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Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system

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Abstract

We present a global dynamical analysis of the following quadratic differential system \(\dot{x}{=}a(y{-}x), \dot{y}\!=\!dy-xz, \dot{z}\!=\!-bz+fx^2+gxy\), where \((x,y,z)\in {\mathbb {R}}^3\) are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in \({\mathbb {R}}^3\), alike Lorenz, Rössler, Chen and other. By using the Poincaré compactification for a polynomial vector field in \({\mathbb {R}}^3\), we study the dynamics of this system on the Poincaré ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.

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Acknowledgments

The first and third authors were partially supported by CAPES and FAPESP. The second author was supported by FAPESP Projects 2012/18413-7 and 2013/24541-0 and by CNPq Project 308315/2012-0. The authors would like to thank the referees for their valuable comments and suggestions, which enabled them to improve the presentation of the paper.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto de Biociências, Letras e Ciências Exatas - IBILCE, Univ Estadual Paulista (UNESP), São José do Rio Preto, SP, Brazil

    Márcio R. A. Gouveia & Claudio Pessoa

  2. Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia - FCT, Univ Estadual Paulista (UNESP), Presidente Prudente, SP, Brazil

    Marcelo Messias

Authors
  1. Márcio R. A. Gouveia
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  2. Marcelo Messias
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  3. Claudio Pessoa
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Correspondence to Marcelo Messias.

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Gouveia, M.R.A., Messias, M. & Pessoa, C. Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system. Nonlinear Dyn 84, 703–713 (2016). https://doi.org/10.1007/s11071-015-2520-4

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  • DOI: https://doi.org/10.1007/s11071-015-2520-4

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