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On Topological Entropy of Piecewise Smooth Vector Fields 2

Non-smooth vector fields does not have necessarily the property of uniqueness of solution passing through a point and this is responsible to enrich the behavior of the system. Even on the plane non-smooth vector fields can be chaotic, a feature impossible for the smooth or continuous case. We propose a new approach towards a better understanding of chaos for non-smooth vector fields and this is done by studying the entropy of the system. In this work we set the ground for one to begin the study of entropy for non-smooth vector fields. We construct a metric space of all possible trajectories of a non-smooth vector field, where we define a flow inherited by the vector field and then define the topological entropy in this scenario. As a consequence, we are able to obtain some general results of this theory and give some examples of planar non-smooth vector fields with positive (finite and infinite) entropy.


Introduction
A large range of real problems can be modelled using systems of Ordinary Differential Equations (ODEs for short) (or, equivalently, vector fields). Some of these problems are governed by two, or more, systems; for example: in a cancer treatment, there exist a system of ODEs that model the dynamic when the patient is submmited to a severe drug treatment (and the infected cells decrease according to a law of evolution in this process) and there is another system of ODEs governing the dynamic when this treatment is abruptly interrupted in order to provide the immune system recovery ( [5,6]). The same holds is recent protocols of HIV treatment ( [3]). There is also examples in ecology, modelling predator preys populations when predator suddenly change its food preference ( [7]); and electric engineering( [8,17]). These vector fields governed by two or more laws are known as Nonsmooth vector fields (NSVFs) or Piecewise smooth vector fields (PSVF). The general mathematical theory about them is being constructed in recent years. There are some similarities with the smooth vector fields (e.g. the Poincaré-Bendixson Theorem [2]), but there are many behaviours which are typical to NSVFs that are not possible in the smooth case. Describing this has been a main source of research for the mathematicians dedicated to this area. For example: it is possible to construct a NSVF on the plane that is chaotic ( [1]), whereas this feature is impossible for the smooth case.
The main aspect that leads to these differences is the non uniqueness of trajectories passing through a point of the domain of the NSVF.
Although NSVF have already a long history it is a field with still many natural open questions. Even basic questions concerning topological transitivity have not been fully answered. For instance it is known that continuous vector field cannot be transitive on the S 2 which recently has been shown not to be true in the case for NSVF [10]. In a recent work [11] the authors establish some basic questions concerning topological transitivity and the existence of a trajectory which is dense, also that topological transitivity is equivalent to chaotic behavior even in some general two dimensional case. These results illustrate that NSVF has a long way ahead to be better understood. What we do in this work is to apply the basic concepts Ergodic Theory to the NSVF. A first question question one must solve is the invariant measures, this has already been done (see [16] and references therein). Entropy plays a major role in Ergodic Theory as it is a very successful tool, therefore and that is what we are going to explore here.
In Section 2, we provide the main ideas and general definitions concerning NSVFs that we use along this text; for more details, [9] is a good textbook for general theory and some applications, and also [13] and [15] are very comprehensive on theory of NSVF.
In this work, we construct a metric space of all possible trajectories of a NSVF, where we can define a flow inherited by the non-smooth vector field and use it to define the topological entropy of the NSVF. The difficult in doing it is that the classical definition of entropy for a flow relies on uniqueness of trajectory to construct an appropriate transformation of the phase space and then using it to define the entropy. To the best of our knowledge, there is no work related to the topological entropy of NSVFs. As consequence, we are able to obtain planar NSVFs with positive (and even, infinite) entropy. In some sense, this explain the complicated dynamic obtained in NSVFs, even on the plane.

