Chaos Induced by Sliding Phenomena in Filippov Systems

Abstract

In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit $\Gamma$. More specifically we prove that the first return map, defined nearby $\Gamma$, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each $m\in \mathbb {N}$ it has infinitely many periodic points with period m. We also study the perturbed system and obtain similar results.

1 Introduction

Real world problems have been the main motivation on discontinuous differential systems. They are very useful to model phenomena presenting abrupt switches such as electronic relays, mechanical impact, mitosis of living cells, and Neuronal networks. That is one of the reasons for this area to have such a variety of rich examples [2, 8, 9, 22, 25]. Therefore the further we understand discontinuous differential systems the more one is prepared to analyse real world problems.

When facing a discontinuous differential system, defining a consistent concept of solution is the first natural issue one has to deal with. One important paradigm to tackle this issue is due to Filippov. In his famous book [11] Filippov studied these systems taking advantage of the well developed theory of differential inclusions [1]. Then, for a class of discontinuous vector fields Z, he provided a branch of rules for what would be a local trajectory of $\dot{u}=Z(u)$ nearby a point of discontinuity. For instance, consider

$$\begin{aligned} Z(u)=\left\{ \begin{array}{l} X(u),\quad \text {if}\quad g(u)>0,\\ Y(u),\quad \text {if}\quad g(u)<0, \end{array}\right. \end{aligned}$$

(1)

where $u\in K,$ being K a closure of an open subset of $\mathbb {R}^n,$ X, Y are $\mathcal {C}^r$ vector fields, and $g:K\rightarrow \mathbb {R}$ has 0 as a regular value. The rules stated by Filippov may be applied to establish the notion of local solution of the discontinuous differential system $\dot{u}=Z(u)$ at a point of discontinuity $\xi \in M=g^{-1}(0).$ Nowadays these rules are known as the Filippov’s conventions, and it turns out that for many physical models these conventions are the ones which have physical meaning [9]. Accordingly, discontinuous differential systems rulled by Filippov’s conventions are called Filippov systems. Due to their importance, not only from the mathematical point of view as well as the physical point of view, we shall assume the Fillipov’s conventions throughout the paper. Under this convention the switching manifold M can be generically decomposed in three regions with distinct dynamical behaviours, namely: crossing $M^c$, sliding $M^s$, and escaping $M^e.$ Concisely, the system $\dot{u}=Z(u)$ may admit solutions either side of the discontinuity M that can be joined continuously, forming a solution that crosses $M^c\subset M$. Alternatively, solutions might be found to impinge upon M, after which they join continuously to solutions that slide inside $M^{s,e}=M^s\cup M^e\subset M$. See items (i)–(v) of Sect. 2.1 for the precise definition of the Filippov’s conventions.

Nonlinear systems may present intricate and complex behaviours such as chaotic motions. Roughly speaking, chaos can be understood as the existence of an invariant compact set $\Lambda$ of initial conditions for which their trajectories are transitive and exhibit sensitive dependence on $\Lambda$ [10, 16, 26]. Each phenomenon from the ordinary theory of differential systems finds its analogous in discontinuous differential systems. However Filippov systems admit a richer variety of behaviours. New chaotic modes rising in discontinuous differential systems have been recently investigated. For instance, in [6, 7] it was studied chaotic set-valued trajectories (nondeterministic chaos), and in [3, 4] the Melnikov ideas were applied to determine the existence of chaos in nonautonomous Filippov systems. Here we shall study deterministic chaos in autonomous Filippov systems.

In this article we analyse 3D Filippov systems admitting a sliding Shilnikov orbit $\Gamma$ (see Fig. 1), which is an entity inherent to Filippov systems. It was first studied in [17]. In the classical theory a Shilnikov homoclinic orbit of a smooth vector field is a trajectory connecting a hyperbolic saddle–focus equilibrium to itself, bi-asymptotically. It is well known that a chaotic behaviour may rise when the Shilnikov homoclinic orbit is perturbed [14, 18, 20, 21]. In the Filippov context pseudo-equilibria are special points contained in $M^{s,e}$ that must be distinguished and treated as typical singularities (see Definition 1). These singularities give rise to the definition of the sliding homoclinic orbit, that is a trajectory, in the Filippov sense, connecting a pseudo-equilibrium to itself in an infinity time at least by one side, forward or backward. Particularly a sliding Shilnikov orbit (see Definition 2) is a sliding homoclinic orbit connecting a hyperbolic pseudo saddle–focus $p_0\in M^s$ to it self. This trajectory intersects the boundary $\partial M^s$ of $M^s$ at a point $q_0$ (see Fig. 1).

Fig. 1

The point $p_0\in M^s$ is a hyperbolic pseudo saddle–focus. The trajectory $\Gamma$, called Shilnikov sliding orbit, connects $p_0$ to itself passing through the point $q_0\in \partial M^s$. We note that the flow leaving $q_0$ reaches the point $p_0$ in a finite positive time, and approaches backward to $p_0$, asymptotically

In dynamics one is concerned to get the most possible complexity from a given system, that is why often for systems which exhibit some complexity one is able to find a Bernoulli shift as a factor. For smooth dynamical systems one may benefit from certain geometrical structures (hyperbolicity) which imply the existence of stable and unstable manifolds. That has been the case in many studies, we mention two classical works done by Bowen [5] and Tresser [24]. The systems we shall study in this paper do not benefit from these geometrical structures, so we have to use the properties of the Filippov systems themselves to overcome this lacking of structures (e.g. the well-posedness of the $\eta _*$ function, see Sect. 4).

Let Z be a 3D discontinuous vector field like (1) defined on $K\subset \mathbb {R}^3$. Assume that the Filippov system $\dot{u}=Z(u)$ admits a sliding Shilnikov orbit $\Gamma$ connecting a hyperbolic pseudo saddle–focus $p_0\in M^s$ to it self and intersecting the curve $\partial M^s$ at a point $q_0$. Consider a small neighbourhood I of $q_0$ in $\partial M^s$, and denote by $\Lambda \subset I$ the points which return infinitely often by the forward orbit of the flow to this neighbourhood of $q_0$. If $\pi$ denote the first return map defined on a subset ${\mathcal {U}}$ of I, then $\Lambda$ is the maximal set in ${\mathcal {U}}$ which is $\pi$ invariant (see Sect. 2.2 for more information). As we shall see, this set is non vanishing. The complexity of a flow is interpreted as the complexity of its returning map $\pi |\Lambda$. In this context our main result (see Theorem A in Sect. 3) states that $\pi$ can be as much chaotic as one wishes by using symbolic dynamics.

