Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

TL;DR

This paper studies the behavior of piecewise smooth vector fields near discontinuities, introducing nonlinear regularization and linking sliding dynamics to singular perturbation theory.

Contribution

It extends the analysis of sliding dynamics to nonlinear regularizations, connecting them to singular perturbation problems and critical manifolds.

Findings

Sliding dynamics are determined by reduced singular perturbation systems.

Nonlinear regularization broadens the class of smoothings beyond convex combinations.

The approach generalizes Fenichel theory to discontinuous vector fields.

Abstract

We consider piecewise smooth vector fields (PSVF) defined in open sets $M\subseteq R^n$ with switching manifold being a smooth surface $\Sigma$. The PSVF are given by pairs $X = (X_+, X_-)$, with $X = X_+$ in $\Sigma_+$ and $X = X_-$ in $\Sigma_-$ where $\Sigma _+$ and $\Sigma _-$ are the regions on $M$ separated by $\Sigma.$ A regularization of $X$ is a 1-parameter family of smooth vector fields $X^{\epsilon},\epsilon>0,$ satisfying that $X^{\epsilon}$ converges pointwise to $X$ on $M\setminus\Sigma$, when $\epsilon\rightarrow 0$. Inspired by the Fenichel Theory , the sliding and sewing dynamics on the discontinuity locus $\Sigma$ can be defined as some sort of limit of the dynamics of a nearby smooth regularization $X^{\epsilon}$. While the linear regularization requires that for every $\epsilon>0$ the regularized field $X^{\epsilon}$ is in the convex combination of $X_+ $ and $X_- $ the nonlinear regularization requires only that $X^{\epsilon}$ is in a continuous combination of $X_+ $ and $X_- $. We prove that for both cases, the sliding dynamics on $\Sigma$ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. \end{abstract}

Equations

$$X=\left(\frac{1+\mathrm{sgn}(h)}{2}\right)X_{+}+\left(\frac{1-\mathrm{sgn}(h)}{2}\right)X_{-},$$

$$[X_{-},X_{+}]=\{(1/2+\lambda /2)X_{+}+(1/2-\lambda /2)X_{-}:\lambda \in [-1,1]\}.$$

$$X^{\epsilon }=(1/2+\phi (h/\epsilon )/2)X_{+}+(1/2-\phi (h/\epsilon )/2)X_{-}.$$

$$\tilde{X}(-1,p)=X_{-}(p),\tilde{X}(1,p)=X_{+}(p),\forall p\in M.$$

$$X.h(p)=\nabla h(p).X(p).$$

$$X^{{\lambda }_1}=\left((-3+4x+\sqrt{1+8x})/4,0\right),X^{{\lambda }_2}=\left((-3+4x-\sqrt{1+8x})/4,0\right).$$

$$r\dot{\theta }=\alpha (r,\theta ,p)=-\sin \theta {\alpha }_0(r,\theta ,p),\dot{p}=\beta (r,\theta ,p),$$

$${\theta }^{\prime }=\alpha (r,\theta ,p),p^{\prime }=r\beta (r,\theta ,p)$$

$${\Sigma }_n^{sl}=\{p=(x,0)\in \Sigma :\exists \lambda \in (-1,1),\tilde{X}(\lambda ,x,0)(0,...,0,1)=s_n(\lambda ,x,0)=0\}.$$

$$\tilde{X}(\phi (y/\epsilon ),x,y)=(s_1(\phi (y/\epsilon ),x,y),...,s_n(\phi (y/\epsilon ),x,y)).$$

$$x_i^{\prime }=s_i(\phi (y/\epsilon ),x,y)y^{\prime }=s_n(\phi (y/\epsilon ),x,y).,i=1,...,n-1.$$

$$r\dot{\theta }=-\sin \theta s_n(\psi (\theta ),x,r\cos \theta ),\dot{x_i}=s_i(\psi (\theta ),x,r\cos \theta ),i=1,...,n-1.$$

