Abstract
This paper deals with bifurcation of limit cycles for perturbed piecewise-smooth systems. Concentrating on the case in which the vector fields are defined in four domains and the discontinuity surfaces are codimension-2 manifolds in the phase space. We present a generalization of the Poincaré map and establish some novel criteria to create a new version of the Melnikov-like function. Naturally, this function is designed corresponding to a system trajectory that interacts with two different discontinuity surfaces. This provides an approach to prove the existence of special type of invariant manifolds enabling the reduction of dynamics of the full system to the two-dimensional surfaces of the invariant cones. It is shown that there exists a novel bifurcation concerning the existence of multiple invariant cones for such system. Further, our results are then used to control the persistence of limit cycles for two- and three-dimensional perturbed systems. The theoretical results of these examples are illustrated by numerical simulations.
Similar content being viewed by others
References
Alexander, J.C., Seidman, T.: Sliding modes in intersecting switching surfaces, I: Blending. Houston J. Math. 24, 545–569 (1998)
Awrejcewicz, J., Lamarque, C.: Bifurcation and Chaos in Nonsmooth Mechanical Systems. World Scientific, Singapore (2000)
Awrejcewicz, J., Fečkan, M., Olejnik, P.: Bifurcations of planar sliding homoclinics. Math. Probl. Eng. Article ID 85349 (2006)
Awrejcewicz, J., Holicke, M.M.: Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods. World Scientific, Singapore (2007)
Calvo, M., Montijano, J.I., Randez, L.: DISODE45: a Matlab Runge Kutta solver for piecewise smooth IVPs of Filippov type. ACM Trans. Math. Softw. 43(3), 25 (2017)
Carmona, V., Fernández-García, S., Freire, E.: Saddle-node bifurcation of invariant cones in 3d piecewise linear systems. Phys. D: Nonlinear Phenom. 241, 623–635 (2012)
Carmona, V., Freire, E., Fernández-García, S.: Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discr. Contin. Dyn. Syst. A 35, 59–72 (2015)
Chicone, C.: Lyapunov–Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators. J. Differ. Equ. 112, 407–447 (1994)
di Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, London (2008)
Dieci, L., Difonzo, F.: The moments sliding vector field on the intersection of two manifolds. J. Dyn. Differ. Equ. 29(1), 169–201 (2017)
Dieci, L., Lopez, L.: Sliding motion in Filippov differential systems: theoretical results and a computational approach. SIAM J. Numer. Anal. 47, 2023–2051 (2009)
Dieci, L., Lopez, L.: Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field. Numer. Math 117, 779–811 (2011)
Dieci, L., Elia, C., Lopez, L.: A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of- attractivity analysis. J. Differ. Equ. 254, 1800–1832 (2013)
Fečkan, M., Pospíšil, M.: Poincaré–Andronov–Melnikov analysis for non-smooth systems. Academic Press is an imprint of Elsevier, London (2016)
Filippov, A.F.: Differential equations with discontinuous right-hand side. Am. Math. Soc. Transl. 2, 199–231 (1964)
Hosham, H.A.: Cone-Like Invariant Manifolds for Nonsmooth Systems. Ph.D. Thesis, Universität zu Köln, Germany (2011)
Hosham, H.A.: Bifurcation of periodic orbits in discontinuous systems. Nonlinear Dyn. 87, 135–148 (2017)
Hosham, H.A., Abou Elela, A.: Discontinuous phenomena in bioreactor system. Discrete Cont. Dyn. B 24(6), 2955–2969 (2019)
Hosham, H.A.: Bifurcations in four-dimensional switched systems. Adv. Differ. Equ. 2018, 388 (2018)
Huan, S.M.: Existence and stability of invariant cones in 3-dim homogeneous piecewise linear systems with two zones. Int. J. Bifurc. Chaos 27(1), 1750007 (2017)
Jeffrey, M.R.: Dynamics at a switching intersection: hierarchy, isonomy, and multiple-sliding. SIADS 13(3), 1082–1105 (2014)
Jeffrey, M.R., Kafanas, G., Simpson, D.J.W.: Jitter in dynamical systems with intersecting discontinuity surfaces. Int. J. Bifurc. Chaos 28(6), 1830020 (2018)
Kalmár-Nagy, T., Csikja, R., Elgohary, T.A.: Nonlinear analysis of a 2-DOF piecewise linear aeroelastic system. Nonlinear Dyn. 85(2), 739–750 (2016)
Küpper, T., Hosham, H.A.: Reduction to invariant cones for nonsmooth systems. Math. Comput. Simul. 81, 980–995 (2011)
Küpper, T., Hosham, H.A., Dudtschenko, K.: The dynamics of bells as impacting system. J. Mech. Eng. Sci. 225, 2436–2443 (2011)
Küpper, T., Hosham, H.A., Weiss, D.: Bifurcation for nonsmooth dynamical systems via reduction methods. In: Recent Trends in Dynamical Systems, Proceedings in Mathematics and Statistics, vol. 35, pp. 79–105. Springer (2013)
Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-smooth Mechanical Systems. Springer, Berlin (2004)
Li, Y., Du, Z.: Applying Battelli–Fečkan’s method to transversal heteroclinic bifurcation in piecewise smooth systems. Discrete Cont. Dyn. B 24(11), 6025–6052 (2019)
Li, S., Gong, X., Zhang, W., Hao, Y.: The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold. Nonlinear Dyn. 89(2), 939–953 (2017)
Li, S., Zhao, S.: The analytical method of studying subharmonic periodic orbits for planar piecewise-smooth systems with two switching manifolds. Int. J. Dyn. Control 7(1), 23–35 (2019)
Llibre, J., Teixeira, M.A.: Piecewise linear differential systems without equilibria produce limit cycles? Nonlinear Dyn. 88(1), 157–164 (2017)
Olejnik, P., Awrejcewicz, J.: Application of Hénon method in numerical estimation of the stick-slip transitions existing in Filippov-type discontinuous dynamical systems with dry friction. Nonlinear Dyn. 73(1–2), 723–736 (2013)
Olejnik, P., Awrejcewicz, J., Fečkan, M.: Modeling, Analysis and Control of Dynamical Systems. World Scientific, Singapore (2017)
Weiss, D., Küpper, T., Hosham, H.A.: Invariant manifolds for nonsmooth systems. Phys. D: Nonlinear Phenom. 241, 1895–1902 (2012)
Weiss, D., Küpper, T., Hosham, H.A.: Invariant manifolds for nonsmooth systems with sliding mode. Math. Comput. Simul. 110, 15–32 (2015)
Wiercigroch, M., Pavlovskaia, E.: Engineering applications of non-smooth dynamics. In: Nonlinear Dynamic Phenomena in Mechanics. Solid Mechanics and Its Applications, vol. 181, pp. 211–269. Springer, Dordrecht (2012)
Acknowledgements
The author is thankful to the reviewers for their useful corrections and suggestions, which improved the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hosham, H.A. Bifurcation of limit cycles in piecewise-smooth systems with intersecting discontinuity surfaces. Nonlinear Dyn 99, 2049–2063 (2020). https://doi.org/10.1007/s11071-019-05400-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-05400-z
Keywords
- Piecewise-smooth systems
- Poincaré map
- Melnikov-like theory
- Limit cycles
- Invariant cones
- Sliding bifurcation