-wise linear observers have been synthetized : a Luenberger, and an inverse lag observer, and the necessary conditions for synchronization have been derived. Finally, for the second order system it has been demonstrated that the exact synchronization can be achieved in two iterations for any initial conditions

Observers design for a new weakly coupled map function

An improved version of this paper will soon be publishd in Journal of Nonlinear Sytems and Applications (2009)International audienceIn some engineering applications, such as chaotic encryption, chaotic maps have to exhibit required statistical and spectral properties close to those of random signals. However, most of the papers dealing with synchronization and observer synthesis consider maps exhibiting poor statistical and spectral properties. Moreover, most of the time these properties, however essential for the chaotic encryption, are simply neglected. Unlike these papers, in our work we present the analysis of a new ultra weakly coupled maps system introduced by Lozi. The model is a deterministic one, but exhibits spectral properties (spectrum, correlation and autocorrelation) close to those of random signals, and successfully passed all the statistical tests for closeness to random signals (NIST). Two different observers have been designed. The convergence rate has been discussed in the case of affine maps, and the conditions to decrease the convergence rate by a factor of 16 have been presented, based on the locally linear behaviour of the weakly coupled map

A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities

7 pages; 3 figures.International audienceIn this article the authors describe the method of construction of approximate chaotic solutions of dynamical model equations with quadratic nonlinearities in a general form using a new accurate numerical method. Numerous systems of chaotic dynamical systems of this type are studied in modern literature.The authors find the region of convergence of the method and offer an algorithm of construction and several criteria to check the accuracy of the approximate chaotic solutions

Exploring some topologies of coupled chaotic networks

International audienceThis paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two topologically coupled maps are studied: T T L RC non-alternate, and T T L SC alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim two new nonlinear CPRNG: M T T L SC 2 and N T T L2 are proposed. The maps successfully passed numerous statistical, graphical and numerical tests, due to proposed ring coupling and injection mechanisms

DYNAMICAL AND STATISTICAL ANALYSIS OF A NEW LOZI FUNCTION FOR RANDOM NUMBERS GENERATION

International audienceThis paper presents the first results of the statistical and dynamical analysis of a new function showing random properties firstly proposed by Lozi. The phase plane analysis via the critical lines tool allowed to delimit analytically the holes in the chaotic attractor and to follow their evolution. In addition, the results of the statistical NIST tests for pseudo-randomness showed to be successful and significantly improved after an under-sampling of the output signal

Modified projective synchronization of fractional-order hyperchaotic memristor-based Chua’s circuit

This paper investigates the modified projective synchronization (MPS) between two hyperchaotic memristor-based Chua circuits modeled by two nonlinear integer-order and fractional-order systems. First, a hyperchaotic memristor-based Chua circuit is suggested, and its dynamics are explored using different tools, including stability theory, phase portraits, Lyapunov exponents, and bifurcation diagrams. Another interesting property of this circuit was the coexistence of attractors and the appearance of mixed-mode oscillations. It has been shown that one can achieve MPS with integer-order and incommensurate fractional-order memristor-based Chua circuits. Finally, examples of numerical simulation are presented, showing that the theoretical results are in good agreement with the numerical ones

