Hyperchaos and quasiperiodicity from a four-dimensional system based on the Lorenz system

Regular Article


This paper reports on numerically computed parameter plane plots for a dynamical system modeled by a set of five-parameter, four autonomous first-order nonlinear ordinary differential equations. The dynamical behavior of each point, in each parameter plane, is characterized by Lyapunov exponents spectra. Each of these diagrams indicates parameter values for which hyperchaos, chaos, quasiperiodicity, and periodicity may be found. In fact, each diagram shows delimited regions where each of these behaviors happens. Moreover, it is shown that some of these parameter planes display organized periodic structures embedded in quasiperiodic and chaotic regions.


Statistical and Nonlinear Physics 


  1. 1.
    E. Lorenz, J. Atmos. Sci. 20, 130 (1963) ADSCrossRefGoogle Scholar
  2. 2.
    C. Sparrow, The Lorenz equations: bifurcations, chaos, and strange attractors (Springer-Verlag, New York, 1982) Google Scholar
  3. 3.
    A. Zarei, S. Tavakoli, Appl. Math. Comput. 291, 323 (2016) MathSciNetGoogle Scholar
  4. 4.
    S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos (Springer, New York, 2003) Google Scholar
  5. 5.
    M.J. Correia, P.C. Rech, Appl. Math. Comput. 218, 6711 (2012) MathSciNetGoogle Scholar
  6. 6.
    H.G. Schuster, W. Just, Deterministic chaos an introduction (Wiley-VCH, Weinheim, 2005) Google Scholar
  7. 7.
    F. Doveil, A. Macor, Y. Elskens, Chaos 16, 033103 (2006) ADSCrossRefGoogle Scholar
  8. 8.
    F.G. Prants, P.C. Rech, Math. Comput. Simul. 136, 132 (2017) CrossRefGoogle Scholar
  9. 9.
    H.A. Albuquerque, R.M. Rubinger, P.C. Rech, Phys. Lett. A 372, 4793 (2008) ADSCrossRefGoogle Scholar
  10. 10.
    C. Bonatto, J.A.C. Gallas, Phys. Rev. E 75, 055204 (2007) ADSCrossRefGoogle Scholar
  11. 11.
    T.S. Krüger, P.C. Rech, Eur. Phys. J. D 66, 12 (2012) ADSCrossRefGoogle Scholar
  12. 12.
    F.G. Prants, P.C. Rech, Eur. Phys. J. B 87, 196 (2014) ADSCrossRefGoogle Scholar
  13. 13.
    P.C. Rech, Int. J. Bifurc. Chaos 25, 1530035 (2015) MathSciNetCrossRefGoogle Scholar
  14. 14.
    P.C. Rech, Phys. Scr. 91, 075201 (2016) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Departamento de Física, Universidade do Estado de Santa CatarinaJoinvilleBrazil

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