A class of discontinuous systems exhibit perturbed period doubling bifurcation

Hany A. Hosham; Alaa A. Alzulaibani; Tarek Sellami; Khaled Sioud; Thoraya N. Alharthi; Hany A. Hosham; Alaa A. Alzulaibani; Tarek Sellami; Khaled Sioud; Thoraya N. Alharthi

doi:10.3934/math.20241223

AIMS Mathematics

2024, Volume 9, Issue 9: 25098-25113. doi: 10.3934/math.20241223

Previous Article Next Article

Research article Special Issues

A class of discontinuous systems exhibit perturbed period doubling bifurcation

1.
Department of Mathematics, Faculty of Science, Taibah University, Yanbu, 41911, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Sfax University, Sfax, 3029, Tunisia
3.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia

Received: 17 May 2024 Revised: 01 August 2024 Accepted: 13 August 2024 Published: 27 August 2024
MSC : 34A36, 34D23, 37G15, 34H10

This research considers discontinuous dynamical systems, which have related vector fields that shift over a discontinuity surface. These systems appear in a variety of applications, including ecology, medicine, neuroscience, and nonsmooth mechanics. The purpose of this paper is to develop a perturbation technique that measures the effect of a nonsmooth perturbation on the period doubling bifurcation of an unperturbed system. The unperturbed system is assumed to be close to a period doubling orbit, such that when the bifurcation parameter varies, the response changes from a period one to a period two limit cycle. The generalized determination of the Poincaré map associated with perturbed systems subjected to nonsmooth transitions is derived. The main techniques used in the proof of the results are normal forms and Melnikov functions, which are defined in two zones. Various examples are presented to show that non-smoothness is responsible for period doubling. To illustrate the interesting period doubling phenomenon that emerges from an existing flat periodic orbit via the non-smooth perturbation, a simple and novel discontinuous system is provided. An additional example is provided to show the emergence of a perturbed period doubling orbit near an unperturbed one.
- nonsmooth perturbation,
- Melnikov function,
- discontinuous systems,
- perturbed period doubling,
- stability
Citation: Hany A. Hosham, Alaa A. Alzulaibani, Tarek Sellami, Khaled Sioud, Thoraya N. Alharthi. A class of discontinuous systems exhibit perturbed period doubling bifurcation[J]. AIMS Mathematics, 2024, 9(9): 25098-25113. doi: 10.3934/math.20241223

Related Papers:

Abstract

This research considers discontinuous dynamical systems, which have related vector fields that shift over a discontinuity surface. These systems appear in a variety of applications, including ecology, medicine, neuroscience, and nonsmooth mechanics. The purpose of this paper is to develop a perturbation technique that measures the effect of a nonsmooth perturbation on the period doubling bifurcation of an unperturbed system. The unperturbed system is assumed to be close to a period doubling orbit, such that when the bifurcation parameter varies, the response changes from a period one to a period two limit cycle. The generalized determination of the Poincaré map associated with perturbed systems subjected to nonsmooth transitions is derived. The main techniques used in the proof of the results are normal forms and Melnikov functions, which are defined in two zones. Various examples are presented to show that non-smoothness is responsible for period doubling. To illustrate the interesting period doubling phenomenon that emerges from an existing flat periodic orbit via the non-smooth perturbation, a simple and novel discontinuous system is provided. An additional example is provided to show the emergence of a perturbed period doubling orbit near an unperturbed one.