This paper is organized as follows. On Sect. 2 we present some basic notions on Filippov systems and symbolic dynamics. On Sect. 3 we state our main result (Theorem A) and some of their consequences (Corollaries A, B and C). The main result is proved on Sect. 4. Some final words appear on Sect. 5 as well as a brief discussion of further directions.

2 Basic Notions and Preliminary Results

This section is devoted to present some basic notions needed to state our main result. On Sect. 2.1 we introduce the basic concepts of Filippov systems as well the definition of sliding Shilnikov orbit. Then on Sect. 2.2 we look carefully at the first return map defined nearby the sliding Shilnikov orbit. Finally, on Sect. 2.3, we present the basic notions about symbolic dynamics.

2.1 Filippov System and the Sliding Shilnikov Orbit

We remark that a major part of this section is constituted by a well known theory and may be found in other works (see for instance [11, 12, 17]).

Let K be the closure of an open subset of $\mathbb {R}^n$, and let $X,Y\in \mathcal {C}^r(K,\mathbb {R}^3)$, be $\mathcal {C}^ r$ vector fields defined on $K\subset \mathbb {R}^3$. We denote by $\Omega _g^r(K,\mathbb {R}^3)$ the space of piecewise vector fields

$$\begin{aligned} Z(u)=\left\{ \begin{array}{l} X(u),\quad \text {if}\quad g(u)>0,\\ Y(u),\quad \text {if}\quad g(u)<0, \end{array}\right. \end{aligned}$$

(2)

defined on K, being 0 a regular value of the differentiable function $g:K\rightarrow \mathbb {R}$. As usual, system (2) is denoted by $Z=(X,Y)$ and the surface of discontinuity $g^{-1}(0)$ by M . So $\Omega _g^r(K,\mathbb {R}^3)=\mathcal {C}^r(K,\mathbb {R}^3)\times \mathcal {C}^r(K,\mathbb {R}^3)$ is endowed with the product topology, while $\mathcal {C}^r(K,\mathbb {R}^3)$ is endowed with the $\mathcal {C}^r$ topology. We concisely denote $\Omega _g^r(K,\mathbb {R}^3)$ and $\mathcal {C}^r(K,\mathbb {R}^3)$ only by $\Omega ^r$ and $\mathcal {C}^r$, respectively.

In order to understand the Filippov’s conventions for the discontinuous differential system $\dot{u}=Z(u)$ we need to distinguish some regions on M. The points on M where both vectors fields X and Y simultaneously point outward or inward from M define, respectively, the escaping $M^e$ or sliding $M^s$ regions, and the interior of its complement in M defines the crossing region $M^c$. The complementary of the union of those regions is the set of tangency points between X or Y with M.

The points in $M^c$ satisfy $Xg(\xi )\cdot Yg(\xi ) > 0$, where $Xg(\xi )=\langle \nabla g(\xi ), X(\xi )\rangle$. The points in $M^s$ (resp. $M^e$) satisfy $Xg(\xi )<0$ and $Yg(\xi ) > 0$ (resp. $Xg(\xi )>0$ and $Yg(\xi ) < 0$). Finally, the tangency points of X (resp. Y) satisfy $Xg(\xi )=0$ (resp. $Yg(\xi )=0$).

Now we define the sliding vector field

$$\begin{aligned} \widetilde{Z}(\xi )=\dfrac{Y g(\xi ) X(\xi )-X g(\xi ) Y(\xi )}{Y g(\xi )- Xg(\xi )}. \end{aligned}$$

(3)

Definition 1

A point $\xi ^*\in M^{s,e}$ is called a pseudo-equilibrium of Z if it is a singularity of the sliding vector field, i.e. $\widetilde{Z}(\xi ^*)=0$. When $\xi ^*$ is a hyperbolic singularity of $\widetilde{Z}$, it is called a hyperbolic pseudo-equilibrium. Particularly if $\xi ^*\in M^s$ (resp. $\xi ^*\in M^e$) is an unstable (resp. stable) hyperbolic focus of $\widetilde{Z}$ then we call $\xi ^*$ a hyperbolic pseudo saddle–focus.

Let $\varphi _W$ denotes the flow of a smooth vector field W. The local trajectory $\varphi _Z(t,p)$, $t\in I_p\subset \mathbb {R}$, of $\dot{u}= Z(u)$ passing through a point $p\in \mathbb {R}^3$ is given by the Filippov’s conventions (see [11, 12]). Here $0\in I_p\subset \mathbb {R}$ denotes a interval of definition of $\varphi _Z(t,p)$. Following straightly [12], the Filippov’s conventions is summarized as:

(i):

for $p\in \mathbb {R}^3$ such that $g(p)>0$ (resp. $g(p)<0$) and taking the origin of time at p, the trajectory is defined as $\varphi _Z(t,p)=\varphi _X(t,p)$ (resp. $\varphi _Z(t,p)=\varphi _Y(t,p)$) for $t\in I_p$.

(ii):

for $p\in M^c$ such that $(Xg)(p),(Yg)(p)>0$ and taking the origin of time at p, the trajectory is defined as $\varphi _Z(t,p)=\varphi _Y(t,p)$ for $t\in I_p\cap \{t<0\}$ and $\varphi _Z(t,p)=\varphi _X(t,p)$ for $t\in I_p\cap \{t>0\}$. For the case $(Xg)(p),(Yg)(p)<0$ the definition is the same reversing time.

(iii):

for $p\in M^{s,e}$ and taking the origin of time at p, the trajectory is defined as $\varphi _Z(t,p)=\varphi _{\widetilde{Z}}(t,p)$ for $t\in I_p.$

(iv):

For $p\in \partial M^c\cup \partial M^s\cup \partial M^e$ such that the definitions of trajectories for points in M in both sides of p can be extended to p and coincide, the trajectory through p is this limiting trajectory. These points are called regular tangency points.

(v):

any other point is called singular tangency points and $\varphi _Z(t,p)=p$ for all $t\in \mathbb {R}$.

Examples of regular tangency points are the regular-fold points. A tangency point $p\in M$ is called a visible fold of X (resp. Y) if $X^2g(p)>0$ (resp. $Y^2g(p)<0)$. Analogously, reversing the inequalities, we define an invisible fold. A fold p of X (resp. Y), visible or invisible, such that $Yg(p)\ne 0$ (resp. $Xg(p)\ne 0)$ is called a regular-fold point. The next result provides the dynamics of the sliding vector field near regular-fold points. A proof of that can be find in [23].

