Periodic Solution of a Piecewise Linear System of Difference Equations with Initial Condition in Positive X - Axis
Abstract
In this paper, we study a piecewise linear system of difference equations with initial condition in positive x-axis which remains an open problem. We find that there exist 5-cycles and equilibrium point. We use some direct iterative calculations and mathematical induction to prove the behaviors of solutions of the system. We separate positive x-axis into subintervals and investigate the behaviors of solutions in each of subintervals. We also find that for such initial condition the attractors are only 5-cycles and equilibrium point. Moreover, we reveal the boundary of basins of attractions for 5-cycles and equilibrium point. Keywords : piecewise linear system ; periodic solution ; equilibrium point ; difference equationReferences
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Border-Collision Normal Form. Int. J. Bifurcation Chaos, 24(9), 1450118
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Kyungpook Mathematical Journal, 60(2), 401-413.
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difference equations. Advances in Difference Equations67 (10 pages); doi:10.1186/s13662-017-1117-2
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Difference Equations and . Advances in Difference Equations,
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Tikjha, W., Lapierre, E. G. & Lenbury, Y. (2015). Periodic solutions of a generalized system of piecewise linear
difference equations. Advances in Difference Equations 2015:248 doi:10.1186/s13662-015-0585-5.
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 – 261.
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Nonlinear Science A, Vol. 44, World Scientific.
Zhusubaliyev Zh. T, Mosekilde E. and Banerjee S. (2008). Multiple-attractor bifurcations and quasiperiodicicity in
piecewise-smooth maps. Int. J. Bifurcation and Chaos, 18(6), 1775–1789.
Bifurcations, Chaos, and Nonlinear Control. IEEE Press.
Botella-Soler, V., Castelo, J.M., Oteo, J.A., & Ros, J. (2011). Bifurcations in the Lozi map. Journal of Physics A, 44,
305101.
Brogliato, B. (1999). Nonsmooth mechanics models, dynamics and control. New York: Springer-Verlag.
Devanney, R.L., (1984). A piecewise linear model of the the zones of instability of an area-preservingmap.
Physica., 10D, 387-393.
Gardini, L., & Tikjha, W. (2019). The role of virtual fixed points and center bifurcations in a Piecewise Linear Map.
Int. J. Bifurcation and Chaos, 29 (14) doi: https://doi.org/10.1142/S0218127419300416
Gardini, L., & Tikjha, W. (2020). Dynamics in the transition case invertible/non-invertible in a 2D Piecewise Linear
Map. Chaos, Solitons and Fractals, 137 (2020) https://doi.org/10.1016/j.chaos.2020.109813
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, 125–166.
Ing, J., Pavlovskaia, E., Wiercigroch, M., & Banerjee, S. (2010). Bifurcation analysis of an impact oscillator with a
one-sided elastic constraint near grazing, Physica D, 239, 312-321.
Jittbrurus, U., & Tikjha, W. (2020). Existence of Coexisting Between 5-cycle and Equilibrium Point on Piecewise
Linear Map. Science and Technology NSRU Journal, 12(15), 39-47.
Krinket, S. and Tikjha, W. (2015). Prime period solution of cartain piecewise linear system of difference equation.
Proceedings of the Pibulsongkram Research. (pp 76-83). (in thai)
Lozi, R. (1978). Un attracteur etrange du type attracteur de Henon. J. Phys. (Paris) 39, 9-10.
Ma, Y., Agarwal, M. & Banerjee, S. (2006) Border collision bifurcations in a soft impact system. Phys. Lett. A,
354 (4), 281--287.
Simpson, D. J. W. (2010). Bifurcations in piecewise-smooth continuous systems. World Scientific.
Simpson, D. J. W. (2014a) Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the
Border-Collision Normal Form, Int. J. Bifurcation and Chaos, 24(6), 1430018
Simpson, D. J. W. (2014b) Scaling Laws for Large Numbers of Coexisting Attracting Periodic Solutions in the
Border-Collision Normal Form. Int. J. Bifurcation Chaos, 24(9), 1450118
Tikjha, W. & Gardini, L. (2020). Bifurcation sequences and multistability in a two-dimensional Piecewise Linear
Map, Int. J. Bifurcation and Chaos, 30(6), doi:S0218127420300141
Tikjha, W., & Lapierre, E. (2020). Periodic solutions of a system of piecewise linear difference equations.
Kyungpook Mathematical Journal, 60(2), 401-413.
Tikjha, W., Lapierre, E.G., & Sitthiwirattham T. (2017). The stable equilibrium of a system of piecewise linear
difference equations. Advances in Difference Equations67 (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 (14 pages ); doi:10.1155/2010/573281
Tikjha, W., Lapierre, E. G. & Lenbury, Y. (2015). Periodic solutions of a generalized system of piecewise linear
difference equations. Advances in Difference Equations 2015:248 doi:10.1186/s13662-015-0585-5.
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 – 261.
Zhusubaliyev, Zh.T., & Mosekilde, E. (2003). Bifurcations and Chaos in piecewise-smooth dynamical systems,
Nonlinear Science A, Vol. 44, World Scientific.
Zhusubaliyev Zh. T, Mosekilde E. and Banerjee S. (2008). Multiple-attractor bifurcations and quasiperiodicicity in
piecewise-smooth maps. Int. J. Bifurcation and Chaos, 18(6), 1775–1789.
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2021-09-14
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