$${\alpha }_0(r,\theta ,x)=s_n(\psi (\theta ),x,r\cos \theta )$$

$$\beta (r,\theta ,x)=(s_1(\psi (\theta ),x,r\cos \theta ),...,s_{n-1}(\psi (\theta ),x,r\cos \theta ))$$

$$\Gamma :{\mathbb{R}}^{n+1}\to {\mathbb{R}}^{n+1},(x,y,\epsilon )=\Gamma (x,\overline{y},\overline{\epsilon })=(x,\overline{\epsilon }.\overline{y},\overline{\epsilon }).$$

$$\Lambda :[0,+\infty )\times [0,\pi ]\times {\mathbb{R}}^{n-1}\to {\mathbb{R}}^{n+1},(x,y,\epsilon )=\Lambda (r,\theta ,x)=(x,r\cos \theta ,r\sin \theta ).$$

$$C\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$x^{\prime }=-1+2\phi (y/\epsilon {)}^2,y^{\prime }=-\phi (y/\epsilon )+2\phi (y/\epsilon {)}^2-x.$$

$$r\dot{\theta }=-\psi (\theta )+2\psi (\theta {)}^2-x,\dot{x}=-1+2\psi (\theta {)}^2.$$

$$\tilde{X}=\frac{1+\lambda }{2}{X}_{+}(x,y)+\frac{1-\lambda }{2}{X}_{-}(x,y)+(P(\lambda ),Q(\lambda ))$$

$$\tilde{X}=(a/2+c/2+(a/2-c/2)\lambda +P(\lambda ),b/2+d/2+(b/2-d/2)\lambda +Q(\lambda )).$$

$$\Delta =\Delta (\lambda ):=(b/2+d/2)+(b/2-d/2)\lambda +Q(\lambda ).$$

$$r\dot{\theta }$$

$$\dot{x}$$

$$0$$

$$\dot{x}$$

$$\tilde{X}=(1/2-\lambda /2+{\lambda }^2,1-\lambda -2{\lambda }^2+{\lambda }^3).$$

$$r\dot{\theta }=-\sin \theta (1-\lambda -2{\lambda }^2+{\lambda }^3),\dot{x}=1/2-\lambda /2+{\lambda }^2,$$

$$\tilde{X}=(P(\lambda )+x,-\lambda +Q(\lambda )-y).$$

$$r\dot{\theta }=-\sin \theta \left(-\psi (\theta )+Q(\psi (\theta ))-r\cos \theta \right),\dot{x}=P(\psi (\theta ))+x.$$

$$0=-\psi (\theta )+Q(\psi (\theta )),\dot{x}=P(\psi (\theta ))+x.$$

Full text

1. Introduction

Piecewise-smooth vector fields have been investigated at least since 1930. This kind of systems is present in many physical phenomenons, for instance, collisions between rigid bodies, in mechanical friction and impacts, electrical circuits and so ones. They also appear in control theory, economy, cell mitosis, predator-prey and climate problems, etc. See [4, 7] for a general scope of these topics.

A piecewise-smooth vector field (PSVF) is determined by three elements: a set $M\subset\mathbb{R}^{n}$, a switching set $\Sigma\subset M$ and a vector field $X:M\rightarrow\mathbb{R}^{n}$. The simplest case is when $M$ is an open set and $\Sigma=h^{-1}(0)$ for some smooth function $h:M\rightarrow\mathbb{R}$ having [math] as a regular value. So $\Sigma$ divides $M$ in two regions $\Sigma^{+},\Sigma^{-}$ and

[TABLE]

for some smooth vector fields $X^{+}$ and $X^{-}$on $M$.