Proposition 1

([23]) Given $Z=(X,Y)\in \Omega ^r$ if $p\in \partial M^{e,s}$ is a fold-regular point of Z then the sliding vector field $\widetilde{Z}$ (3) is transverse to $\partial M$ at p.

The above conventions provide the unicity of the trajectories passing through a point. This property plays an important whole in establishing the notion of local equivalence between two Filippov systems (see [12]). However if one consider, for instance, a point $p\in \Sigma ^s\cup \Sigma ^e$, besides the trajectory defined above, there are two other trajectories (of X and Y) which arrive to p in finite time. Therefore in the study of global behavior the matching of these distinct trajectories must be taken into account.

In [17] it has been introduced the concept of sliding Shilnikov orbits, and some of their properties were studied. In what follows we give the definition of this object and two results. The first one is about the co-dimension of the sliding Shilnikov orbit $\Omega ^r$ and the second one is about the existence of sliding periodic orbits nearby a sliding Shilnikov orbit.

Definition 2

(Sliding Shilnikov orbit) Let $Z=(X,Y)$ be a 3D discontinuous vector field having a hyperbolic pseudo saddle–focus $p_0\in M^{s}$ (resp. $p_0\in M^{e})$. We assume that there exists a tangential point $q_0\in \partial M^s$ (resp. $q_0\in \partial M^e)$ which is a visible fold point of the vector field X such that

(j):

the orbit passing through $q_0$ following the sliding vector field $\widetilde{Z}$ converges to $p_0$ backward in time (resp. forward in time);

(jj):

the orbit starting at $q_0$ and following the vector field X spends a time $t_0>0$ (resp. $t_0<0)$ to reach $p_0$.

So through $p_0$ and $q_0$ a sliding loop $\Gamma$ is characterized. We call $\Gamma$ a sliding Shilnikov orbit (see Fig. 1). Accordingly we denote $\Gamma ^+=\Gamma \cap \{u\in K:\, g(u)>0\}$ and $\Gamma ^s=\Gamma \cap M^s$.

Theorem 1

([17]) Assume that $Z_0=(X_0,Y_0)\in \Omega ^r$ (with $r\ge 1)$ has a sliding Shilnikov orbit $\Gamma _0$ and let $W\subset \Omega ^r$ be a small neighbourhood of $Z_0$. Then there exists a $\mathcal {C}^1$ function $g:W\rightarrow \mathbb {R}$ having 0 as a regular value such that $Z\in W$ has a sliding Shilnikov orbit $\Gamma$ if and only if $g(Z)=0$.

Theorem 2

([17]) Assume that $Z_0=(X_0,Y_0)\in \Omega ^r$ (with $r\ge 0)$ has a sliding Shilnikov orbit $\Gamma _0$ and let $Z_{\alpha }=(X_{\alpha },Y_{\alpha })\in \Omega ^r$ be an 1-parameter family of Filippov systems breaking the sliding Shilnikov orbit for $|\alpha |\ne 0$, this family is called a Splitting of $\Gamma _0$ (Fig. 2). Then the following statements hold:

(a):

for $\alpha =0$ every neighbourhood $G\subset \mathbb {R}^3$ of $\Gamma _0$ contains countable infinitely many sliding periodic orbits of $Z_0$;

(b):

for every neighbourhood $G\subset \mathbb {R}^3$ of $\Gamma _0$ there exists $|\alpha _0|\ne 0$ sufficiently small such that G contains a finite number $N_G(\alpha _0)>0$ of sliding periodic orbits of $Z_{\alpha }$. Moreover $N_G(\alpha )\rightarrow \infty$ when $\alpha \rightarrow 0$.

Fig. 2

Representation of a Splitting of $\Gamma _0$, $Z_{\alpha }\in \Omega ^r$

In what follows we present an example of a 2-parameter family of discontinuous piecewise linear differential system $Z_{a,b}$ admitting, for every positive real numbers $\alpha$ and b, a sliding Shilnikov orbit $\Gamma _{\alpha ,\beta }$. This family was studied in [17].

$$\begin{aligned} Z_{a,b}(x,y,z)=\left\{ \begin{array}{ll} X_{a,b}(x,y,z)=\left( \begin{array}{c} -a\\ x-b\\ y-\dfrac{3b^2}{8a} \end{array} \right) &{}\quad \text {if}\quad z>0,\\ Y_{a,b}(x,y,z)=\left( \begin{array}{c} a\\ \dfrac{3a}{b}y+b\\ \dfrac{3b^2}{8a} \end{array} \right)&\quad \text {if}\quad z<0. \end{array}\right. \end{aligned}$$

(4)

The switching manifold for system (4) is given $M=\{z=0\}$, and decomposed as $M=\overline{M^c}\cup \overline{M^s}\cup \overline{M^e}$ being

$$\begin{aligned} \begin{array}{llll} M^c=\left\{ (x,y,0):\,y>\dfrac{3\beta ^2}{8a}\right\} ,&M^s=\left\{ (x,y,0):\,y<\dfrac{3\beta ^2}{8a}\right\}&\text {and}&M^e=\emptyset . \end{array} \end{aligned}$$

The origin $p_0=(0,0,0)$ is a hyperbolic pseudo saddle–focus of system $Z_{a,b}$ (4) in such way that its projection onto M is an unstable hyperbolic focus of the sliding vector field $\widetilde{Z}_{a,b}$ (3) associated with (4). Moreover there exists a sliding Shilnikov orbit $\Gamma _{a,b}$ connecting $p_0=(0,0,0)$ to itself and passing through the fold-regular point $q_0=\big (3b/2,3b^2/(8a),0\big )$.

2.2 The First Return Map

The behaviour of a system close to a sliding Shilnikov orbit can be understood by studying the first return map in a small neighbourhood $I\subset \partial M^s$ of $q_0$, wherever it is defined. In what follows we shall define this map.

Let $Z_0\in \Omega ^r$ be a Filippov system admitting a sliding Shilnikov orbit $\Gamma _0$, and let $Z_{\alpha }$ be a splitting of $\Gamma _0$. For sake of simplicity we shall denote $Z=Z_0$ and $\Gamma =\Gamma _0$.

For $\xi \in M^s$ and $z\in \mathbb {R}^3$, let the functions $\varphi _s(t,\xi )$ and $\varphi _X(t,z)$ denote the solutions of the differential systems induced by $\widetilde{Z}$ and X, respectively, such that $\varphi _s(0,\xi )=\xi$ and $\varphi _X(0,z)=z$.

