Quantifying Stability and Chaoticity of One-dimensional and Two-dimensional Discrete Dynamical Systems using Stability Analysis and Lyapunov Exponents
Abstract
Discrete dynamical system is a system that evolve dynamically with discrete time. In this paper, we consider two discrete systems which exhibit chaotic behaviour. We show that the chaoticity of a system is depend on the values of parameter in the system. The objective of this paper is to investigate both stability and chaoticity of the systems using stability analysis and Lyapunov exponent, respectively. The results show that there is only one fixed point for the one-dimensional system, while for the two-dimensional system, there exist four possible fixed points. We have proved the stability conditions for each fixed point obtained. The Lyapunov results show that the one-dimensional system is stable when r<π/2 and chaotic when r>π/2. Whereas for two-dimensional systems considered, the system is chaotic for the whole range of parameter α∈(0,1]. This study shows that it is significant to consider various values of parameter to study the stability of a dynamical systems in particular to control the chaos in a system.
References
[2] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems. Berlin: Springer, 1996.
[3] U. A. M. Roslan, and P. Ashwin, “Local and global stability indices for a riddled basin attractor of a piecewise linear map. Dynamical Systems: An International Journal, vol. 31, pp. 375-392, 2016.
[4] U. A. Mohd Roslan, and S. Samsuddin, “On the comparison of stability analysis with phase portrait for a discrete prey-predator system”, International Journal of Applied Engineering Research, vol. 12, no. 24, pp. 15273-15277, 2017.
[5] J. Chen, and X. Xie, “Stability analysis of a discrete competitive system with nonlinear interinhibition terms”, Advances in Difference Equations, vol. 294, pp. 1-19, 2017.
[6] O. Obi, “Stability of autonomous and non-autonomous differential equations”, M. Sc. thesis, Eastern Mediterranean University, North Cyprus, 2013.
[7] P. Glendinning, “Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations”, Cambridge: Cambridge University Press, 1994.
[8] U. A. Mohd Roslan and M. T. Mohd Lutfi, “Global stability index for an attractor with riddled basin in a two-species competition system”, in Dynamical Systems, Bifurcation Analysis and Applications, M. H. Mohd, N. Abdul Rahman, N. Abd Hamid and Y. Mohd Yatim, Eds. Singapore: Springer Proceedings in Mathematics & Statistics, 2019, pp. 135-146.
[9] R. L. Devaney, An Introduction to Chaotic Dynamical Systems. 2nd ed. USA: Westview Press, 2003.
[10] T. -Y. Li, and J. A. Yorke, “Period three implies chaos”, The American Mathematical Society, vol. 82, no. 10, pp. 985-992, 1975.
[11] U. A. Mohd Roslan, “Some contributions on analysis of chaotic dynamical systems”, M. Sc. thesis, Unversiti Malaysia Terengganu, Terengganu, 2010.
[12] A. -C. Dascalescu, R. E. Boriga, and A. -V. Diaconu, “Study of a new chaotic dynamical system and its usage in a novel pseudorandom bit generator”, Mathematical Problems in Engineering. pp. 1-10, 2013.
[13] R. Lopez-Ruiz, and D. Fournier-Prunaret, “Periodic and chaotic events in a discrete model of logistic type for the competitive interaction of two species,” Chaos, Solitons & Fractals, vol. 41, pp. 334–347, 2009.
[14] I. Kan, “Open sets of diffeomorhpisms having two attractors, each with an everywhere dense basin”, American Mathematical Society, vol. 31, no. 1, July, pp. 68-74, 1994.
[15] J. Buescu, “Exotic Attractors: From Liapunov Stability to Riddled Basins”, Switzerland: Birkhauser-Verlag. 1997.
[16] L. Xu, B. Chen, Y. Zhao, and Y. Cao, “Normal Lyapunov exponents and asymptotically stable attractors”, Dynamical Systems: An International Journal, vol. 23, no. 2, pp. 207-218.
