Mecanismos de Criação de Atratores Estranhos no Segundo Sistema de Rössler
DOI:
https://doi.org/10.5540/tema.2008.09.02.0275Abstract
Neste trabalho fazemos uma análise das bifurcações locais que ocorrem nos pontos de equilíbrio do Segundo Sistema de Rössler, que é um sistema quadrático tridimensional de equações diferenciais ordinárias, dependendo de três parâmetros reais, a, b e c. Determinamos as superfícies no espaço de parâmetros, para as quais o sistema apresenta bifurcações de Hopf. Mostramos numericamente que para valores dos parâmetros próximos aos de bifurcação de Hopf o sistema possui atratores estranhos. Além disso, para a = 0 o sistema possui uma família formada por infinitos ciclos heteroclínicos singularmente degenerados, que consistem de conjuntos invariantes formados por uma linha de equilíbrios, juntamente com uma órbita heteroclínica conectando dois destes equilíbrios. Mostramos numericamente que pequenas perturbações do sistema, tomando-se a > 0 pequeno,levam à quebra destes ciclos degenerados e à criação de atratores estranhos.References
[1] Jiin-Po Yeh, Kun-Lin Wu, A simple method to synchronize chaotic systems and its application to secure communications, Mathematical and Computer Modelling 47 (2008), 894-902.
H. Kokubu, R. Roussarie, Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: Part I, J. Dyn. Diff. Equat. 16 (2004), 513-557.
Y.A. Kuznetsov, “Elements of Applied Bifurcation Theory”, Second Edition, Springer-Verlag, New York, 2004.
J. Llibre, M. Messias, P.R. da Silva, On the global dynamics of the Rabinovich system, J. Phys. A: Math. Theor. 41 (2008), 275210-31.
E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141.
L.F. Mello, M. Messias, D.C. Braga, Bifurcation analysis of a new Lorenz-like chaotic system, Chaos, Solitons and Fractals 37 (2008), 1244-1255.
M. Messias, “Dynamic at Infinity and Existence of Singularly Degenerate Heteroclinic Cycles in the Lorenz System”, Relat´orio T´ecnico - DMEC-FCTUNESP 01 (2008), 1-25.
K. Murali, M. Lakshmanan, Secure communication using a compound signal from generalized synchronizable chaotic systems, Physics Letters A 241, No. 6 (1998), 303-310.
L.S. Pontryagin, “Ordinary Differential Equations”, Addison-Wesley Publishing Company Inc., Reading, 1962.
O.E. R¨ossler, An equation for continuous chaos, Physics Letters, 57A, No. 5 (1976), 397-398.
O.E. R¨ossler, Continuous chaos – four prototype equations, em “Bifurcation Theory and Applications in Scientific Disciplines”, Ann. New York Acad. Sci., 316 (1979), 376-392.
C. Sparrow, “The Lorenz equations: bifurcations, chaos and strange attractors”, Springer–Verlag, New York, 1982.
S.H. Strogatz, “Nonlinear Dynamics and Chaos: with applications in Physics, Biology, Chemistry and Engineering”, Addison Wesley Publishing Company Inc., Cambridge, USA, 1994.
M. Viana, What’s new on Lorenz strange attractors?, Math. Intelligencer, 22, No. 3 (2000), 6-19.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.