Take $\gamma _r:=\overline{B_r(q_0)\cap \partial M^s}$. Here $B_r(q_0)\subset M$ is the planar ball with center at $q_0$ and radius r. Of course $\gamma _r$ is a branch of the fold line contained in the boundary of the sliding region $\partial M^s$. From Definition 2, $\varphi _X(t_0,q_0)=p_0\in M^s$, moreover, the intersection between $\Gamma ^+$ and M at $p_0$ is transversal. So taking $r>0$ sufficiently small, we find a function $\tau (\xi )>0$, defined for $\xi \in \gamma _r$, such that $\tau (q_0)=t_0$ and $\varphi _s(\tau (\xi ),\xi )\in M^s$ for every $\xi \in \gamma _r$.

The forward saturation of $\gamma _r$ through the flow of X meets M in a curve $\mu _r$, that is $\mu _r=\{\varphi _X(\tau (\xi ),\xi ):\,\xi \in \gamma _r\}$. So let $\theta :\gamma _r\rightarrow \mu _r$ denote the diffeomorphism $\theta (\xi )=\varphi _s(\tau (\xi ),\xi )$. A diffeomorphism $\theta _{\alpha }:\gamma _r\rightarrow \mu _r^{\alpha }$ can be constructed in a similar way, but now the pseudo saddle-focus is not contained in $\mu _r^{\alpha }$.

Now, Proposition 1 implies that the intersection between $\Gamma ^s$ and $\partial M^s$ at $q_0$ is transversal. So in addition, taking $r>0$ small enough, the backward saturation $S_r$ of $\gamma _r$ through the flow of $\widetilde{Z}$ converges to $p_0$. Therefore

$$\begin{aligned} S_r\cap \mu _r=\bigcup _{i=1}^{\infty } J_i, \end{aligned}$$

where $J_i\cap J_j=\emptyset$ if $i\ne j$ and $J_i\rightarrow \{p_0\}$. For each $i=1,2,\ldots$, we take $I_i=\theta ^{-1}(J_j)\subset \gamma _r$. Clearly $I_i\cap I_j=\emptyset$ if $i\ne j$ and $I_i\rightarrow \{q_0\}$. Set $I=\gamma _r$ and

$$\begin{aligned} \mathcal {U}_r=\bigcup _{i=1}^{\infty } I_i \subset I. \end{aligned}$$

(5)

Therefore the first return map $\pi :\mathcal {U}_r\rightarrow I$ is well defined. Moreover it can be taken as

$$\begin{aligned} \pi (\xi )=\varphi _s\big (\tau _s(\varphi _X(\tau (\xi ),\xi )),\varphi _X(\tau (\xi ),\xi )\big ), \end{aligned}$$

(6)

where, for each $\xi '\in \mu _r{\setminus }\{p_0\}$, $\tau _s(\xi ')>0$ denotes the time such that $\varphi _s(\tau _s(\xi '),\xi ')\in I$.

For $|\alpha |\ne 0$ sufficiently small one could proceed as above to construct a first return map $\pi _{\alpha }:\mathcal {U}^{\alpha }_r\rightarrow I$, with respect to the system $Z_{\alpha }$. We notice that, since the backward saturation of I through the flow of $\widetilde{Z}_{\alpha }$ intersects $\mu _r$ in a finite number $n_{\alpha }<\infty$ of connected components $J_i^{\alpha }$, the set $\mathcal {U}^{\alpha }_r$ will be given by a union of $n_{\alpha }$ intervals $I_i^{\alpha }\subset I$:

$$\begin{aligned} \mathcal {U}_r^{\alpha }=\bigcup _{i=1}^{n_{\alpha }} I_i^{\alpha } \subset I, \end{aligned}$$

where $I_i^{\alpha }=\theta ^{-1}_{\alpha }(J_i^{\alpha })$.

The next result estimates the derivative of the first return map.

Proposition 2

Consider $\mathcal {U}_r$ as defined in (5). There exists $r>0$ sufficiently small such that $|\pi '(\xi )|>1$ for every $\xi \in \mathcal {U}_r$. Consequently, for $|\alpha |\ne 0$ sufficiently small, $|\pi '_{\alpha }(\xi )|>1$ for every $\xi \in \mathcal {U}^{\alpha }_r$.

Proof

For each $R>0$, the focus $p_0\in M^s$ of the sliding vector field $\widetilde{Z}$ is contained in $\mu _{R}$ which is traversal to the flow of $\widetilde{Z}$. So there exists $R_0>0$ such that, for every $0<R\le R_0$, it is well defined a first return map from $\mu _{R}$ into $\mu _{\overline{R}}$, for some big $\overline{R}>0$, which we denote by $\rho :\mu _{R}\rightarrow \mu _{\overline{R}}$. Since $p_0$ is a hyperbolic unstable fixed point of $\rho$, $\rho$ admits a $C^1$ linearization in a neighborhood of $p_0$ (see [13, 19]), that is, there exists a neighborhood $U\subset \mu _{R_0}$ and a $\mathcal {C}^1$ diffeomorphism $H:U\rightarrow U$ such that $\rho (\zeta )=H(\lambda H^{-1}(\zeta ))$, with $|\lambda |>1$. So choose $R>0$ sufficiently small such that $\mu _{R}\subset U$. Therefore if $\rho ^{k-1}(\zeta )\in U$ then $\rho ^k(\zeta )=H(\lambda ^k H^{-1}(\zeta ))$

The backward saturation of $\gamma _{R}$ through the flow of $\widetilde{Z}$ intersects U many times, indeed it converges to $p_0$. So denote by S the first connected component of this intersection which is entirely contained in U. The flow of $\widetilde{Z}$ induces a diffeomorphism $\widetilde{\rho }$ between S and $\gamma _{R}$. Moreover the flow of X induces a diffeomorphism $\rho _X$ between $\gamma _R$ and $\mu _{R}$.

Since $\widetilde{\rho }$ and $\rho _X$ are diffeomorphism, there exists $\widetilde{\alpha }>0$ and $\alpha _X>0$ such that $\widetilde{\alpha }=\min \{|\widetilde{\rho }\,'(\zeta )|:\,\zeta \in S\}$ and $\alpha _X=\min \{|\rho _X'(\xi )|:\,\xi \in \gamma _R\}$.

Now given $k_0\in \mathbb {N}$, there exists a sufficiently small $r\in (0,R)$ such that $\rho ^{k}(\mu _r)\cap S=\emptyset$ for every $0<k<k_0$. In particular, we can assume that $\alpha _X\widetilde{\alpha }|\lambda |^{k_0}>1$.

