The Melnikov Method and Subharmonic Orbits in a Piecewise-Smooth System

Abstract

Abstract. In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold x = 0. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of x = 0. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. When considering a non-autonomous (T -periodic) Hamiltonian perturbation of amplitude ε, using an impact map, we rigorously prove that, for every n and m relatively prime and ε > 0 small enough, there exists a nT -periodic orbit impacting 2m times with the switching manifold at every period if a modified subharmonic Melnikov function possesses a simple zero. We also prove that, if the orbits are discontinuous when they cross x = 0, then all these orbits exist if the relative size of ε > 0 with respect to the magnitude of this jump is large enough.We also obtain similar conditions for the splitting of the heteroclinic connections.

The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

“…Proceeding as in [Hog89,GHS12], we define the direct sequence of impactsω i associated with the sectionΣ as…”

“…The single block model was first introduced in [Hou63]; further details of its dynamics can be found in [YCP80,SK84,Hog89,GHS12]. Both blocks are rigid, of mass m i and with semi-diagonal of length R i .…”

“…This turned out to be a modified Melnikov function, whose zeros give rise to the existence of heteroclinic connections for the perturbed system. The general results in [GHS12] were then applied to the case of the rocking block [Hou63,Hog89] and excellent agreement was obtained with the results of [Hog89].…”

“…In [GHS12] these classical results were rigorously extended to a general class of piecewise-smooth differential equations, allowing for a general periodic Hamiltonian perturbation, with no symmetry assumptions. For such systems, the unperturbed system is defined in two domains, separated by a switching manifold Σ, each possessing one hyperbolic critical point either side of Σ.…”

Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate

“…Yagasaki [13] extended a refined version of the subharmonic Melnikov method for piecewise-smooth systems. Granados et al [14] considered the dynamics of a twodimensional piecewise-smooth system by means of the subharmonic Melnikov method.…”