The regularization method which we consider keeps the phase-portrait unchanged outside $O(\varepsilon)$-neighborhoods of the discontinuity set and it involves transition functions. A regularization of a PSVF $X:M\rightarrow\mathbb{R}^{n}$ is a 1-parameter family of smooth vector fields $X^{\varepsilon}:M\rightarrow\mathbb{R}^{n},\varepsilon>0,$ satisfying that $X^{\varepsilon}$ converges pointwise to $X$ on $M\setminus\Sigma$, when $\varepsilon\rightarrow 0.$

We denote by $[X_{-},X_{+}]$ the convex combination of $X_{-}$ and $X_{+}$:

[TABLE]

We say that a regularization $X^{\varepsilon}:M\rightarrow\mathbb{R}^{n}$ of $X$ is linear if $X^{\varepsilon}(p)\in[X_{-}(p),X_{+}(p)]$, for any $p\in M.$

The main example of linear regularization is the Sotomayor-Teixeira regularization proposed in [19, 22]. It bases on replacing the two adjacent fields by an $\varepsilon$-parametric field built as a linear convex combination of them in an $\varepsilon$-neighborhood of the discontinuity. A transition function $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is used, that is, a function satisfying $\varphi(x)=1$ for $x\geq 1$, $\varphi(x)=-1$ for $x\leq 1$ and $\varphi^{\prime}(x)>0$ if $x\in(-1,1)$, in order to get a family of continuous vector fields that approximates the discontinuous one.

Definition 1.1**.**

The $\varphi$–Sotomayor-Teixeira regularization of $X_{-}$ and $X_{+}$ is the one parameter family

[TABLE]

Note that it is necessary that $X_{+}$ and $X_{-}$ to be defined on both of sides of $\Sigma$.

Different regularizations can lead to different ways of defining the sliding solutions on $\Sigma$. The way chosen will depend on suitability to model the problem. Linear regularizations are not sufficiently general for physical or biological switching processes, such as mechanical chatter or electrical heating, or the energy required to activate the switch at $h=0$. A vast expanse of non-equivalent but no less valid dynamical systems can be unconsidered.

In [9] the authors considers a general model of discontinuous dynamics in terms of nonlinear sliding modes, along with its perturbation by smoothing and its response to errors, and applied it to a heuristic model that captures some key characteristics of dry friction.

A large bibliography can be found about regularization of discontinuous systems. We refer [1, 11, 12, 15, 16, 17, 18, 20] for instance.

Our purpose is to study the sliding dynamics emerging from nonlinear regularization.

A continuous combination of $X_{-}$ and $X_{+}$ is a 1–parameter family of smooth vector fields $\widetilde{X}(\lambda,.)$, with $\lambda\in[-1,1]\,$ and satisfying that

[TABLE]

Definition 1.2**.**

A regularization $X^{\varepsilon}$ is of the kind nonlinear if there exists a continuous combination $\widetilde{X}$ such that $X^{\varepsilon}(p)\in\{\widetilde{X}(\lambda,p),\lambda\in[-1,1]\}$ for any $p\in M.$

In the same way we define the * $\varphi$–nonlinear regularization* of $X_{-}$ and $X_{+}$.

Definition 1.3**.**

A $\varphi$–nonlinear regularization of $X_{-}$ and $X_{+}$ is the $1$–parameter family given by $X^{\varepsilon}(p)=\widetilde{X}(\varphi(h/\varepsilon),p).$

Note that if $h>\varepsilon$ then $\varphi(h/\varepsilon)=1$ and $X^{\varepsilon}=X_{+}$; and if $h<-\varepsilon$ then $\varphi(h/\varepsilon)=1$ and $X^{\varepsilon}=X_{-}$.

The paper is organized as follows. In Section (2) we recall the usual definitions of sewing and sliding regions and generalize them for the nonlinear case. We compare the classical and the new sliding regions. We also define the nonlinear sliding vector field. Finally we state the main theorem which establishes a connection between the nonlinear sliding vector fields and singular perturbation problems. In Section (3) we prove our main result and present an illustrative example. In Sections (4) and (6) we apply our techniques in the description of the nonlinear regularization of normal forms of PSVF in $\mathbb{R}^{2}$ and in $\mathbb{R}^{3}$ .