References

[1]	H. Svensmark, M. R. Samuelsen, Perturbed period-doubling bifurcation. Ⅰ. Theory, Phys. Rev. B, 41 (1990), 4181–4188. https://doi.org/10.1103/PhysRevB.41.4181 doi: 10.1103/PhysRevB.41.4181
[2]	A. J. Sojahrood, R. Earl, M. C. Kolios, R. Karshafian, Investigation of the 1/2 order subharmonic emissions of the period-2 oscillations of an ultrasonically excited bubble, Phys. Lett. A, 384 (2020), 126446. https://doi.org/10.1016/j.physleta.2020.126446 doi: 10.1016/j.physleta.2020.126446
[3]	C. Athanasouli, K. Kalmbach, V. Booth, C. G. D. Behn, Nrem-rem alternation complicates transitions from napping to non-napping behavior in a three-state model of sleep-wake regulation, Math. Biosci., 355 (2023), 108929. https://doi.org/10.1016/j.mbs.2022.108929 doi: 10.1016/j.mbs.2022.108929
[4]	N. F. Tehrani, M. R. Razvan, Bifurcation structure of two coupled FHN neurons with delay, Math. Biosci., 270 (2015), 41–56. https://doi.org/10.1016/j.mbs.2015.09.008 doi: 10.1016/j.mbs.2015.09.008
[5]	L. Gyllingberg, D. J. T. Sumpter, Å. Brännström, Finding analytical approximations for discrete, stochastic, individual-based models of ecology, Math. Biosci., 365 (2023), 109084. https://doi.org/10.1016/j.mbs.2023.109084 doi: 10.1016/j.mbs.2023.109084
[6]	M. D. Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, Piecewise-smooth dynamical systems theory and applications, Springer Science & Business Media, 163 (2018).
[7]	T. Küpper, H. A. Hosham, K. Dudtschenko, The dynamics of bells as impacting system, Proc. I. Mech. Eng. Part, 225 (2011), 2436–2443. https://doi.org/10.1177/0954406211413855 doi: 10.1177/0954406211413855
[8]	J. Awrejcewicz, M. Fečkan, P. Olejnik, Bifurcations of planar sliding homoclinics, Math. Probl. Eng., 2006 (2006), 1–13. https://doi.org/10.1155/MPE/2006/85349 doi: 10.1155/MPE/2006/85349
[9]	H. A. Hosham, Bifurcation of periodic orbits in discontinuous systems, Nonlinear Dynam., 87 (2017), 135–148. https://doi.org/10.1007/s11071-016-3031-7 doi: 10.1007/s11071-016-3031-7
[10]	H. A. Hosham, T. N. Alharthi, Bifurcation and chaos in simple discontinuous systems separated by a hypersurface, AIMS Math., 9 (2024), 17025–17038. https://doi.org/10.3934/math.2024826 doi: 10.3934/math.2024826
[11]	M. R. Jeffrey, T. I. Seidman, M. A. Teixeira, V. I. Utkin, Into higher dimensions for nonsmooth dynamical systems, Physica D, 434 (2022), 133222. https://doi.org/10.1016/j.physd.2022.133222 doi: 10.1016/j.physd.2022.133222
[12]	T. Küpper, H. A. Hosham, Reduction to invariant cones for non-smooth systems, Math. Comput. Simulat., 81 (2011), 980–995. https://doi.org/10.1016/j.matcom.2010.10.004 doi: 10.1016/j.matcom.2010.10.004
[13]	D. Weiss, T. Küpper, H. A. Hosham, Invariant manifolds for nonsmooth systems, Physica D, 241 (2012), 1895–1902. https://doi.org/10.1016/j.physd.2011.07.012 doi: 10.1016/j.physd.2011.07.012
[14]	D. Weiss, T. Küpper, H. A. Hosham, Invariant manifolds for nonsmooth systems with sliding mode, Math. Comput. Simulat., 110 (2015), 15–32. https://doi.org/10.1016/j.matcom.2014.02.004 doi: 10.1016/j.matcom.2014.02.004
[15]	H. A. Hosham, Bifurcation of limit cycles in piecewise-smooth systems with intersecting discontinuity surfaces, Nonlinear Dynam., 99 (2020), 2049–2063. https://doi.org/10.1007/s11071-019-05400-z doi: 10.1007/s11071-019-05400-z
[16]	H. A. Hosham, Nonlinear behavior of a novel switching jerk system, Int. J. Bifurc. Chaos, 2020. https://doi.org/10.1142/S0218127420502028
[17]	F. Luo, Z. D. Du, Complicated periodic cascades arising from double grazing bifurcations in an impact oscillator with two rigid constraints, Nonlinear Dynam., 111 (2023), 13829–13852. https://doi.org/10.1007/s11071-023-08600-w doi: 10.1007/s11071-023-08600-w