Finally take $\mathcal {U}_r$ as defined in (5). For $\xi \in \mathcal {U}_r$, let $\overline{k}$ be a positive integer such that $\rho ^{\overline{k}}\big (\rho _X(\xi )\big )\in S$. From the continuity of the map $\rho$, there exists a neighborhood $W\subset \mathcal {U}_r$ of $\xi$ such that $\rho ^{\overline{k}}\big (\rho _X(w)\big )\in S\subset U$ for every $w\in W$. Since $\rho _X(\xi )\in \mu _r$, $\overline{k}\ge k_0$. Therefore, for every $w\in W$, the first return map reads

$$\begin{aligned} \begin{array}{rl} \pi (w)&{}=\widetilde{\rho }\circ \rho ^{\overline{k}}\circ \rho _X(w)\\ &{}=\widetilde{\rho }\circ H(\lambda ^{\overline{k}} H^{-1}(\rho _X(w))). \end{array} \end{aligned}$$

Hence $|\pi '(\xi )|\ge \alpha _X\widetilde{\alpha }|\lambda |^{\overline{k}}\ge \alpha _X\widetilde{\alpha }|\lambda |^{ k_0}>1.$ $\square$

2.3 Basic Facts on Bernoulli Shifts

On what follows the reader may find a good introduction to the subject on [15] and the references therein.

Let $(X,{\mathcal {A}},\mu )$ be a probability space and $f:X \rightarrow X$ be a measurable function. We say that a measurable set $B \subset X$ is f-invariant if

$$\begin{aligned} f^{-1}(B)=B {\text {mod}} 0, \end{aligned}$$

where ${\text {mod}} 0$ means that except a measure zero set both sets are equal. We say that f preserves the measure $\mu$, or that f is $\mu$-invariant, when

$$\begin{aligned} \mu (f^{-1}(B)) = \mu (B) \end{aligned}$$

for every measurable set $B \subset X$.

Given a measurable preserving function $f:X \rightarrow X$ in a probability space $(X,\mu )$, we say that f is ergodic if, and only if, for every f-invariant measurable set $B\subset X$ we have

$$\begin{aligned} \mu (B) = 0 \text { or } \mu (B)=1. \end{aligned}$$

Ergodicity is a very important property in Dynamical Systems and it roughly means that the dynamics cannot be broken in smaller simple dynamics. Hence it actually implies a certain type of chaos for a system with respect to a given measure. Bernoulli shifts have a fairly simple description and still amazingly they are the most chaotic possible examples. We now describe the Bernoulli shifts.

Throughout this paper we denote ${\mathbb {N}}^*:={\mathbb {N}} {\setminus } \{0\}$. Given any natural number $k \in {\mathbb {N}}^*$, we define the space of all sequences of natural numbers between 0 and $k-1$ by

$$\begin{aligned} \Sigma _k = \{0,1,\ldots ,k-1 \}^{{\mathbb {N}}}. \end{aligned}$$

Due to a more intuitive approach in the proof of our main result (see Theorem A of Sect. 3) we will need the set

$$\begin{aligned} \Sigma ^{*}_k = \{1,\ldots ,k \}^{{\mathbb {N}}}, \end{aligned}$$

which we point out it is not a standard notation on symbolic dynamics.

These are a countable product space where each coordinate is a discrete compact space. By Tychonoff’s theorem $\Sigma _k$ (respectively $\Sigma ^*_k$) is compact with the product topology induced by the discrete topology of $\{0,1,\ldots ,k-1\}$ (respectively $\{1,\ldots ,k\}$). A metric in this space, which generates the product topology, is given by

$$\begin{aligned} d:\Sigma _k \times \Sigma _k\rightarrow & {} {\mathbb {R}}\\ d(\alpha , \beta )= & {} \left\{ \begin{array}{c} 0 \; \text { se } \alpha =\beta \\ \left( \frac{1}{2}\right) ^n ,\; n=max\{ a \in {\mathbb {N}}: \alpha (i)=\beta (i), |i|\le a \} \end{array} \right. \end{aligned}$$

Acting on $\Sigma _k$ we have the so called one-sided Bernoulli shift $\sigma : \Sigma _k \rightarrow \Sigma _k$, which simply operates a left-translation on each sequence, that is, given any sequence $(x_n)_{n \in {\mathbb {N}}}$ the image if this sequence is the sequence

$$\begin{aligned} \sigma ((x_n)_{n \in {\mathbb {N}}}) = (x_{n+1})_{n \in {\mathbb {N}}}. \end{aligned}$$

Definition 3

Given $n\in {\mathbb {N}}$ and m values $a_1,a_2,..,a_m \in \{ 0,1,\ldots ,k-1\}$. We denote by $C(n; a_1,a_2,\ldots ,a_m)$ the set defined by

$$\begin{aligned} C(n; a_1,a_2,\ldots ,a_m) = \{ (x_i)_{i\in {\mathbb {N}}} : x_{n+1} = a_1, x_{n+2} = a_2,\ldots ,x_{n+m}=a_m\}. \end{aligned}$$

The sets of this form are called cylinders.

Let ${\mathfrak {C}}$ be the family of all cylinders in $\Sigma _k$. This family generates a $\sigma$-algebra ${\mathcal {C}}$, which will be the standard $\sigma$-algebra to work with on $\Sigma _k$.

To define a measure on $\Sigma _k$ we take any probability vector $p=(p_0,\ldots ,p_{k-1})$ (i.e. $p_i \in [0,1]$ and $\sum _i p_i = 1$). The probability vector p defines, in a trivial way, a measure p on $\{0,1,\ldots ,k-1\}$. Thus, we can take $\mu$ as the product measure $\mu = p^{{\mathbb {N}}}$ on $\Sigma _k$. This measure is characterized by its values on cylinders. Given a cylinder $C(n; a_1,a_2,\ldots ,a_m)$, one can easily see that

$$\begin{aligned} \mu (C(n; a_1,a_2,\ldots ,a_m)) = p_{a_1}\cdot p_{a_2} \cdot \cdots \cdot p_{a_m}. \end{aligned}$$

The measure $\mu$ is called a Bernoulli measure. It is easy to see that $\mu$ is $\sigma$ invariant for any Bernoulli shift $\sigma :\Sigma _k \rightarrow \Sigma _k$. Also, the system $(\sigma ,\mu )$ is ergodic. A measurable automorphism $f:X \rightarrow X$ of a probability space $(X,\mu )$ is called a Bernoulli automorphism if it is isomorphic to a Bernoulli shift $\sigma :\Sigma _k \rightarrow \Sigma _k$ for some $k\in {\mathbb {N}}$. By an isomorphism we mean a bimeasurable function that conjugates the dynamics and takes $\mu$ to a Bernoulli measure. That is a Bernoulli automorphism preserves all ergodic properties of a Bernoulli shift.

