2.
Douglas D. Novaes; Régis Varão
A note on invariant measures for Filippov systems Journal Article
Em: Bulletin des Sciences Mathématiques, vol. 167, pp. 102954, 2021.
@article{NovVar2021,
title = {A note on invariant measures for Filippov systems},
author = {Douglas D. Novaes and Régis Varão},
url = {http://arxiv.org/abs/1706.04212},
doi = {10.1016/j.bulsci.2021.102954},
year = {2021},
date = {2021-01-18},
journal = {Bulletin des Sciences Mathématiques},
volume = {167},
pages = {102954},
abstract = {We are interested in Filippov systems which preserve a probability measure on a compact manifold. We define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a consequence, the volume preserving Filippov systems are the refractive piecewise volume preserving ones. We conjecture that if a Filippov system admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. We prove this conjecture for Lipschitz differential inclusions. Then, in light of our previous results, we analyze the existence of invariant measures for many examples of Filippov systems defined on compact manifolds.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We are interested in Filippov systems which preserve a probability measure on a compact manifold. We define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a consequence, the volume preserving Filippov systems are the refractive piecewise volume preserving ones. We conjecture that if a Filippov system admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. We prove this conjecture for Lipschitz differential inclusions. Then, in light of our previous results, we analyze the existence of invariant measures for many examples of Filippov systems defined on compact manifolds.
1.
Douglas D. Novaes; Gabriel Ponce; Régis Varão
Chaos induced by sliding phenomena in Filippov systems Journal Article
Em: Journal of Dynamics and Differential Equations, pp. 1-15, 2017.
@article{NovPonVar2017,
title = {Chaos induced by sliding phenomena in Filippov systems},
author = {Douglas D. Novaes and Gabriel Ponce and Régis Varão},
url = {http://dx.doi.org/10.1007/s10884-017-9580-8},
doi = {10.1007/s10884-017-9580-8},
year = {2017},
date = {2017-02-16},
journal = {Journal of Dynamics and Differential Equations},
pages = {1-15},
abstract = {In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit G. More specifically we prove that the first return map, defined nearby G, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each natural number m it has infinitely many periodic points with period m.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit G. More specifically we prove that the first return map, defined nearby G, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each natural number m it has infinitely many periodic points with period m.