PROPOSTAS PARA O ENSINO DE CIÊNCIAS NATURAIS POR INVESTIGAÇÃO A PARTIR DE OBSERVAÇÕES EM UM LICEU FRANCÊS

International audienceThis study focused on observations of Life and Earth Sciences lessons taught in a French public high school and aims to discuss the conditions necessary for the teaching of natural sciences based on research to help the students to appropriating of scientific knowledge effectively. This approach, not to be restrictive from the didactic point of view, should be supported in the practice of a hypothetico-deductive experimental type, in particular in the mediation of the use of models and the modeling, the choice content and the organization of class time. Three questions addressed to the students were always present in the didactical situations: What do you observe? What do you already know? What can we deduce? The lesson plans were developed following ten keys steps, including the problem-situation of research, the scientific debate in the classroom in order to design a resolution strategy, practical work and using or constructing models, the presentation and argumentation of the results from the students, the correction and validation of these results under the teacher's guidance and a final summary scheme. For classes with this approach to achieve their goals in the context of Brazilian education, it is essential to prepare teachers to apply them and reorganize the school environment. The good level of quality of teacher - student - knowledge interactions depends, therefore, on the effective investment in teachers training in the didactic modality of the use of models and modeling, essential to the success of natural sciences education.Cette étude porte sur l'observation de cours de Sciences de la Vie et de la Terre dispensés dans un lycée public français et a pour objectif de discuter des conditions nécessaires à un enseignement des sciences de la nature reposant sur des recherches visant à aider les étudiants à s'approprier les connaissances scientifiques de manière efficace. Cette approche, pour ne pas être restrictive d’un point de vue didactique, se devrait d’être soutenue par la pratique d'un type expérimental hypothético-déductif, en particulier dans la médiation de l'utilisation de modèles et de la modélisation, le contenu du choix et l'organisation du temps de classe. Trois questions adressées aux élèves ont toujours été présentes dans les situations didactiques: Qu'observez-vous ? Que savez-vous déjà ? Que pouvons-nous déduire ? Les plans de cours ont été élaborés en suivant dix étapes clés, incluant le problème de la situation de la recherche, le débat scientifique en classe afin de concevoir une stratégie de résolution, le travail pratique et l’utilisation ou la construction de modèles, la présentation et la discussion des résultats obtenus par les élèves, la correction et la validation de ces résultats sous la direction de l’enseignant et un schéma de synthèse final.Pour que les classes ayant cette approche atteignent leurs objectifs dans le contexte de l'éducation brésilienne, il est essentiel de préparer les enseignants à les appliquer et à réorganiser l'environnement scolaire. Le bon niveau de qualité des interactions enseignant-élève-savoir dépend donc d’un investissement effectif dans la formation des enseignants à la modalité didactique de l’utilisation des modèles et de la modélisation, essentiels au succès de l’enseignement des sciences naturelles.Este estudo centrou-se em observações de aulas de Ciências da Vida e da Terra ministradas em uma escola pública francesa de ensino médio e objetiva discutir as condições necessárias para que o ensino de ciências naturais baseado em investigação possa ajudar efetivamente os alunos a se apropriarem dos saberes científicos. Esta abordagem, para não ser restritiva do ponto de vista didático, deve estar apoiada na prática de um procedimento experimental do tipo hipotético-dedutivo, sobretudo na mediação do uso de modelos e modelizações, na escolha dos conteúdos e na organização do tempo de aula. Três questões dirigidas aos alunos estavam sempre presentes nas situações didáticas: O que vocês observam? O que vocês já sabem? O que podemos deduzir? Os planos de aulas eram desenvolvidos seguindo dez etapas indispensáveis, entre elas: a situação-problema desencadeadora da investigação, o debate científico na classe visando conceber uma estratégia de resolução, o trabalho prático e autônomo dos alunos utilizando ou construindo modelos, a apresentação e argumentação dos resultados por parte dos alunos, a correção e validação desses resultados sob a pilotagem do professor e um esquema recapitulativo final. Para que aulas com esta abordagem tenham seus objetivos atingidos no contexto da educação brasileira são indispensáveis o preparo dos professores em aplicá-las e a reorganização do ambiente escolar. O bom nível de qualidade das interações professor - aluno – saber, depende, portanto, do investimento efetivo na formação de professores na modalidade didática do uso de modelos e modelizações, indispensável para o sucesso do ensino de ciências naturais

Some elements for a history of the dynamical systems theory

Leon Glass would like to thank the Natural Sciences and Engineering Research Council (Canada) for its continuous support of curiosity-driven research for over 40 years starting with the events recounted here. He also thanks his colleagues and collaborators including Stuart Kauffman, Rafael Perez, Ronald Shymko, Michael Mackey for their wonderful insights and collaborations during the times recounted here. R.G. is endebted to the following friends and colleagues, listed in the order encountered on the road described: F. T. Arecchi, L. M. Narducci, J. R. Tredicce, H. G. Solari, E. Eschenazi, G. B. Mindlin, J. L. Birman, J. S. Birman, P. Glorieux, M. Lefranc, C. Letellier, V. Messager, O. E. Rössler, R. Williams. U.P. would like to thank the following friends and colleagues who accompanied his first steps into the world of nonlinear phenomena: U. Dressler, I. Eick, V. Englisch, K. Geist, J. Holzfuss, T. Klinker, W. Knop, A. Kramer, T. Kurz, W. Lauterborn, W. Meyer-Ilse, C. Scheffczyk, E. Suchla and M. Wisenfeldt. The work by L. Pecora and T. Carroll was supported directly by the Office of Naval Research (ONR) and by ONR through the Naval Research Laboratory’s Basic Research Program. C.L. would like to thank Jürgen Kurths for his support to this project.Peer reviewedPostprintPublisher PD