Basic Notions on NSVFs
Let U ⊂ R 2 an open set, and consider a codimension one manifold Σ of R 2 given by Σ = f −1 (0)∩U, where f : R 2 → R is a C r function (with r > 1 large enough for our purposes) having 0 ∈ R as a regular value. We call Σ the switching manifold and it splits U in two open sets: where X and Y are C r vector fields in Σ + and Σ − . In this, we are using the standard convention that for a function to be smooth in a non-open domain D, it means that it can be extended to a smooth function on an open set containing D.
We denote (1) simply as Z = (X, Y ). And let Z r denote the space of all vector fields of this type. We consider this space with the product topology between topologies of X r (Σ + ) and X r (Σ − ).
In order to define rigorously the flow of Z we distinguish whether this point is at Σ ± or Σ. For the first two regions, the local trajectory is defined by X and Y respectively, as usual, but for Σ we divide this into three regions based on the contact between the vector fields X, Y and Σ characterized by the Lie derivative Xf (p) = ∇f (p), X(p) where · , · is the usual inner product: • Crossing Region: These regions are relatively open in Σ and their definitions exclude the tangency points, where Xf (p)·Y f (p) = 0. These points are on the boundary of those regions.
it is a fold point for both X and Y , and it is visible-visible if visible for both (respectively invisible-visible and invisible-invisible).
In addition, a tangency point p is singular if p is a invisible tangency for both X and Y . On the other hand, a tangency point p is regular if it is not singular.
In Σ s,e , the definition of local orbit is given by Filippov convention [12], described in the following. Definition 2.2. Given a point p ∈ Σ s ∪Σ e ⊂ Σ, we define the sliding vector field at p as the convex combination of X(p) and Y (p) that is tangent to Σ. In the planar case, it is given by the expression Moreover, the sliding vector field can be extend to Σ e ∪ Σ s . Now we establish the classical convention on the trajectories of a PSVF: through p ∈ U is defined as follows: (i) For p ∈ Σ + and p ∈ Σ − the trajectory is given by φ Z (t, p) = φ X (t, p) and φ Z (t, p) = φ Y (t, p) respectively, where t ∈ I. (ii) For p ∈ Σ c+ and taking the origin of time at p, the trajectory is defined as φ Z (t, p) = φ Y (t, p) for t ∈ I ∩ {t ≤ 0} and φ Z (t, p) = φ X (t, p) for t ∈ I ∩ {t ≥ 0}. If p ∈ Σ c− the definition is the same reversing time. (iii) For p ∈ Σ e and taking the origin of time at p, the trajectory is For p ∈ Σ s the definition is the same reversing time. (iv) For p a regular tangency point and taking the origin of time at p, the trajectory is defined as

Basic Notions on Entropy
Let f : X → X a continuous map of a compact metric space (X, d) and .
< ε} the open ball centered at x, with radius ε with respect to metric d f n .
, is the minimal cardinality of a (n, ε)−dense set, equivalently, the cardinality of a minimal (n, ε)−dense set.

Definition 3.2. Consider the exponential growth rate
and define the topological entropy of f , denoted h(f ) by: Definition 3.3. Let X is a vector field, and ϕ is the flow defined by this field, we define the time-one map of this field by X 1 (x) = ϕ(x, 1). And the topological entropy of the flow is defined by h(ϕ) = h(X 1 ).
We may also refer to f being topologically semi-conjugate to g by H.
The map H is called topological conjugacy if it is a semi-conjugacy and (iv) H is one to one and onto and has a continuous inverse (so H is a homeomorphism). We also say that f and g are topologically conjugate by H, or simply that f and g are conjugate.
The following Propositions give some useful properties of topological entropy. Their proofs can be found at [14].

General Theory on Entropy for NSVFs
In this section, we adapt Definition 3.3 for the non-smooth case and prove some preliminary results of this theory. As far as the authors know, this is the first work to cover these aspects of non-smooth vector fields.
Let Z = (X, Y ) a NSVF defined over a compact 2-dimensional surface M and Ω = {γ : global trajectory of Z}.
is bounded for all t ∈ R, thus the series above converges for any γ 1 , γ 2 .
If ρ(γ 1 , γ 2 ) = 0 then And, finally, for the triangle inequality part it is enough to notice that

The Propositions 4.2, 4.3 give some important insights about how this metric behaves.
Proposition 4.2. Given t 0 , ε 0 , then there exists δ > 0 such that, for all Moreover, we know that γ 1 , γ 2 are differentiable by parts. Hence it makes sense to talk about their derivatives (which could differ if taken by left or right), and we know that because the vector fields that generate the Filippov system are all bounded, these derivatives are all uniformly bounded. Now, Which implies that β δ − α δ goes to zero as δ → 0, and then the derivative of this function would explode at some point of J, but it can not happen, since the derivatives are bounded. Proposition 4.3. Given t 0 > 0, and (γ n ) ⊂ Ω such that γ n → γ ∈ Ω, then for all |t| < t 0 , γ n (t) → γ(t).
In this space of all orbits, we can define a natural flow determined by the NSVF Z as F : . And then we have the time-one map, F 1 (γ) = γ(. + 1).
Using the relation above, we obtain: Hence F 1 is continuous. The proof of continuity of the inverse is analogous. Note that, if we consider Z = (X, X) then Z is a smooth vector field. In this case, the value given by Definition 4.2 should coincide with the value obtained in Definition 3.3. The next proposition assures it.
Moreover, for all x ∈ M and all t ∈ R, f (  Proof. Let Ω represent the set of all trajectories of Z over M . Since K ⊂ M is closed and Z−invariant, the set Ω K ⊂ Ω, of all trajectories through points of K is invariant for F 1 . Moreover, given any sequence (γ n ) ⊂ Ω K , such that γ n → γ, we have that, γ ∈ Ω K , for if it is not, then γ ⊂ M \ K, which is open, then for every point x of γ, there is ε x such that B(x, ε x ) ⊂ M \ K, which implies γ cannot be limit of trajectories of Ω K . Thus Ω K ⊂ Ω is closed. Now, Ω K ⊂ Ω is closed and F 1 −invariant, then h(Z K ) = h(F 1 | Ω K ) h(F 1 ) = h(Z), by Proposition 3.2.

