Abundance of periodic orbits for typical impulsive flows
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929337590087680 |
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| author | Siqueira, Jaqueline Torres, Maria Joana Varandas, Paulo |
| author_facet | Siqueira, Jaqueline Torres, Maria Joana Varandas, Paulo |
| contents | Impulsive dynamical systems, modeled by a continuous semiflow and an impulse function, may be discontinuous and may have non-intuitive topological properties, as the non-invariance of the non-wandering set or the non-existence of invariant probability measures. In this paper we study dynamical features of impulsive flows parameterized by the space of impulses. We prove that impulsive semiflows determined by a C1-Baire generic impulse are such that the set of hyperbolic periodic orbits is dense in the set of non-wandering points which meet the impulsive region. As a consequence, we provide sufficient conditions for the non-wandering set of a typical impulsive semiflow (except the discontinuity set) to be invariant. Several applications are given concerning impulsive semiflows obtained from billiard, Anosov and geometric Lorenz flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Abundance of periodic orbits for typical impulsive flows Siqueira, Jaqueline Torres, Maria Joana Varandas, Paulo Dynamical Systems Impulsive dynamical systems, modeled by a continuous semiflow and an impulse function, may be discontinuous and may have non-intuitive topological properties, as the non-invariance of the non-wandering set or the non-existence of invariant probability measures. In this paper we study dynamical features of impulsive flows parameterized by the space of impulses. We prove that impulsive semiflows determined by a C1-Baire generic impulse are such that the set of hyperbolic periodic orbits is dense in the set of non-wandering points which meet the impulsive region. As a consequence, we provide sufficient conditions for the non-wandering set of a typical impulsive semiflow (except the discontinuity set) to be invariant. Several applications are given concerning impulsive semiflows obtained from billiard, Anosov and geometric Lorenz flows. |
| title | Abundance of periodic orbits for typical impulsive flows |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2405.04667 |