2. Preliminaries and Main Result

Let $X$ be a PSVF like (1). We use the notation $X(p)=(X_{+}(p),X_{-}(p))$ or simply $X=(X_{+},X_{-})$ and assume that $X$ is bi-valuated on the switching manifold $\Sigma=h^{-1}(0)$. We may not have unicity of trajectories by points $p\in\Sigma$.

Denote

[TABLE]

The points $p\in\Sigma$ are classified as regular if $(X_{-}.h)(p)\neq 0$ and $(X_{+}.h)(p)\neq 0;$ and as singular if $(X_{-}.h)(p)=0$ or $(X_{+}.h)(p)=0.$

There are two kinds of singular points: the equilibrium points of $X_{-}$ or $X_{+}$ on $\Sigma$ and the points where the trajectories of $X_{-}$ or $X_{+}$ are tangent to $\Sigma$.

The regular points are classified as sewing if $(X_{-}.h).(X_{+}.h)>0$ or sliding if $(X_{-}.h).(X_{+}.h)<0.$ The sets of sewing and sliding points are denoted by $\Sigma^{sw}$ and $\Sigma^{sl}$ respectively.

Consider the PSVF given by (1), a continuous combination $\widetilde{X}(\lambda,p)$ of $X_{-}$ and $X_{+}$ and a regular point $p\in\Sigma$.

• (a)

We say that $p$ is a nonlinear sewing point and denote $p\in\Sigma^{sw}_{n}$ if $(\widetilde{X}.h)(\lambda,p)\neq 0$ for all $\lambda\in[-1,1].$

• (b)

We say that $p$ is a nonlinear slidding point and denote $p\in\Sigma^{sl}_{n}$ if there exists $\lambda\in[-1,1],$ such that $(\widetilde{X}.h)(\lambda,p)=0$.

Proposition 2.1**.**

Consider a PSVF given by (1). We have $\Sigma^{sw}_{n}\subseteq\Sigma^{sw}$ and $\Sigma^{sl}\subseteq\Sigma^{sl}_{n}.$

Proof.

Suppose that $p\in\Sigma^{sw}_{n}$. Since $(\widetilde{X}.h)(p)\neq 0$ for all $\lambda\in[-1,1]$ then both $X_{+}(p)$ and $X_{-}(p)$ point toward either $\Sigma^{+}$ or to $\Sigma^{-}$. Thus $(X_{-}.h).(X_{+}.h)>0$ in $p$. It follows that $p\in\Sigma^{sw}$. Suppose now that $p\in\Sigma^{sl}.$ It implies that $X_{+}(p)$ and $X_{-}(p)$ are directed to opposite sides. Thus all continuous path connecting them intersects $\Sigma$. It means that $p\in\Sigma^{sl}_{n}.$ ∎

Consider the PSVF given by (1) and a continuous combination of $X_{-}$ and $X_{+}$. For each $p\in\Sigma^{sl}_{n}$ there exists $\lambda(p)\in[-1,1]$ such that $(\widetilde{X}.h)(p)=0$. We say that $\widetilde{X}(\lambda(p),p)$ is a nonlinear sliding vector field.

Example 1. Let $X=(X_{+},X_{-})$ be a PSVF defined on $\mathbb{R}^{2}$ with $h(x,y)=y$, $X_{+}=(1,1-x)$ and $X_{-}=(1,3-x).$ Consider the continuous combination of $X_{-}$ and $X_{+}$ given by $\widetilde{X}=(-1+2\lambda^{2},-\lambda+2\lambda^{2}-x)$. Solving $-\lambda+2\lambda^{2}-x=0$ in the variable $\lambda$ we determine two possible sliding vector fields

[TABLE]

Note that $x\geq-1/8$ is a necessary condition. Moreover, we also require $|\lambda_{1}|<1$ and $|\lambda_{2}|<1.$ It implies that $\Sigma^{sl}=(1,3)$ and $\Sigma^{sl}_{n}=[-1/8,1)\cup(1,3)$. In $[-1/8,1)$ are defined two nonlinear sliding vector fields ($X^{\lambda_{1}}$ and $X^{\lambda_{2}}$ ) and on $(1,3)$ is defined only $X^{\lambda_{2}}.$

Consider a PSVF as given by (1), a transition function $\varphi$ and the $\varphi$– nonlinear regularization of $X_{-}$ and $X_{+}$. Our main result is the following.

