Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence

TL;DR

This paper provides a comprehensive mathematical analysis of a forced vibro-impact oscillator with Coulomb friction, revealing complex coexistence of Hamiltonian and dissipative dynamics, invariant tori, bifurcations, and stability properties.

Contribution

It introduces a complete mathematical framework for the system, including symplectic structure, existence of KAM tori, bifurcation analysis, and extensions to multi-particle systems, correcting prior universality claims.

Findings

Existence of symmetric T-periodic orbits and saddle-center bifurcation.

Persistence of invariant KAM tori near elliptic orbits.

Destruction of conservative structure by damping, leading to a single attracting basin.

Abstract

The forced vibro-impact oscillator with Amonton-Coulomb friction and elastic walls was shown by Gendelman et al. (2019) to exhibit a coexistence of Hamiltonian stability islands and dissipative attractors in a single phase space. We provide a complete mathematical analysis of this phenomenon. We prove global well-posedness of the associated Filippov flow and construct a global lift to a piecewise smooth Hamiltonian system on a covering manifold. On the maximal forward-invariant non-sticking set, we show that the time-$T$ stroboscopic map is exact symplectic, within the formalism of symplectic dynamics. We derive a closed-form existence equation for symmetric $T$-periodic orbits and establish a parameter-dependent saddle-center bifurcation at $f_{\rm sc}(F,\omega,R)$, correcting a universality claim in prior work. Using Moser's twist theorem, we prove the existence of invariant Cantor families (KAM tori) near elliptic non-sticking periodic orbits, while a Melnikov analysis yields hyperbolic dynamics conjugate to a Bernoulli shift near the associated saddle. We further show that any positive restitution defect or viscous damping destroys the conservative structure: elliptic periodic orbits persist but become asymptotically stable, replacing Hamiltonian islands by a single attracting basin. The approach extends to multi-particle systems with elastic collisions, where a symplectic structure and higher-dimensional KAM tori are obtained. A computer-assisted proof verifies the existence and ellipticity of a non-sticking periodic orbit at a specific parameter point.