We study bifurcations of bounded solutions from homoclinic orbits for time-perturbed discontinuous systems. Functional analytic method is used. An illustrative example of a periodically perturbed piecewise linear differential equation in \(\mathbb{R}^3\) is presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Awrejcewicz, J., M. Fečkan and Olejnik, P. (2006). Bifurcations of planar sliding homoclinics. Math. Problems Eng., 1–13.
Battelli, F., Fečkan, M. (2002). Some remarks on the Melnikov function. Elect. J. Differential Equations, 1–29.
Battelli F., Lazzari C. (1990). Exponential dichotomies, heteroclinic orbits, and Melnikov functions. J. Differential Equations 86, 342–366
Brogliato B. (1996). Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences 220. Springer, Berlin
Chua L.O., Komuro M., Matsumoto T. (1986). The double scroll family. IEEE Trans. CAS 33, 1073–1118
Coppel, W. A. (1978). Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York.
Gruendler J. (1995). Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations. J. Differential Equations 122, 1–26
Guckenheimer J., Holmes P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer–Verlag, New York
Kukučka P. (2007). Jumps of the fundamental solution matrix in discontinuous systems and applications. Nonlinear Analysis Th. Meth. Appl. 66, 2529–2546
Kukučka P. (2007). Melnikov method for discontinuous planar systems. Nonlinear Analysis Th. Meth. Appl. 66, 2698–2719
Kunze, M., and Küpper, T. (2001). Non-smooth dynamical systems: an overview, Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems, Springer, Berlin, 431–452.
Kuznetsov Yu. A., Rinaldi S., Gragnani A. (2003). One-parametric bifurcations in planar Filippov systems. Int. J. Bif. Chaos 13, 2157–2188
Leine R.I., Van Campen D.H., Van de Vrande B.L. (2000). Bifurcations in nonlinear discontinuous systems. Nonlinear Dynamics 23, 105–164
Llibre J., Ponce E., Teruel A.E. (2007). Horseshoes near homoclinic orbits for piecewise linear differential systems in \(\mathbb{R}^3\) . Int. J. Bif. Chaos 17, 1171–1184
Palmer K.J. (1984). Exponential dichotomies and transversal homoclinic points. J. Differential Equations 55, 225–256
Rudin, W. (1974). Real and Complex Analysis, McGraw-Hill, Inc., New York.
Zou, Y. K., and Küpper, T. (2004). Melnikov method and detection of chaos for non-smooth systems (preprint).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Battelli, F., Fečkan, M. Homoclinic Trajectories in Discontinuous Systems. J Dyn Diff Equat 20, 337–376 (2008). https://doi.org/10.1007/s10884-007-9087-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-007-9087-9