[18]	Z. Fan, Z. D. Du, Bifurcation of periodic orbits crossing switching manifolds multiple times in planar piecewise smooth systems, Int. J. Bifurcat. Chaos, 29 (2019). https://doi.org/10.1142/S0218127419501608
[19]	M. Feckan, M. Pospíšil, Poincaré-Andronov-Melnikov analysis for non-smooth systems, Oxford: Academic Press, 2016.
[20]	J. Llibre, D. D. Novaes, C. A. B. Rodrigues, Bifurcations from families of periodic solutions in piecewise differential systems, Physica D, 404 (2020), 132342. https://doi.org/10.1016/j.physd.2020.132342 doi: 10.1016/j.physd.2020.132342
[21]	X. Guo, R. Tian, Q. Xue, X. Zhang, Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system, Chaos Soliton. Fract., 164 (2022), 112629. https://doi.org/10.1016/j.chaos.2022.112629 doi: 10.1016/j.chaos.2022.112629
[22]	J. L. R. Bastos, C. A. Buzzi, J. Llibre, D. D. Novaes, Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold, J. Differ. Equations, 267 (2019), 3748–3767. https://doi.org/10.1016/j.jde.2019.04.019 doi: 10.1016/j.jde.2019.04.019
[23]	K. S. Andrade, O. A. R. Cespedes, D. R. Cruz, D. D. Novaes, Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve, J. Differ. Equations, 287 (2021), 1–36. https://doi.org/10.1016/j.jde.2021.03.039 doi: 10.1016/j.jde.2021.03.039
[24]	Y. Li, Z. Du, Applying Battelli-Fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems, Discrete Cont. Dyn.-S., 24 (2019), 6025–6052.
[25]	M. Wang, Z. Wei, J. Wang, X. Yu, T. Kapitaniak, Stochastic bifurcation and chaos study for nonlinear ship rolling motion with random excitation and delayed feedback controls, Physica D, 462 (2024), 134147. https://doi.org/10.1016/j.physd.2024.134147 doi: 10.1016/j.physd.2024.134147
[26]	Z. Wei, Y. Li, T. Kapitaniak, W. Zhang, Analysis of chaos and capsizing of a class of nonlinear ship rolling systems under excitation of random waves, Chaos Int. J. Nonlinear Sci., 34 (2024), 043106. https://doi.org/10.1063/5.0187362 doi: 10.1063/5.0187362
[27]	S. H. Strogatz, Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, 3 Eds., CRC press, 2024.
[28]	R. L. Devaney, An introduction to chaotic dynamical systems, 2 Eds., CRC press, 2018. https://doi.org/10.4324/9780429502309
[29]	J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, New York: Springer, 1984. https://doi.org/10.1007/978-1-4612-1140-2
[30]	C. Chicone, Lyapunov-schmidt reduction and melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differ. Equations, 112 (1994), 407–447. https://doi.org/10.1006/jdeq.1994.1110 doi: 10.1006/jdeq.1994.1110
[31]	P. Kowalczyk, M. D. Bernardo, Two-parameter degenerate sliding bifurcations in Filippov systems, Physica D, 204 (2005), 204–229. https://doi.org/10.1016/j.physd.2005.04.013 doi: 10.1016/j.physd.2005.04.013
[32]	V. Acary, B. Brogliato, Numerical methods for nonsmooth dynamical systems: Applications in mechanics and electronics, Springer Science & Business Media, 2008. https://doi.org/10.1007/978-3-540-75392-6
[33]	L. Dieci, L. Lopez, Numerical solution of discontinuous differential systems: Approaching the discontinuity surface from one side, Appl. Numer. Math., 67 (2013), 98–110. https://doi.org/10.1016/j.apnum.2011.08.010 doi: 10.1016/j.apnum.2011.08.010
[34]	N. Guglielmi, E. Hairer, An efficient algorithm for solving piecewise-smooth dynamical systems, Numer. Algorithms, 89 (2022), 1311–1334. https://doi.org/10.1007/s11075-021-01154-1 doi: 10.1007/s11075-021-01154-1

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)