The following proposition is a very well-known fact from the theory of Bernoulli shifts (e.g. [15]).

Proposition 3

For any $k \in {\mathbb {N}}$, the Bernoulli shift $\sigma :\Sigma _k \rightarrow \Sigma _k$ has periodic orbits of all periods and the set of transitive points is a residual set.

Along the paper we will work with the spaces $\Sigma _2 \times \Sigma _k^{*}$. For each $k \in {\mathbb {N}}^{*}$, on each $\Sigma _2 \times \Sigma _k^{*}$ we have the shift on two-coordinates

$$\begin{aligned} \sigma ((x_n)_n ,(y_m)_m) = ((x_{n+1})_n, (y_{m+1})_m). \end{aligned}$$

This shift on two coordinates is isomorphic to a standard shift on $\Sigma _{2\cdot k}$, so it is also a Bernoulli automorphism. It is also a direct fact that the two-coordinates shift above is topologically conjugate to the standard shift on $\Sigma _{2\cdot k}$, thus the conclusions of Proposition 3 are also true for the two-coordinates shift.

Let us define

$$\begin{aligned} \Sigma ^{b} := \bigcup _{k \in {\mathbb {N}}^*}\Sigma ^{*}_k=\{ \{ x_i\}_i| \exists L \in {\mathbb {R}} \text { s.t. }|x_i|\le L, x_i \in {\mathbb {N}}^* \; \forall i \}. \end{aligned}$$

We consider the two-coordinates shift

$$\begin{aligned} \sigma : \Sigma _2 \times \Sigma ^{b} \rightarrow \Sigma _2 \times \Sigma ^{b}. \end{aligned}$$

To make notations easier, we will denote by $\sigma _k$ the restriction of $\sigma$ to the space $\Sigma _2 \times \Sigma ^{*}_k$. Hence $\sigma _k: \Sigma _2 \times \Sigma ^{*}_k \rightarrow \Sigma _2 \times \Sigma ^{*}_k.$

The space of sequences $\Sigma _2 \times \Sigma ^{*}_k$ is naturally endowed with the product topology, which is the coarsest topology for which the cylinders are open set. And the topology on $\Sigma _2 \times \Sigma ^b$ is the coarset topology having the cylinders of $\Sigma _2 \times \Sigma ^{*}_k$ as open sets $\forall k \in {\mathbb {N}}^*$. We note that $\sigma _k$ is defined on a compact space while $\sigma$ is not.

One of the most useful invariants on Dynamical Systems and Ergodic Theory is the topological entropy. One may think of topological entropy as a measurement of chaos. Topological entropy has a not so straight definition (which we recomend the reader to take a look [15]) but fortunately to our context it can be associated to the growth of periodic points. Therefore to our purpose we consider the topological entropy as follows. For a compact $\pi$-invariant set $\Omega \subset \Sigma _2 \times \Sigma ^b$ we define the topological entropy of $\pi |\Omega$ as

$$\begin{aligned} h_{\sigma |\Omega }:= \lim _{n \rightarrow \infty }\frac{1}{n} \# Per_n(\sigma |\Omega ), \end{aligned}$$

where $\#Per_n(f)$ means the number of periodic point of period n. It is not difficult to prove that $h_{\sigma _k}=log(2k)$.

3 Statement of Results

Using the same notations as above, our main result is:

Theorem A

Let $Z=(X,Y)\in \Omega ^r$ be a Filippov system (2). Assume that Z admits a sliding Shilnikov orbit $\Gamma$ and let $\pi$ denote the first return map (6) defined on ${\mathcal {U}}_r$ nearby $q_0=\Gamma \cap \partial M^s$. Then, for $r>0$ sufficiently small, there is a set $\Lambda \subset {\mathcal {U}}_r$ such that:

• (a) for each $k \in {\mathbb {N}}$ there exists a $\pi$-invariant cantor set $\Lambda _k \subset \Lambda$ such that $\pi |\Lambda _k$ is conjugate to the shift on $\Sigma _2 \times \Sigma _k^*$, that is

  $$\begin{aligned} h_k\circ \sigma _k = \pi \circ h_k \end{aligned}$$

  where $h_k: \Sigma _2 \times \Sigma _k^* \rightarrow \Lambda _k$ is a homeomorphism. In particular the dynamics on $\Lambda _k$ is transitive, sensitive to initial conditions and has dense periodic points.

• (b) There is a homeomorphism $h: \Sigma _2 \times \Sigma ^b \rightarrow \Lambda := \bigcup _{k}\Lambda _k$ such that h conjugates the dynamics of $\sigma$ and $\pi$ and $\Lambda \cup \{ q_0 \}$ is a compact set. In particular the topological entropy of $\pi$ is infinite.

Hence, given any natural number $m\ge 1$ we can find infinitely many periodic points for the first return map with period m and, consequently, infinity many closed orbits of $\dot{u}=Z(u)$ nearby $\Gamma$. Indeed, given $k\ge 1$, each periodic point of period m for $\sigma _k$ is mapped by $h_k$ in a periodic point of period m for $\pi$, thus varying $k \ge 1$ we obtain infinitely many periodic points of a fixed period m for $\pi$.

We also obtain another two consequences from Theorem A. The first one states that we are able to understand any compact invariant set for the first returning map (or flow) by some dynamics on a symbolic dynamics.

Corollary A

Given a compact set $K \subset {\mathcal {U}}_r$ $\pi$-invariant, then $\pi |K$ is conjugate for some k to a $\sigma _k|\Omega$ where $\Omega \subset \Sigma _2 \times \Sigma _k^*$ is an $\sigma _k$-invariant set.

Proof

Notice that for $\xi \in \Lambda ,$ there exists a positive number $M_\xi <\infty$ such that $n_*(\xi )<M_\xi$ for $* \in \{0,1\}.$ By continuity of $\pi$ there is an neighbourhood $U_{\xi }$ of ${\xi }$ such that $\forall z \in U_{\xi }$ $n_*(z)\le M_{\xi }$. Since K is a compact set take a finite cover of K and consider the maximum of $n_*$ for these finite cover. Let $k_0$ be this maximum. This means that for all points in K if we catalogue its trajectory it has to be given by a sequence in $\Sigma _2 \times \Sigma _{k_0}^*$. Proving the corollary. $\square$

We are able to fully characterize the ergodic properties of the system:

Corollary B

For $r>0$ sufficiently small if $(\pi ,\mu )$ is ergodic, then there exist $k\in {\mathbb {N}}$ such that

• $\mu (\Lambda _k)=1$;

• there exist a measure $\nu$ which is $\sigma _k$-invariant for which $(\pi |\Lambda _k), \mu )$ is isomorphic to $(\sigma _k, \nu )$.

