A new construction for Melnikov chaos in piecewise-smooth planar systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Calamai, Alessandro Franca, Matteo Pospisil, Michal |
| author_facet | Calamai, Alessandro Franca, Matteo Pospisil, Michal |
| contents | In this paper we consider a piecewise smooth $2$-dimensional system \[ \dot{\vec{x}}=\vec{g} (\vec{x})+\varepsilon\vec{g}(t,\vec{x},\varepsilon) \] where $\varepsilon>0$ is a small parameter and $\vec{f}$ is discontinuous along a curve $Ω^0$. We assume that $\vec{0}$ is a critical point for any $\varepsilon \geq 0$, and that for $\varepsilon=0$ the system admits a trajectory $\vecγ(t)$ homoclinic to $\vec{0}$ and crossing transversely $Ω^0$ in $\vecγ(0)$.
In a previous paper we have shown that, also in an $n$-dimensional setting, the classical Melnikov condition is enough to guarantee the persistence of the homoclinic to perturbations, but more recently we have found an open condition, a geometric obstruction which is not possible in the smooth case, which prevents chaos for $2$-dimensional systems when $\vec{g}$ is periodic in $t$.
In this paper we show that when this obstruction is removed we have chaos as in the smooth case. The proofs involve a new construction of the set $Σ$ from which the chaotic pattern originates.
The results are illustrated by examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15543 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new construction for Melnikov chaos in piecewise-smooth planar systems Calamai, Alessandro Franca, Matteo Pospisil, Michal Dynamical Systems 34A36, 37G20, 37G30, 34C37, 34E10, 34C25, 37C60 In this paper we consider a piecewise smooth $2$-dimensional system \[ \dot{\vec{x}}=\vec{g} (\vec{x})+\varepsilon\vec{g}(t,\vec{x},\varepsilon) \] where $\varepsilon>0$ is a small parameter and $\vec{f}$ is discontinuous along a curve $Ω^0$. We assume that $\vec{0}$ is a critical point for any $\varepsilon \geq 0$, and that for $\varepsilon=0$ the system admits a trajectory $\vecγ(t)$ homoclinic to $\vec{0}$ and crossing transversely $Ω^0$ in $\vecγ(0)$. In a previous paper we have shown that, also in an $n$-dimensional setting, the classical Melnikov condition is enough to guarantee the persistence of the homoclinic to perturbations, but more recently we have found an open condition, a geometric obstruction which is not possible in the smooth case, which prevents chaos for $2$-dimensional systems when $\vec{g}$ is periodic in $t$. In this paper we show that when this obstruction is removed we have chaos as in the smooth case. The proofs involve a new construction of the set $Σ$ from which the chaotic pattern originates. The results are illustrated by examples. |
| title | A new construction for Melnikov chaos in piecewise-smooth planar systems |
| topic | Dynamical Systems 34A36, 37G20, 37G30, 34C37, 34E10, 34C25, 37C60 |
| url | https://arxiv.org/abs/2507.15543 |