Finite Positive Topological Entropy
In this section we prove results about the existence of NSVFs of positive topological entropy, but finite.  0) is an invisible-invisible two-fold, and there is a closed trajectory that goes through the origin (0, 0). Let α = 3, and consider six rays from the origin (one at each multiple of π 3 ). In that way, the plane is divided into six different regions. Number each region from 1 to 6, counter clockwise. We define X in region 1, and Y in region 6. Now, define a vector field in each one of these regions, such that in regions 3 and 5 we have a phase portrait that is the one of X rotated, and in regions 2 and 4 the phase portrait is symmetric to the one of Y (see Figure 2). Now there are three closed arcs that goes through the origin. That defines a NSVF with six different regions such that, apart from three invisible two-folds, and the origin, every other point is either regular or crossing, and every trajectory that does not goes through the origin is closed. The union of the curves I 1 , I 2 , I 3 highlighted in Figure 2 is invariant to the NSVF and a trajectory in it is the amalgamation of these three different arcs in every possible combination.
For any γ, ψ ∈ Ω * and for all i ∈ Z, we have that γ(i) = ψ(i) = 0, and the integral i+1 i |γ 1 (t) − γ 2 (t)|dt vanishes when the same arc is used at the same time in the construction of γ and ψ or it equals a constant µ, if different arcs occur (because of the symmetry of phase portrait this value µ is the same for any pair I j , I k with j = k).
Since there are three options I j , it is possible to construct ψ * that does not coincide with either ψ 1 or ψ 2 at any given time. So, ρ(ψ * , ψ 1 ) = ρ(ψ * , ψ 2 ) = 3µ and, therefore, Thus S ρ (F 1 , 2µ, 1) = 3. Now fix γ 11 a trajectory that begins with the concatenation of I 1 twice in a row; γ 12 for one starting with I 1 and followed by I 2 and so on. In general, fix γ jk for a trajectory beginning with the concatenation of I j followed by I k , for j, k = 1, 2, 3.
Consider the figure with three arcs (Figure 2), in demonstration of Theorem 5.1. If we consider Z the restriction of Z to I 1 ∪ I 2 (the union of only two arcs), then h( Z) = log 2.
Corollary 5.1. More generally, given Z with 2α regions, as in proof of Theorem 5.1 (with α closed arcs), we can consider Z as the restriction of Z to any number β < α of closed arcs, and then h( Z) = log β.