Proof

We know that $\Lambda _k \subset \Lambda _{k+1}$ and $\Lambda = \bigcup _{k=1}^\infty \Lambda _k$ and $\Lambda _k$ is $\pi$-invariant. Hence, by ergodicity $\mu (\Lambda _k)\in \{ 0,1\}$, if $\mu (\Lambda _k)=0$, $\forall k \in {\mathbb {N}}$ then $\mu (\Lambda )=0$ which is an absurd. Therefore, there exist $k_0$ such that $\mu (\Lambda _{k_0})=1$. Since there is a conjugacy from $\pi |\Lambda _{k_0}$ to $\sigma _{k_0}$ the theorem is done with $\nu := (h_{k_0})_*\mu$. $\square$

Corollary C

Let $k \in {\mathbb {N}}$ be fixed. Then for any sufficiently small $\alpha$ the first return map $\pi _\alpha$ is defined in a neighborhood of $\Lambda _k$ and there is a cantor set $\Lambda _{k,\alpha }$ for which $\pi |_{\Lambda _k}$ and $\pi _{\alpha }|_{\Lambda _{k,\alpha }}$ are topologically conjugate. In particular one has that any sufficiently small $\alpha$ the system $Z_{\alpha }$ exhibits infinitely many periodic orbits.

Proof

The first return map $\pi$ is defined on the open (in the line topology) set ${\mathcal {U}}_r$, given k as in the corollary we know that $\Lambda _k \subset {\mathcal {U}}_r$. Since $\Lambda _k$ is compact we know that we may restrict to a smaller neighborhood of $\Lambda _k$ let us call ${\mathcal {V}}_r$ for which for a sufficiently small parameter $\alpha$ the first return map of the perturbation is also defined on ${\mathcal {V}}_r$. Hence, because $\Lambda _k$ is a hyperbolic repeller, it is structurally stable (e.g. [15, Chapter 18]), therefore there exists $\Lambda _{k,\alpha }$ which is $\pi _\alpha$ invariant such that $\pi |_{\Lambda _k}$ and $\pi _{\alpha }|_{ \Lambda _{k,\alpha }}$ are topologically conjugate. In particular, it has infinitely many periodic orbits. $\square$

4 Proof of Theorem A

Consider the Filippov system $\dot{u}=Z(u)=(X,Y)(u)$ given by (2), and denote by $\varphi (t,v)$ its (Filippov) solution such that $\varphi (0,v)=v$. Assume that Z contains a sliding Shilnikov orbit $\Gamma ,$ and let $p_0\in M^s$ and $q_0\in \partial M^s$ be as in Definition 2. We consider a neighbourhood $I\subset \partial M^s$ of $q_0$ for which the first return map $\pi$ is well defined. We assume that I has end points $q_1$ and $q_2$ and we denote $I=[q_1,q_2]$. The forward saturation of I through the flow of X intersects M in a curve J.

Let us call by $\Lambda$ the set of points in I which return infinitely often to I through the forward flow of Z. That is

$$\begin{aligned} \Lambda = \{\xi \in I \; | \; \exists \{t_n\}_{n \in {\mathbb {N}}}, t_n \rightarrow \infty , \varphi (t_n,\xi )\in I \}. \end{aligned}$$

We note that Theorem 2 guarantees that $\Lambda \ne \emptyset$.

Call $I_0 := [q_1,q_0]$, $I_1 := [q_0,q_2]$, $J_0$ and $J_1$ denote the intersection of M with the forward saturation of $I_0$ and $I_1$, respectively. Given a point $\xi \in \Lambda$ we denote by $\eta _*(\xi )$, $* \in \{0,1 \}$ the number of intersections that the forward flow orbit of $\xi$ has with $J_*$ before returning to $\Lambda \subset I$, that is,

$$\begin{aligned} \eta _*(\xi ) := \# \{\varphi (t,\xi ) \cap J_* : 0<t<t_{\xi } \} . \end{aligned}$$

where $t_\xi$ is the first return time of $\xi$ on $\Lambda$. The intersections detected by the function $\eta _*(\xi )$ are occurring on the sliding region $M^s$ of the switching surface M. That means, for a point $\xi \in \partial M^s$ sufficiently close to $q_0$, the flow starting at $\xi$ travels forward in time following the vector field X. After a finite time it reaches transversally the switching surface at a point of $J\subset M^s$ close to $p_0$. Then the flow follows the sliding vector field $\widetilde{Z}$ [see (3)], spiralling outward around $p_0$ until reaching the fold line $\partial M^s$. Since the curve J is transversal to the sliding vector field $\widetilde{Z}$ and contains the pseudo saddle–focus $p_0$, the number $\eta _*(\xi )$ is well defined. Notice that $\eta _*(\xi )$ counts the amount of times that the flow of $\widetilde{Z}$ intersects $J_*$, it could, of course, turn around $p_0$ several times more before reaching $\Lambda$ without intersect J.

We will construct a map

$$\begin{aligned} h_k:\Sigma _2 \times \Sigma _{k}^* \rightarrow \Lambda \end{aligned}$$

that will conjugate the dynamics of $\sigma _k$ with $\pi$ (i.e. $h_k \circ \sigma _k = \pi \circ h_k$), where $\Sigma _k^* = \{ 1,2, \ldots , k \}^{{\mathbb {N}}}$.

Fix a natural number $k>0$ and take a point

$$\begin{aligned} (X,N) = ((x_i)_{i\in {\mathbb {N}}} , (n_i)_{i\in {\mathbb {N}}}) \in \Sigma _2 \times \Sigma _k^* \end{aligned}$$

We will define $h_k((X,N))$ through a limit process.

Define $P_0(X,N)$ as the points which are in $I_{x_0}$, that is $P_0(X,N) = I_{x_0}$. Define $P_1(X,N)$ as the points which are in $I_{x_0}$ and before arriving by the first return maps to $I_{x_1}$ touches $n_0$ times the segment $J_{x_1}$, that is:

$$\begin{aligned} P_1(X,N) = \{ \xi \in P_0(X,N)\; | \; \eta _{x_1}(\xi )=n_0, \pi (\xi )\in I_{x_1} \}. \end{aligned}$$

In general we define

$$\begin{aligned} P_{m+1}(X,N) = \{ \xi \in P_m(X,N) | \eta _{x_{m+1}}(\pi ^m(\xi ))=n_m, \; \pi ^{m+1}(\xi ) \in I_{x_{m+1}}\}. \end{aligned}$$

Now consider the following set

$$\begin{aligned} P(X,N) : = \bigcap _{i\in {\mathbb {N}}} P_i(X,N). \end{aligned}$$

(7)

Notice that $P_i(X,N) \subset P_{i-1}(X,N)$ and each $P_i(X,N)$ is a closed interval. Hence P(X, N) is a point or a non-degenerated interval, we want to rule out the non-degenerate interval case. We start with the following.

