Melnikov Function Of Discontinuous Systems

This is a survey for the Melnikov function in discontinuous systems with discon-tinuous right-hand sides.The discontinuous system reads where f:Rn→Rn,t∈R, and x∈Rn\Σand h∈Cr(Rn, R), r≥1. Rn is split into two partsIn the discontinuous planar system, let: (i)f has a singular point x1.(ii) In the respective domains(iii) Let the perturbation is a T-periodic function with respect to the variable t:Theorem 1. A kind of O(∈)-measure of the distance between the stable and un-stable manifolds where In this formula M1 is the similar to the smooth case, so it is smooth part, M2 is the unstable part, M3 is the stable part. These three parts are the so called Melnikov function.Then consider some examples: A one-degree-of-freedom mechanical system: where Ff is a discontinuous functionA friction problem: whereμ1,μ2∈R is a small parameter. Specially, letThe unperturbed system has a fix point 0, andφ∈C2(R, R) is a periodic perturbation. Suppose the unperturbed system has homoclinc solution, i.e., there exists a q, such that Then And through some alternates, Melnikov function takes the form:It has a simple zero at some points. Then ifδbig enough, the perturbed system exhibits chaotic behavior.A slide equation:The perturbed system where x=(z,y)∈RxRn-1, f±:S→Rn, f±∈Cbr(S), and g:R×S×R→Rn, g∈Cbr(R×S×R), S is a bounded open subset of Rn that has nonempty intersection with the hyperplane z=0. We also assume that the r-th order derivatives of f±, and not like the system above, we need that g(x) is uniformly continuous.The Melnikov function is:...