Infinite Topological Entropy
In this section, we prove the existence of a NSVF of infinite topological entropy with an example, and next we give sufficient conditions for a NSVF Z to have h(Z) = ∞. Example 6.1. Consider the NSVF: It has a compact invariant set K = {(x, y) ∈ R 2 : −1 x 1 and x 4 /2 − x 2 /2 y 1 − x 2 }. Furthermore, K is a chaotic non-trivial minimal set for Z (see [2]). As before, we will consider Ω = {γ global trajectory of Z| Λ } the set of all possible trajectories contained in the set Λ (see Figure 3).
We will show that h(Z) = ∞.
Definition 6.1. We say that a trajectory γ escapes Σ e at a time s 0 if γ(s 0 ) ∈ Σ e and there is s 1 > s 0 , such that γ(s) / ∈ Σ e , for all s 0 < s < s 1 . We may also add that γ escapes to Σ + or Σ − , depending on which of these regions the points γ(s) are located in. Alternatively, we may say that γ(s 0 ) is a escape point of γ.
Since every trajectory of Ω intersects Σ e in positive time, define the function τ : Ω → R + , τ (γ) = τ γ , as the minimum t > 0, for which γ escapes Σ e , in other words, the time τ γ is the "next escape time" for γ (see Figure 4).  Now, consider a subset R ⊂ K, bounded by two distinct simple closed trajectories, and let S = {γ ∈ Ω | ∀t, γ(t) ∈ R and γ escapes to Σ − at t = 0} that is, S is the set of trajectories entirely contained in R that escapes at s 0 = 0 (see Figure 5). We may restrict R in order to obtain some positive integer k > 1 for which we have k − 1 < τ (γ) < k for every γ ∈ S. And define the map: where P (γ)( · ) = γ( · + τ γ ).
In the following, we prove some properties of τ and P , that will be useful next.
Proposition 6.2. The map P : S → S as defined above is continuous.
Let ε > 0 then there is t 0 > 0 such that Let ε 0 > 0 such that Now, by Propositions 4.3 and 6.1, and considering the fact that γ is piecewise differentiable with bounded derivatives, we have that, there is N large enough such that if n > N : Hence, if n > N , ρ(P (γ n ), P (γ)) < ε.
Proposition 6.3. The map P has infinite topological entropy.
Proof. In order to prove this, we show that the following diagram commutes: where k > 1 is the integer given above, such that k − 1 < τ γ < k, for all γ ∈ S.
Given γ ∈ S, and t, on one side we have and on the other side Then it is enough to show that, for any γ ∈ S: In fact, on the one hand, τ P (γ) + τ γ is exactly the time at which γ escapes Σ e for the second time. And since k − 1 < τ γ < k, we have k 2k − 2 < τ P (γ) + τ γ < 2k.
That is, On the other hand: From the above equation and (6), we get (5). Since P is onto and continuous it works as a semi-conjugacy between F k 1 and P , then by Proposition 3.1, h(F 1 ) h(P ) = ∞. Thus h(Z) = ∞.
In the following theorem, we give sufficient conditions for a NSVF Z to have infinity topological entropy, but first one lemma that will be useful to complete the theorem. Lemma 6.1. Let Z and Z two NSVF defined over a compact invariant set Proof. Let γ be a trajectory of Z. It is not difficult to see that γ(t) = γ( 1 c t) is trajectory of Z. If Ω and Ω represent the spaces of trajectories of Z and Z, respectively, we can define a continuous bijection between them H : Ω → Ω, simply by H(γ)(t) = γ(t) = γ( 1 c t). Consider the following diagram: The diagram above commutes. In fact, for any t ∈ R, we have: On the other hand: Theorem 6.2. Let Z = (X, Y ) be a NSVF defined over a compact invariant set K ⊂ R 2 , with Σ e = ∅. Assume there exists a closed interval J ⊂ Σ e , such that for all x ∈ J, there is a trajectory γ x that escapes through x to Σ + (or Σ − ) and γ x intersects J in finite time (see Figure 6). Then h(Z) = ∞. Figure 6. Every point of J (in blue) has a trajectory that escapes to Σ + and returns to J.
Proof. From hypothesis, we get that for every x, y ∈ J there is an arc of trajectory escaping at x and going to y. Consider S to be the set of every possible trajectory γ that is formed by a concatenation of these arcs and γ(0) ∈ J is a escape point. On this set S, define the functions τ and P in the same way as defined before.
Note that Propositions 6.1 to 6.3 can be repeated word by word, since they do not depend on the vector field.
In the example, we required an integer k > 1 such that k − 1 < τ γ < k for all γ ∈ S. If we can restrict to J 0 ⊂ J in which this inequality holds, then the Theorem 6.1 can also be repeated which concludes the proof.
If it is not the case, we must do some considerations first: since J is closed, τ has a minimum value M > 0, in this case less than 1, so let c > 0 be the smallest integer such that 1 < c · M and define Z = 1 c Z. Each one of the hypothesis are still true for Z, and now we can find an integer k satisfying that inequality. Hence h( Z) = ∞ and the Lemma 6.1 concludes the proof.

Examples
Example 7.1. Consider the NSVF Z = (X, Y ) where X = 1, x 2 − 4x 3 and Y = −1, x 2 − 4x 3 and Σ = {y = 0}, we have that h(Z) = log 2. In fact, there is a set Λ that is Z−invariant, that is the union of two symmetric integral curves of X and Y (see Figure 7) in which we can distinguish two arcs I 1 , I 2 and apply the same calculations from the proof of Theorem 5.1. Note that this example presents a polynomial NSVF that satisfies Theorem 5.1 for α = 2 with only two regions, using a visible two-fold. Example 7.2. In [4], the authors present in details a NSVF Z tangent to S 2 , without equilibrium points that has a non-trivial minimal set diffeomorphic to that presented in Example 6.1 (see Figure 8). Thus, by Theorem 4.1, h(Z) = ∞. Example 7.3. Let Z be a NSVF defined over S 2 , with discontinuity on S 1 , and Σ + , Σ − are the northern and southern hemispheres. Consider X + being a vector field without singularities, and X − a vector field with a visible attracting node. Then it has a escaping region in some segment on S 1 and  all global orbits tend to p in positive time. And in negative time it reaches the escaping region (see Figures 9 and 10). Now, for all ε > 0, there exists m ∈ N such that given any γ ∈ Ω, for any n > m, |F n 1 (γ)(0) − p| < ε, because every trajectory tends to p. This implies that ρ(F n 1 (γ), γ p ) < ε. Then the ε-capacity, S ρ (F 1 , ε, n) does not grow with n, then lim sup 1 n log S ρ (F 1 , ε, n) = 0. Hence h(Z) = 0.