Theorem 2.1**.**

Consider a PSVF as given by (1) with a continuous combination $\widetilde{X}$ of $X_{+}$ and $X_{-}$ and nonlinear sliding region $\Sigma^{sl}_{n}$. There exists a singular perturbation problem

[TABLE]

with $r\geq 0,\theta\in[0,\pi],p\in\Sigma^{sl}_{n}$ and critical manifold $\mathcal{S}$ satisfying the following.

• (a)

For any normally hyperbolic $(\theta,p)\in\mathcal{S}$ there exist homeomorphic neighborhoods $p\in I_{p}\subset\Sigma^{sl}_{n}$ and $\mathcal{S}_{p}\subset\mathcal{S}$ and a sliding vector field $\widetilde{X}(\lambda(p),p)$ defined in $I_{p}$ which is conjugated to the slow flow of (2) on $\mathcal{S}_{p}\subset\mathcal{S}$.

• (b)

For any $p\in int(\Sigma^{sl}_{n})$ consider $\ell=\#\{\theta\in(0,\pi):(\theta,p)\in\mathcal{S}\}.$ There exist $\ell$ choices of sliding vector fields defined in $p$.

• (c)

If all points on $\mathcal{S}$ are normally hyperbolic then there exists only one choice for the sliding vector field in $\Sigma^{sl}_{n}$.

The zero set $\mathcal{S}=\left\{(\theta,p):\alpha_{0}(0,\theta,p)=0\right\}$ is called slow manifold and a point $q=(\theta,p)\in\mathcal{S}$ is normally hyperbolic is $\dfrac{\partial\alpha_{0}}{\partial\theta}(0,q)\neq 0.$

If we consider a time-rescaling, system (2) becomes

[TABLE]

In general we refer to systems (2) and (3) as * slow* and fast systems respectively. If $r=0$ in (2) we have the reduced system and if $r=0$ in (3) we have the layer system. Note that $\mathcal{S}$ is a set of equilibrium points of the layer system which has a vertical flow. Slow and fast systems are equivalent for $\varepsilon>0$, that is, they have the same phase portrait. It means that the phase portrait of system (2) for $\varepsilon\sim 0$ approaches two limit situations: the phase portrait of reduced and layer systems.

Our theorem establishes a connection between a singular perturbation problem and the sliding system in $\Sigma$. We obtain that the possible sliding dynamics are identified with reduced dynamics on the slow manifold of the singular perturbation problem. More specifically, the possible sliding systems are projections ($\mathcal{S}\rightarrow\Sigma$) of the slow flow. This is only an initial step in the analysis of the regularization. The advantage of our approach is that we can use GSP-theory techniques. However, the major obstacle to the overall understanding of the regularized system is the existence of non–normally hyperbolic points in the slow manifold. In this case additional efforts need to be made. See for instance [5, 10, 13].

In [21] the autors study the regularization of an oriented 1-foliation $\mathcal{F}$ on $M\setminus\Sigma$ where $M$ is a smooth manifold and $\Sigma\subset M$ is a closed subset, which can be interpreted as the discontinuity locus of $\mathcal{F}$.

3. Proof of the Main Theorem

In this section we prove our main result and consider an illustrative example.

Proof of the main Theorem..

