Characterization of Non-Deterministic Chaos in Two-Dimensional Non-Smooth Vector Fields

Abstract

Our context is Filippov systems defined on two-dimensional manifolds having a finite number of tangency points. We prove that topological transitivity is a necessary and sufficient condition for the occurrence of non-deterministic chaos when the Filippov system has non-empty sliding or escaping regions. A fundamental result for continuous flows is the equivalence of topological transitivity and existence of a dense orbit. We prove in our setting that topological transitivity for Filippov systems is indeed equivalent to the existence of a dense Filippov orbit, although, in contrast to the continuous case, we are not able to garantee that the dense orbit implies the existence of a residual set of dense orbits. Finally we prove that, in this context, topological transitivity implies strictly positive topological entropy for the Filippov system. This calculation is made using techniques similar to those from symbolic dynamics.

Figures (6)

• Figure 1: A Filippov system on $\mathbb{S}^2$. The continuous path between the tangency points is a stable sliding region $\Sigma^{\mathrm{ss}}$ (notice the pseudo-equilibrium reached by the two dashed orbits).
• Figure 2: Defining a vector field on the switching manifold $\Sigma$. The vector tangent to $\Sigma$ is a convex combination of the other two vectors.
• Figure 3: An open set $U$ that reaches $\Sigma^{\mathrm{ss}}$ flowing forward and an open set $V$ that reaches $\Sigma^{\mathrm{us}}$ flowing backwards.
• Figure 4: All the $4$ possibilities of connection of $2$ points $q_0$ and $q_1$ on the sliding region $\Sigma^{\mathrm{s}}$. The dashed orbit in the center of the figure represents that each orbit segment on the left can connect to each on the right.
• Figure 5: Given a point $q_0 \in \Sigma^{\mathrm{s}}$ and an open set $U$, we can find a periodic orbit segment staring at $q_0$ that intersects $U$. In the figure we depict the case in which $\Sigma^{\mathrm{ss}} \neq \emptyset$.

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