Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case

Flaviano Battelli,Michal Fečkan

doi:10.3390/math9192449

Abstract

We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation.

Highlights

In [1] a system like x = f (x, y) y = εg(x, y, ε), ε ∈ R (1)has been considered, where x ∈ R2, x = f (x, y) is Hamiltonian for any y ∈ R and has a one-parameter family of periodic solutions q(t − θ, y, α) with period T(y, α) being C1 in (y, α)
We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed equation has a non singular periodic solution
They answered the following question: do any of these periodic solutions persist for ε = 0? They constructed a vector valued function Mp/q(y, α, θ) that they called subharmonic Melnikov function which is a measure of the difference between the starting value and the value of the solution at the time p q in a direction transverse to the unperturbed vector field at the starting point

Summary

Introduction

Introducing the variable θ = t mod T, the perturbed time dependent vector field is reduced to a time independent system on R3 × S1 where S1 is the unit circle They answered the following question: do any of these periodic solutions persist for ε = 0? We emphasize that the results of this paper extend to the case where f±(x, y) is replaced by f±(x, y, ε) = f0,±(x, y) + ε f1,±(x, y, ε) and f0,±(x, y), f1,±(x, y, ε) are smooth outside the singularity manifold {h(x, y) = 0}. In this case in the unperturbed system x = f±(x, η). We observe that our results fit into a general theory of discontinuous differential equations presented in series of works [2–9]

Preliminary Results

Periodic Solutions

Discussion