Existence of Coexisting Between 5-cycle and Equilibrium Point on Piecewise Linear Map

  • Uraiwan - Jittburus Pibulsongkram Rajabhat University
Keywords: 5-cycle, equilibrium point, piecewise linear system, difference equation

Abstract

A piecewise linear system of difference equations is one of the piecewise systems that has special characters like coexisting attracting sets. In this article, we also exhibit the coexisting attractors of 5-cycles and equilibrium point. We use some direct iterative calculations and an inductive statement to explain all behaviors of solutions belonging to the system with initial condition belonging to negative -axis. We also found intervals of initial conditions that solutions become 5-cycles and equilibrium point.

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References

Aharonov, D., Devaney, R.L. & Ellas, U. (1987). The dynamics of a piecewise linear map and its smooth approximation. International Journal of Bifurcation and Chaos, 7(2), 351-372.

Cannings, C., Hoppensteadt, F. C. & Segel, L. A. (2005). Epidemic Modelling: An Introduction. New York: Cambridge University Press.

Cull, P. (2006). Difference Equations as Bilogical Models. Scientiae Mathematicae Japonicae e-2006, 965-981.

Devaney, R.L. (1984). A piecewise linear model for the zones of instability of an area-preserving map. Journal of Physica,10(3), 387-393.

Grove, E.A. & Ladas, G. (2005). Periodicities in Nonlinear Difference Equations. New York: Chapman Hall.

Grove, E.A., Lapierre, E. & Tikjha, W. (2012). On the Global Behavior of and . Cubo Mathematical Journal, 14(2), 125–166.

Krinket, S. & Tikjha, W. (2015). Prime period solution of cartain piecewise linear system of difference equation. Proceedings of the Pibulsongkram Research: Vol. 2015 (pp. 76-83). (in thai)

Lozi, R. (1978). Un attracteur etrange du type attracteur de Henon. Journal of Physics, 39(C5), 9-10.

Simpson, D.J.W. (2010). Bifurcations in piecewise-smooth continuous systems. Canada: World Scientific.

Tikjha, W., Lapierre, E.G. & Sitthiwirattham, T. (2017). The stable equilibrium of a system of piecewise linear difference equations. Advances in Difference Equations 67 (10 pages); doi:10.1186/s13662-017-1117-2

Tikjha, W., Lenbury, Y. & Lapierre, E.G. (2010). On the Global Character of the System of Piecewise Linear Difference Equations and . Advances in Difference Equations, 573281 (14pages ); doi:10.1155/2010/573281

Tikjha, W. & Piasu, K. (2020). A necessary condition for eventually equilibrium or periodic to a system of difference equations. Journal of Computational Analysis and Applications, 28(2), 254-260.

Zhusubaliyev, Zh.T. & Mosekilde, E. (2006). Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system. Physics Letters, 351(3),167-174.

Zhusubaliyev, Zh. T, Mosekilde, E. & Banerjee, S. (2008). Multiple-attractor bifurcations and quasiperiodicity in piecewise-smooth maps. International Journal of Bifurcation and Chaos 18(6), 1775–1789.

Published
2020-06-01
Section
บทความวิจัย