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

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

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 ...

Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies

We study perturbations of a class of analytic two-dimensional autonomous systems with perturbations depending periodically on time; for instance one can imagine a periodically driven or forced system with one degree of freedom. In the first part of the paper, we revisit a problem considered by Chow and Hale on the existence of subharmonic solutions. In the analytic setting, under more general (...

A Method of Piecewise-Smooth Numerical Branching

Many pulses heteroclinic orbits with a Melnikov method and chaotic dynamics of a parametrically and externally excited thin plateMinghui

The multi-pulse heteroclinic orbits with a Melnikov method and chaotic dynamics in a parametrically and externally excited thin plate are studied in this paper for the first time. The thin plate is subjected to transversal and in-plane excitations, simultaneously. The formulas of the thin plate are derived from the von Kármán equation and Galerkin’s method. The method of multiple scales is used...

A verified method for solving piecewise smooth initial value problems

In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is r...