Take a PSVF $X$ and a continuous combination $\widetilde{X}$ as in the statement. Choose local coordinates $(x,y)=(x_{1},...,x_{n-1},y)$ such that $h(x,y)=y$ and $\widetilde{X}(\lambda,x,y)=(s_{1}(\lambda,x,y),....,s_{n}(\lambda,x,y))$. Thus $\Sigma=\{p=(x,0)\}$ and

[TABLE]

Consider a strictly increasing transition function $\varphi$, that is, it tends to $1$ and $-1$ without actually achieve these values, for instance $\varphi(s)=\tanh(s)$. The $\varphi$–nonlinear regularization is given by

[TABLE]

Its trajectories are determined by the system

[TABLE]

The nonlinear sliding vector field is $\widetilde{X}(\lambda,x,y)$ with $\lambda$ satisfying that $s_{n}(\lambda,x,y)=0.$

Consider the blow up $y=r\cos\theta$ and $\varepsilon=r\sin\theta$ with $r\geq 0$ and $\theta\in[0,\pi].$ Denote $\psi(\theta)=\varphi(\cot\theta)$ which is an injective decreasing function with $\psi(0)=1,\psi(\pi)=-1$. Thus the system (4) become

[TABLE]

Then

[TABLE]

and

[TABLE]

determine the singular perturbation desired.

(a). Note that slow manifold and nonlinear sliding region are defined by the same equation : $s_{n}(\psi(\theta),x,0)=0$ for the slow manifold and $s_{n}(\lambda,x,0)=0$ for the sliding. Thus the local diffeomorphism is immediate because $\psi(\theta)$ is decreasing in $(-1,1)$. Moreover the slow flow is determined by the reduced system $\dot{x_{i}}=s_{i}(\psi(\theta),x,0)$ and the nonlinear sliding vector field by $\dot{x_{i}}=s_{i}(\lambda,x,0)$. Since $\psi(\theta)$ and $\lambda$ have the same expression it concludes (a).

(b). The number of possible choices of nonlinear sliding vector fields is the number of possible choises of $\lambda$ such that $s_{n}(\lambda,x,0)=0$. Since the number of choices of $\lambda$ is exactly the same that the number of $\theta=\theta(x,0)$ defined implicitly by $s_{n}(\psi(\theta),x,0)=0$ , the statement (b) is proved.

(c). The normal hyperbolicity of the points on $\mathcal{S}$ implies that the graphic implicitly defined by $s_{n}(\psi(\theta),x,0)=0$ is the graphic of only one $\theta=\theta(x)$. It means that $s_{n}(\lambda,x,0)=0$ defines uniquely $\lambda=\lambda(x).$ So the statement (c) is proved. ∎

We could apply the direcional blow up in the proof of the main theorem. The direcional blow up consists in the following change of coordinates:

[TABLE]

This blow up was considered by the authors in [2] and in [20]. However, geometrically speaking, it is more convenient to consider the polar blow up coordinates

[TABLE]

The map $\Lambda$ induces the vector field on $[0,+\infty)\times[0,\pi]\times\mathbb{R}^{n-1}$. The parameter value $\varepsilon=0$ is now represented by $r=0$. We observe that the direcional blow up and the polar blow up are essentially the same. In fact, if we consider the map $G:C=[0,+\infty)\times[0,\pi]\times\mathbb{R}^{n-1}\rightarrow\mathbb{R}^{n+1}$ given by $G(r,\theta,\pi)=(\cot\theta,y,r\sin\theta)$ then $\Gamma\circ G=\Lambda.$

[TABLE]

**Example 2. ** Let $X=(X_{+},X_{-})$ as in Example 1 We consider the nonlinear regularization given by $X^{\varepsilon}=\widetilde{X}(\varphi(y/\varepsilon),x,y)$. The trajectories of $X^{\varepsilon}$ satisfies the system

[TABLE]

Consider the blow up $y=r\cos\theta$ and $\varepsilon=r\sin\theta$ with $r\geq 0$ and $\theta\in[0,\pi]$. Thus, denoting $\psi(\theta)=\varphi(\cot\theta)$, the system becomes

[TABLE]

