Asymptotically stable equilibrium points in new chaotic systems
PDF

Keywords

Chaotic systems
asymptotically stable equilibrium
non-existence of Shilnikov chaos
Lyapunov exponents Sistemas caóticos
equilibrio asintóticamente estable
no existencia del caos de Shilnikov
ex-ponentes de Lyapunov

How to Cite

Casas-García, K., Quezada Téllez, L. A., Carrillo-Moreno, S., Flores-Godoy, J. J., & Fernández Anaya, G. (2016). Asymptotically stable equilibrium points in new chaotic systems. Nova Scientia, 8(16), 41–58. https://doi.org/10.21640/ns.v8i16.321

Abstract

In this paper ten new chaotic nonlinear autonomous systems are presented. These systems were found by using the Monte Carlo method and they characterized by having one of their equilibrium points asymptotically stable. These new systems does not present chaos in the sense of Shilnikov, but their bifurcation diagrams show a period doubling route towards chaos. Kaplan-Yorke dimensions were also calculated, which is fractional order enclosed in a range of 2-3.

https://doi.org/10.21640/ns.v8i16.321
PDF

References

Carrillo, S., Casas-García, K., Flores-Godoy, J. J., Valencia, F. V., & Fernández-Anaya, G. (2015). Study of new chaotic flows on a family of 3-dimensional systems with quadratic nonlinearities. In Journal of Physics: Conference Series (Vol. 582, No. 1, p. 12016). IOP Publishing.

Chen, G. & Ueta, T (1999). "Yet another chaotic attractor", Int. J. Bifurcation and Chaos 9, 1465-1466.

Chlouverakis K. E., and J.C. Sprott (2005). "A comparison of correlation and Lyapunov dimensions", Physica D 200.

Elhadj Z. and J. C. Sprott (2012). "Non-existence of Shilnikov Chaos in continuos-time systems", Appl. Math. Mech. - Engl. Ed., 33(3), 1-4.

Gómez-Mont, X., Flores Godoy, J. J and Fernandez Anaya G (2013). "Some atractors in the extended complex Lorenz Model", International Journal of Bifurcation and Chaos, Volume 23, Issue 09, September 2013.

Li C. and J.C.Sprott (2014). "Chaotic flows with a single nonquadratic term", Physics Letters A 378, 178-183.

Leonov, G. A., & Kuznetsov, N. V. (2013). Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. International Journal of Bifurcation and Cha os, 23(01), 1330002.

Lorenz E. N. (1963). "Deterministic nonperiodic flow", J. Atmos. Sci. 20, 130-141.

Lü J. and G. Chen (2002). "A new chaotic attractor coined", International Journal of Bifurcation and Chaos, Vol. 12, No. 3, 659-661.

Malihe Molaie, Sajad Jafari, J. C. Sprott and S. M. R. Hashemi Golpayegani (2013). "Simple Chaotic Flows with one stable equilibrium", International Journal of Bifurcation and Chaos, Vol. 23, No. 11.

Medrano-T. Rene O., Murilo S. Baptista, and Ibere L. Caldas (2005). "Basic structures of the Shilnikov homoclinic bifurcation scenario", Chaos 15.

Perko, L (1991). Differential Equations and Dynamical Systems, Springer-Verlag, New York, Inc.

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1996). Numerical recipes in C (Vol. 2). Cambridge: Cambridge university press.

Sajad Jafari and J.C. Sprott (2013). "Simple chaotic flows with a line equilibrium", Chaos, Solitons & Fractals 57, 79-84.

Sifeu Takougang Kingni, Lars Keuninckx, Paul Woafo, Guy Van der Sande and Jan Danckaert (2013). "Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation", Nonlinear Dynam., 73, 1111-1123.

Sprott J. C. (2000). "Algebraically Simple Chaotic Flows", International Journal of Chaos Theory and Applications, Vol. 5, No. 2.

Sprott J. C. (1993). "Automatic Generations of Strange Attractors", Compu & Graphics, Vol. 17, No. 3, pp. 325-332.

Sprott J. C. (1994). "Some simple chaotic flows", Physical Review E, Vol. 50, Num. 2.

Sprott J. C. (1997). "Simplest dissipative chaotic flow", Physics Letters A 228, 271-274.

Sprott J. C., X. Wang and G. Chen (2013). "Coexistence of point, periodic and strange at tractors", International Journal of Bifurcation and Chaos, Vol. 23, No. 5.

Sprott, J. C., & Xiong, A. (2015). Classifying and quantifying basins of attraction. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(8), 083101.

Tianshou Zhou and Guanrong Chen (2006). "Classification of Chaos in 3-D Autonomous Quadratic Systems-I. Basic Framework and Methods", International Journal of Bifurcations and Chaos, Vol. 16, No. 9.

Tingli Xing, Roberto Barrio and Andrey Shilnikov (2014). "Symbolic Quest into Homoclinic Chaos", International Journal of Bifurcations and Chaos, Vol. 24, No. 8.

Xiong Wang and Guanrong Chen. "Constructing a chaotic system with any number of equilibria", http://arxiv.org/abs/1201.5751v1

Xiong Wang & Guanrong Chen (2012). "A chaotic system with only one stable equlibrium", Communications in Nonlinear Science and Numerical Simulation, Volume 17, Issue 3.

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Copyright (c) 2015 Nova Scientia