Lemma 1

If $P(X,N)\cap P(X', N') \ne \emptyset$, then $P(X,N)= P(X', N')$. In particular $(X,N)=(X',N')$.

Proof

That comes directly from the definition of the sets, because P(X, N) is solely defined by stating what the orbit of a point “behaves”, hence if a point is also on $P(X',N')$ that means the sets are the same. $\square$

Lemma 2

$\pi (P(X,N))$ is of the form $P(X',N')$

Proof

In fact one have $\pi (P(X,N)) = P(\sigma (X,N))$, this comes once again from the definition of the set. $\square$

Lemma 3

If $\pi : \Lambda \rightarrow \Lambda$ is such that $|\pi '(\xi )|>1$ for all $\xi \in \Lambda$, then P(X, N) is a point $\forall (X,N) \in \Sigma _2 \times \Sigma ^*_k$.

Proof

Let us consider l as the length measure. Notice that if $l(P(X,N))>0$, then $l(\pi (P(X,N)))>l(P(X,N))$, but since $\pi ^n(P(X,N)) \subset I$ and $l(I)<\infty$ the family $\{\pi ^n(P(X,N)))\}_{n \in {\mathbb {N}}}$ cannot be pairwise disjoint, otherwise one would have

$$\begin{aligned} \infty> l(I) \ge l(\cup _n \pi ^n(P(X,N)))=\sum _n \pi ^n(P(X,N)) > \sum _n l(P(X,N)) = \infty , \end{aligned}$$

which is an absurd. Therefore, there must exist $n_1$ and $n_2$ such that

$$\begin{aligned} \pi ^{n_1}(P(X,N)) \cap \pi ^{n_2}(P(X,N)) \ne \emptyset , \end{aligned}$$

the above lemmas imply that $\pi ^{n_2-n_1}P(X,N)=P(X,N)$, which cannot happen since $|\pi '|>1$. Hence P(X, N) is a point. $\square$

By Proposition 2 we know that if $q_1$ and $q_2$ are sufficiently close to $q_0$, then $|\pi '|>1$ on I. This means that the functions $h_k$ are well defined by the above lemma as

$$\begin{aligned} h_k:\Sigma _2 \times \Sigma ^*_k\rightarrow & {} \Lambda \\ (X,N)\mapsto & {} P(X,N). \end{aligned}$$

Since the domain of $h_{k+1}$ contain the domain of $h_k$ and the two functions by construction coincide on the domain of $h_k$, the function h

$$\begin{aligned} h:\Sigma _2 \times \Sigma ^b\rightarrow & {} \Lambda \\ (X,N)\mapsto & {} h_k(X,N), \text { if } (X,N) \in \Sigma _2 \times \Sigma ^*_k. \end{aligned}$$

is well defined. Recall that $\pi (P(X,N)) = P(\sigma (X,N))$, which implies $\pi \circ h_k = h_k \circ \sigma _k$ as well as $\pi \circ h = h \circ \sigma$.

Lemma 4

The maps $h_k$ and h are continuous.

Proof

Let $(X,N)\in \Sigma _2\times \Sigma _k^*$ and $\epsilon >0$ be given. From (7) and (8) we know that

$$\begin{aligned} h_k(X,N)= \bigcap _{n \in {\mathbb {N}}} P_n(X,N), \end{aligned}$$

hence consider $n_\epsilon$ such that $P_{n_\epsilon }(X,N) \subset (-\epsilon +h_k(X,N), h_k(X,N) + \epsilon ) \subset I$.

Let ${\mathcal {V}}_{(X,N)}$ be a neighborhood of (X, N) in $\Sigma _2 \times \Sigma _k^*$ given by the cylinder

$$\begin{aligned} {\mathcal {V}}_{(X,N)}:= \{(Y,M) | y_i = x_i, \; m_{i+1} =n_{i+1} \; i,j \in \{0,1, \ldots , n_{\epsilon }\}\}. \end{aligned}$$

And the continuity follows, since

$$\begin{aligned} h_k({\mathcal {V}}_{(X,N)}) \subset (-\epsilon +h_k(X,N), h_k(X,N) + \epsilon ). \end{aligned}$$

The same proof serves for h. $\square$

Lemma 5

The map $h_k$ and h are homeomorphisms onto their image.

Proof

We prove $h_k$ is a homeomorphism onto its image, the case for h is analogous. Notice that $h_k$ is injective by Lemma 1. To see that the inverse is continuous, consider a point $h_k(X,N)$ and a neighborhood ${\mathcal {U}}_{(X,N)}$ of (X, N), therefore $(Y,M)\in {\mathcal {U}}_{(X,N)}$ means that the the first digits of both sequences (X, N) and (Y, M) coincide, using the continuity of the flow we get that for points close enough to $h_k(X,N)$ they must have this predefined trajectory and the continuity follows. $\square$

The above lemmas imply Theorem A, where $\Lambda _k := h_k(\Sigma _2 \times \Sigma _k^*)$. $\square$

5 Final Comments

5.1 Conclusion and Further Directions

In this paper we studied Filippov systems admitting a sliding Shilnikov orbit $\Gamma$, which is a homoclinic connection inherent to Filippov systems. This connection has been firstly studied in [17]. Using the well known theory of Bernoulli shifts, we were able to provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit $\Gamma$, answering then some inquiries made in [17]. As our main result, we established the existence of a set $\Lambda \subset \partial M^s$ such that the restriction to $\Lambda$ of the the first return map $\pi$, defined nearby $\Gamma$, is topologically conjugate to a Bernoulli shift with infinite topological entropy. This ensures $\pi$, consequently the flow, to be as much chaotic as one wishes. In particular, given any natural number $m\ge 1$ one can find infinitely many periodic points of the first return map with period m and, consequently, infinitely many closed orbits nearby $\Gamma$ of the Filippov system.

As it has already been observed in [17], a possible direction for further investigations is to consider higher dimensional vector fields, since in higher dimension it is allowed the existence of many other kinds of sliding homoclinic connections. We feel that the techniques applied in this paper may be straightly followed to obtain similar results in higher dimensions.