<p>This paper investigates the periodic oscillatory behaviors in a second-order discontinuous differential equation with a delayed switching boundary. The equation has two kinds of equivalent forms, one simplified first-order system and one common planar system with two real saddle points. Sufficient conditions are presented to ensure that the first-order system has a unique slowly oscillatory periodic solution, up to phase shift; meanwhile, the orbitally asymptotical stability and the reachability in finite time are also shown. As to the planar system, the composition of solutions with respect to the current and historical switching boundaries is described firstly; then, according to the relative position of two saddle points, three sufficient conditions are detailed for the existence and uniqueness of a slowly oscillatory periodic orbit; finally, some criterions on the orbital stability of the slowly oscillatory periodic orbit are given by using the successive function, moreover, an extra curve separating the orbitally asymptotically stable and unstable regions is provided in the parameter space of time delay and slope of the switching line.</p>

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Existence, uniqueness and orbital stability of slowly oscillatory periodic solutions in a second-order discontinuous differential equation with time delay

  • Liping Li

摘要

This paper investigates the periodic oscillatory behaviors in a second-order discontinuous differential equation with a delayed switching boundary. The equation has two kinds of equivalent forms, one simplified first-order system and one common planar system with two real saddle points. Sufficient conditions are presented to ensure that the first-order system has a unique slowly oscillatory periodic solution, up to phase shift; meanwhile, the orbitally asymptotical stability and the reachability in finite time are also shown. As to the planar system, the composition of solutions with respect to the current and historical switching boundaries is described firstly; then, according to the relative position of two saddle points, three sufficient conditions are detailed for the existence and uniqueness of a slowly oscillatory periodic orbit; finally, some criterions on the orbital stability of the slowly oscillatory periodic orbit are given by using the successive function, moreover, an extra curve separating the orbitally asymptotically stable and unstable regions is provided in the parameter space of time delay and slope of the switching line.