Papers by Jean-Marc Malasoma
Contribution à l'étude de modèles chaotiques par approches métriques et topologiques
ABSTRACT
Computation of basin of attraction for three coexisting attractors
Le Centre pour la Communication Scientifique Directe - HAL - Inria, Oct 1, 1992
Modélisation phénoménologique de systèmes complexes non-linéaires à partir de séries chronologiques scalaires
Ces travaux de these visent a developper une methode originale de modelisation du comportement de... more Ces travaux de these visent a developper une methode originale de modelisation du comportement des systemes complexes. L'objectif est de construire un modele phenomenologique d'evolution d'un systeme en regime chaotique, sous la forme d'equations differentielles ordinaires, a partir de la donnee d'une variable scalaire regulierement echantillonnee. La technique de modelisation mise en place s'appuie sur l'utilisation d'une bibliotheque exhaustive de modeles, bibliotheque construite mathematiquement de facon formelle et rigoureuse. La methode theorique d'identification a ete integree au sein d'une methode numerique generale pour laquelle la serie chronologique analysee est la seule donnee d'entree et qui permet de construire des modeles d'approximation du comportement dynamique du systeme etudie. Elle se decompose en trois modules. Dans un premier temps, la serie scalaire est debruitee et ses series derivees, requises pour la reconstruc...
Bistability and hidden attractors in the paradigmatic Rössler'76 system
Chaos, 2020
In this paper, the dynamics of the paradigmatic Rössler system is investigated in a yet unexplore... more In this paper, the dynamics of the paradigmatic Rössler system is investigated in a yet unexplored region of its three-dimensional parameter space. We prove a necessary condition in this space for which the Rössler system can be chaotic. By using standard numerical tools, like bifurcation diagrams, Poincaré sections, and first-return maps, we highlight both asymptotically stable limit cycles and chaotic attractors. Lyapunov exponents are used to verify the chaotic behavior while random numerical procedures and various plane cross sections of the basins of attraction of the coexisting attractors prove that both limit cycles and chaotic attractors are hidden. We thus obtain previously unknown examples of bistability in the Rössler system, where a point attractor coexists with either a hidden limit cycle attractor or a hidden chaotic attractor.
Dynamique complexe d'un système a attracteurs multiples
Cette etude porte sur la dynamique d'un systeme a une dimension presentant une non-linearite ... more Cette etude porte sur la dynamique d'un systeme a une dimension presentant une non-linearite polynomiale avec une composante quadratique et une autre cubique, soumis a une excitation parametrique sinusoidale. Apres une etude des bifurcations de l'unique position d'equilibre de ce systeme menee aussi bien a l'aide de methodes analytiques que numeriques, nous montrons qu'il existe trois familles de solutions periodiques de meme periode que l'excitation parametrique, et nous presentons divers resultats concernant leurs bifurcations et en particulier l'apparition de chaos temporel. L’analyse globale de la dynamique de ce systeme consiste a determiner l'ensemble des comportements asymptotiques possibles pour un meme choix des parametres de controle. La determination des bassins d'attractions de ces solutions et l'analyse de leurs frontieres permet alors de montrer que l'espace des phases possede le plus souvent une structure fractale qui condui...
New Lorenz-like Chaotic Flows with Minimal Algebraic Structure
In 1994, a numerical examination by J.C. Sprott of three-dimensional autonomous ordinary differen... more In 1994, a numerical examination by J.C. Sprott of three-dimensional autonomous ordinary differential equations with quadratic nonlinearities uncovered five new simple examples of chaotic flows with either of the five terms including two nonlinearities on the right hand side of the differential equations. Two of these flows resemble the Lorcnz system as they are equivariant under the action of the rotation through π radians around the z-axis. Moreover, both exhibit, two off-axis fixed points and two-scroll butterfly shaped chaotic attraeiors. In canonical form, these systems have in the second equation the same nonlinearity xz as that of the Lorenz system, and the nonlinearity in the third equation is xy or y2. The present paper introduces the first known examples of Lorenz-like systems with the simplest algebraic structure including the nonlinearity x2 in the third equation when written in canonical form, and allowing chaotic behaviour. Of particular interest is the fact that these...
