Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence
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2026
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| _version_ | 1866909029373050880 |
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| author | Thiam, Abdoulaye |
| author_facet | Thiam, Abdoulaye |
| contents | 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 $f_{\rm sc}(F,ω,R)$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05605 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence Thiam, Abdoulaye Dynamical Systems Mathematical Physics Symplectic Geometry Chaotic Dynamics Primary: 37J40, 70K28, 37C29, Secondary: 37N05, 34A36, 34A60, 70K44 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 $f_{\rm sc}(F,ω,R)$, 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. |
| title | Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence |
| topic | Dynamical Systems Mathematical Physics Symplectic Geometry Chaotic Dynamics Primary: 37J40, 70K28, 37C29, Secondary: 37N05, 34A36, 34A60, 70K44 |
| url | https://arxiv.org/abs/2605.05605 |