Abstract
We investigate a class of planar piecewise smooth systems with a generalized heteroclinic loop (a closed curve composed of hyperbolic saddle points, generalized singular points and regular orbits). We give conditions for the stability of the generalized heteroclinic loop and provide some sufficient conditions for the maximum number of limit cycles that bifurcate from the heteroclinic connection. The discussions rely on the approximation of the Poincaré map, which is constructed near the generalized heteroclinic loop. To obtain it, we introduce the Dulac map and use Melnikov method. By analyzing the fixed point of the Poincaré map, we get the number of limit cycles, which can be produced from the generalized heteroclinic loop. As applications to our theories, we give an example to show that two limit cycles can appear.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Afsharnezhad, Z., Amaleh, M.K.: Continuation of the periodic orbits for the differential equation with discontinuous right hand side. J. Dyn. Differ. Equ. 23, 71–92 (2011)
Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Theory of Bifurcations of dynamic systems on a plane. Israel Program for Scientific Translations, Jerusalem (1971)
Artés, J.C., Llibre, J., Medrado, J.C., Teixeira, M.A.: Piecewise linear differential systems with two real saddles. Math. Comput. Simul. 95, 13–22 (2014)
Brogliato, B.: Nonsmooth Impact Mechanics. Models, Dynamics and Control. Springer, London (1996)
Chen, S., Du, Z.: Stability and perturbations of homoclinic loops in a class of piecewise smooth systems. Int. J. Bifurc. Chaos Appl. Sci. Engrgy 25, 1550114 (2015)
Chow, S.N., Hale, J.K.: Methods of Bifurcations Theory. Springer, New York (1982)
Coll, B., Gasull, A., Prohens, R.: Degenerate Hopf bifurcations in discontinuous planar systems. J. Math. Anal. Appl. 253, 671–690 (2001)
Colombo, A., di Bernardo, M., Hogan, S.J., Jeffrey, M.R.: Bifurcations of piecewise smooth flows: perspectives, methodologies and open problems. Physica D 241, 1845–1860 (2012)
Demidovich, B.P.: Collection of Problems and Exercises on Mathematical Analysis. Nauka, Moscow (2010)
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth Dynamical Systems: Theory and Applications. Springer, London (2008)
Du, Z., Li, Y., Zhang, W.: Bifurcation of periodic orbits in a class of planar Filippov systems. Nonlinear Anal. 69, 3610–3628 (2008)
Fečkan, M., Pospíšil, M.: On the bifurcation of periodic orbits in discontinuous systems. Commun. Math. Anal. 8, 87–108 (2010)
Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic, Dordrecht (1988)
Galvanetto, U., Bishop, S.R., Briseghella, L.: Mechanical stick-slip vibrations. Int. J. Bifurc. Chaos Appl. Sci. Engrgy 5, 637–651 (1995)
Granados, A., Hogan, S.J., Seara, T.M.: The Melnikov method and subharmonic orbits in a piecewise-smooth system. SIAM J. Appl. Dyn. Syst. 11, 801–830 (2012)
Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)
Han, M.: Periodic Solutions and Bifurcation Theory of Dynamical Systems (in Chinese). Science Press, Beijing (2002)
Han, M., Zhang, W.: On Hopf bifurcation in non-smooth planar systems. J. Differ. Equ. 248, 2399–2416 (2010)
Hogan, S.J.: Heteroclinic bifurcations in damped rigid block motion. Proc. R. Soc. Lond. Ser. A 439, 155–162 (1992)
Huan, S., Yang, X.: Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics. Nonlinear Anal. 92, 82–95 (2013)
Joyal, P.: Generalized Hopf bifurcation and its dual generalized homoclinic bifurcation. SIAM. J. Appl. Math. 48, 481–496 (1988)
Kunze, M.: Non-Smooth Dynamical Systems. Springer, Berlin (2000)
Li, L., Huang, L.: Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems. J. Math. Anal. Appl. 411, 83–94 (2014)
Liang, F., Han, M., Zhang, X.: Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems. J. Differ. Equ. 255, 4403–4436 (2013)
Li, S., Shen, C., Zhang, W.: The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application. Nonlinear Dyn. 85, 1091–1104 (2016)
Llibre, J., Novaes, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn. 82, 1159–1175 (2015)
Luo, A.C.J.: Discontinuous Dynamical Systems. Higher Education Press, Beijing (2012)
Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Mosc. Math. Soc. 12, 1–57 (1963)
Shen, J., Du, Z.: Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones. Z. Angew. Math. Phys. 67, 1–17 (2016)
Tsypkin, Y.Z.: Relay Control Systems. Cambridge University Press, Cambridge (1984)
Acknowledgements
The author would like to thank an anonymous referee for his (or her) valuable comments and suggestions which help to improve the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China (11601355, 11671279, 11671280).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, S. Stability and Perturbations of Generalized Heteroclinic Loops in Piecewise Smooth Systems. Qual. Theory Dyn. Syst. 17, 563–581 (2018). https://doi.org/10.1007/s12346-017-0256-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12346-017-0256-x