Architecture of chaotic attractors for flows in the absence of any singular point
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2016
Nonlinear behaviour of a nonlinear mechanical system
ABSTRACT
New Lorenz-like Chaotic Flows with Minimal Algebraic Structure
In 1994, a numerical examination by J.C. Sprott of three-dimensional autonomous ordinary differen... more In 1994, a numerical examination by J.C. Sprott of three-dimensional autonomous ordinary differential equations with quadratic nonlinearities uncovered five new simple examples of chaotic flows with either of the five terms including two nonlinearities on the right hand side of the differential equations. Two of these flows resemble the Lorcnz system as they are equivariant under the action of the rotation through π radians around the z-axis. Moreover, both exhibit, two off-axis fixed points and two-scroll butterfly shaped chaotic attraeiors. In canonical form, these systems have in the second equation the same nonlinearity xz as that of the Lorenz system, and the nonlinearity in the third equation is xy or y2. The present paper introduces the first known examples of Lorenz-like systems with the simplest algebraic structure including the nonlinearity x*x in the third equation when written in canonical form, and allowing chaotic behaviour. Of particular interest is the fact that thes...
Minimal smooth chaotic flows
World Scientific Series on Nonlinear Science Series A, 2013
ABSTRACT
The results of the study of the global behaviour of the convective flow of a binary mixture in a ... more The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.
Chaotic behavior of a parametrically excited nonlinear mechanical system
ABSTRACT The chaotic dynamics of a single-degree-of-freedom nonlinear mechanical system under per... more ABSTRACT The chaotic dynamics of a single-degree-of-freedom nonlinear mechanical system under periodic parametric excitation is investigated. Besides the well known type-I and type-III intermittent transitions to chaos we give numerical evidence that the system can follow an alternative route to chaos via intermittency from an equilibrium state to a chaotic one, which was not found in the previous simulations of the dynamics of the system.
Computation of Basins of Attraction for Three Coexisting Attractors
Bifurcation and Chaos: Analysis, Algorithms, Applications, 1991
Universalities in the chaotic generalized Moore & Spiegel equations
Chaos, Solitons & Fractals, 2014
ABSTRACT By investigating the topology of chaotic solutions to the generalized Moore and Spiegel ... more ABSTRACT By investigating the topology of chaotic solutions to the generalized Moore and Spiegel equations, we address the question how the solutions of nonlinear dynamical systems are dependent on the nature of the nonlinearity. In these generalized jerk equations, the single nonlinear term has a parity which depends on unu̇. The system has an inversion symmetry when n is even and no symmetry property when n is odd. It is shown that the topology of chaotic solutions only depends on the parity of n, that is, on the symmetry properties and not on the degree of nonlinearity. The value of n only affects the possibility to develop the chaotic solution, that is, to increase the number of unstable periodic orbits within the attractor.
Physical Review E, 2001
The simplest equivariant chaotic dynamics is investigated in terms of its image, i.e., under the ... more The simplest equivariant chaotic dynamics is investigated in terms of its image, i.e., under the 2→1 mapping allowing one to obtain a projection of the dynamics without any residual symmetry. The inversion symmetry is therefore deleted. The bifurcation diagram can thus be predicted from the unimodal order although the first-return map computed in the original phase space exhibits three critical points. This feature is the same as the one observed on the Burke and Shaw system although this latter system has a rotation symmetry.
Numerical observation of an intermittent transition from an equilibrium point to a chaotic attractor
Physics Letters A, 1994
ABSTRACT The numerical observation of an intermittent transition from an equilibrium state to a c... more ABSTRACT The numerical observation of an intermittent transition from an equilibrium state to a chaotic one is reported in a nonlinear oscillator with sinusoidal parametric excitation. The transition is due to a subcritical Hopf bifurcation of a fixed point which occurs when the frequency or the amplitude of the forcing is varied.
What is the simplest dissipative chaotic jerk equation which is parity invariant?
Physics Letters A, 2000
... that depends on the instantaneous velocity (viscous damping) but also on the position and vel... more ... that depends on the instantaneous velocity (viscous damping) but also on the position and velocityhistory. ... After the crisis, the asymptotic motion settles down to a larger symmetric chaotic attractor. ... 3 shows the variations of the largest, non zero, exponent λ m versus parameter α. ...
Analysis of nonlinear oscillations by wavelet transform: Lyapunov exponents
Nonlinear Dynamics, 1996
ABSTRACT In this paper we give the definition of exponents which would look like Lyapunov exponen... more ABSTRACT In this paper we give the definition of exponents which would look like Lyapunov exponents in the cases of non-smooth flows of differential equations or iterated maps, and carry back Lyapunov exponents in smooth cases. Here we test our definition by using some simple linear and nonlinear smooth examples.
Journal of Sound and Vibration, 1996
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Papers by Jean-Marc Malasoma