Bifurcation Analysis of a Piecewise Smooth Map with Two Asymptotes

Authors

  • Roya Makrooni University of Sistan and Baluchestan

DOI:

https://doi.org/10.52737/18291163-2022.14.13-1-12

Keywords:

Discontinuous System, Divergence, Flip Bifurcation, Periodic Orbit

Abstract

In this paper, we consider a discontinuous piecewise smooth system involving four parameters and two asymptotes, recently introduced as a model in engineering sciences. We classify and investigate its bifurcation behaviour. A local bifurcation analysis of the system in the range of parameters which has not been studied so far is undertaken and then supported by numerical computations. This reveals the existence of a flip bifurcation depends on the power singularity. Moreover, we state that a set of positive measure of points with divergent dynamic behaviour exists.

References

S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps, Phys. Rev. E, 59 (1991), pp. 4052-4061. https://doi.org/10.1103/physreve.59.4052

S. Banerjee, M. S. Karthik, G. H. Yuan, and J. A. Yorke, Bifurcations in one-dimensional piecewise smooth maps: Theory and applications in switching circuits, IEEE Trans. Circuits Sys. I: Fundam. Theory Appl., 47 (2000), no. 3, pp. 389-394. https://doi.org/10.1109/81.841921

S. Banerjee and G. C. Verghese, textit{Nonlinear phenomena in power electronics: Attractors, bifurcations, chaos, and nonlinear control}, New York: Wiley-IEEE Press, 2001.

C. M. Berger, X. Zhao, D. G. Schaeffer, H. M. Dobrovolny, and D. J. Krassowska Wand Gauthier, Period-doubling bifurcation to alternans in paced cardiac tissue: Crossover from smooth to border-collision characteristics, Phys. Rev. Lett., 99 (2007), 058101. https://doi.org/10.1103/physrevlett.99.058101

H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick–slip oscillations, Physica D, 136 (2000), no. 3-4, pp. 280-302. https://doi.org/10.1016/s0167-2789(99)00161-x

M. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk, Piecewise-smooth dynamical systems: Theory and applications, Applied Mathematical Sciences, 163, Springer-Verlag, London, 2008. https://doi.org/10.1007/978-1-84628-708-4

M. di Bernardo, P. Kowalczyk and A. B. Nordmark, Bifurcations of dynamical systems with sliding: derivation of normal-form mappings, Physica D, 170 (2002), no. 3-4, pp. 170-175. https://doi.org/10.1016/s0167-2789(02)00547-x

L. Gardini, V. Avrutin, I. Sushko, and F. Tramontana, Continuous and discontinuous piecewise-smooth one-dimensional maps: Invariant sets and bifurcation structures, World scientific series on nonlinear science, Series A: vol. 95, 2019. https://doi.org/10.1142/8285

L. Gardini and R. Makrooni, Necessary and sufficient conditions of full chaos for expanding Baker-like maps and their use in non-expanding Lorenz maps, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), pp. 272-289. https://doi.org/10.1016/j.cnsns.2018.06.018

L. Gardini, R. Makrooni and I. Sushko, Cascades of alternating smooth bifurcations and border collision bifurcations in a family of discontinuous linear-power maps, Discrete Contin. Dyn. Sys.: Series B, 23 (2018), no. 2, pp. 701-729. https://doi.org/10.3934/dcdsb.2018039

C. Halse, M. Homer and M. di Bernardo, C-bifurcations and period-adding in one-dimensional piecewise smooth maps, Chaos Solit. Fractals, 18 (2003), no. 5, pp. 953-976. https://doi.org/10.1016/s0960-0779(03)00066-3

M. A. Hassouneh and E. H. Abed, Border collision bifurcation control of cardiac alternans, Int. J. Bifurcat. Chaos, 14 (2004), no. 9, pp. 3303-3315. https://doi.org/10.1142/s0218127404011351

A. Kumar, S. Banerjee and D. P. Lathrop, Dynamics of a piecewise smooth map with singularity, Phys. Lett. A, 337 (2005), no. 1-2, pp. 87-92. https://doi.org/10.1016/j.physleta.2005.01.046

R. Makrooni, N. Abbasi, M. Pourbarat, and L. Gardini, Robust unbounded chaotic attractors in 1D discontinuous maps, Chaos Solit. Fractals, 77 (2015), pp. 310-318. https://doi.org/10.1016/j.chaos.2015.06.012

R. Makrooni, L. Gardini, I. Sushko, Bifurcation structures in a family of 1D discontinuos linear-hyperbolic invertible maps, Int. J. Bifurcat. Chaos, 25 (2015), no. 13, 1530039, 21 pp. https://doi.org/10.1142/s0218127415300396

R. Makrooni, F. Khellat and L. Gardini, Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: Divergence and bounded dynamics, J. Differ. Equ. Appl., 21 (2015), no. 9, pp. 791-824. https://doi.org/10.1080/10236198.2015.1046855

R. Makrooni, F. Khellat and L. Gardini, Border collision and fold bifurcations in a family of one-dimensional discontinuous piecesiwe smooth maps: Unbounded chaotic sets, J. Differ. Equ. Appl., 21 (2015), no. 8, pp. 660-695. https://doi.org/10.1080/10236198.2015.1045893

A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, J. Sound Vib., 145 (1991), no. 2, pp. 279-297. https://doi.org/10.1016/0022-460x(91)90592-8

E. Pavlovskaia, M. Wiercigroch and C. Grebogi, Two-dimensional map for impact oscillator with drift, Phys. Rev. E, 70 (2004), 036201. https://doi.org/10.1103/physreve.70.036201

E. Pavlovskaia and M. Wiercigroch, Low-dimensional maps for piecewise smooth oscillators, J. Sound Vib, 305 (2007), no. 4-5, pp. 750-771. https://doi.org/10.1016/j.jsv.2007.04.044

M. Pourbarat, N. Abbasi and R. Makrooni, Chaos in smooth piecewise dynamical systems with one discontinuous point, J. Adv. Math. Model., 9 (2019), no. 2, pp. 93-105.

R. Sharan and S. Banerjee, Character of the map for switched dynamical systems for observations on the switching manifold, Phys. Lett. A, 372 (2008), no. 23, pp. 4234-4240. https://doi.org/10.1016/j.physleta.2008.03.050

W. T. Shi, C. L. Gooderidge and D. P. Lathrop, Viscous effects in droplet-ejecting capillary waves, Phys. Rev. E, 56 (1997), 4157. https://doi.org/10.1103/physreve.56.472

S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer-Verlag, New York, 2003.

Downloads

Published

2022-10-02

How to Cite

Makrooni, R. (2022). Bifurcation Analysis of a Piecewise Smooth Map with Two Asymptotes. Armenian Journal of Mathematics, 14(13), 1–12. https://doi.org/10.52737/18291163-2022.14.13-1-12