The slow manifold is given by $x=-\psi(\theta)(2\psi(\theta)-1).$ It is easy to see that $x$ is a smooth curve connecting $(\theta,x)=(0,1)$ and $(\theta,x)=(\pi,3)$. Moreover the derivative of $x$ is $x^{\prime}=\psi^{\prime}(4\psi-1)$ and it is zero if $\psi=\dfrac{1}{4}$. In this case $x=-1/8$. The slow flow is given by $x^{\prime}=(-3+4x\pm\sqrt{1+8x})/4$ which is exactly the same expression of the sliding $X^{\lambda_{1}}$ and $X^{\lambda_{2}}.$ See figure (2).

4. Nonlinear regularization of Generic PSVF’s on $\mathbb{R}^{2}$

In this section normal forms of generic PSVF (sewing, sliding, saddle and fold regular) on $\mathbb{R}^{2}$ are discussed. We use singular perturbation techniques to analyze the dynamics of their $\varphi$–nonlinear regularizations. From now on we are considering the continuous combination

[TABLE]

with $P,\,Q$ real polynomials of degree $m$ and $n$ respectively, satisfying the condition $P(\pm 1)=0$ and $Q(\pm 1)=0$.

We recall that $\varphi$ denotes a strictly increasing transition function and $\psi(\theta)=\varphi(\cot\theta)$.

4.1. Sewing

We say that $X_{N}=(X_{+},X_{-})$ is the normal form of the sewing PSVF defined on $\mathbb{R}^{2}$ if $X_{+}(x,y)=(a,b)$, $X_{-}(x,y)=(c,d)$, $a,\,b,\,c,\,d\in\mathbb{R}\setminus\{0\}$, $bd>0$, $h(x,y)=y$ and $\Sigma=h^{-1}(0)$. In this case we have $\Sigma=\Sigma^{sw}$.

Proposition 4.1**.**

Let $X_{N}=(X_{+},X_{-})$ be the normal form of the sewing PSVF defined on $\mathbb{R}^{2}$ and consider the continuous combination

[TABLE]

• (a)

There exists a real continuous function $\Delta(\lambda)$ such that: If $\Delta$ has $k\leq n$ real roots $\lambda_{i}\in[-1,1]$, $i=1,\ldots,\,k$ then $\Sigma=\Sigma^{sl}_{n}$; if $\Delta(\lambda)\neq 0$, for all $\lambda\in[-1,1]$ then $\Sigma^{sw}_{n}=\Sigma$.

• (b)

For the case that $\Sigma^{sl}_{n}=\Sigma$ the singular perturbation problem (2) in the Theorem 2.1 is such that the slow manifold is the union of $k\leq n$ lines $\theta=\theta_{i},\,i=1,\ldots,k$. Moreover the slow flow on $\theta=\theta_{i}$ is determined by $\dot{x}=\operatorname{sgn}((a+c)/2+\psi(\theta_{i})(a-c)/2+P(\psi(\theta_{i}))).$

Proof.

Let $\widetilde{X}$ be the continuous combination of $X$ given by the statement of the theorem. Let $\Delta$ be the real continuous function given by

[TABLE]

According with the definition $p\in\Sigma^{sl}_{n}$ if there exists $\lambda\in[-1,1],$ satisfying $(\widetilde{X}.h)(p)=\Delta=0$. Thus if exists $k\leq n$ real roots of $\Delta$, $\lambda_{i}\in[-1,1]$, $i=1,\ldots,\,k$, the nonlinear sliding region $\Sigma^{sl}_{n}\neq\emptyset$, in fact, $\Sigma^{sl}_{n}=\Sigma$. Otherwise, $\Sigma^{sw}_{n}=\Sigma$ and the statement (a) is proved.

By the Theorem 2.1, there exists a singular perturbation problem, $r\geq 0$, $x\in\Sigma^{sl}_{n}$, given by

[TABLE]

The slow manifold for $\theta\in(0,\pi)$ is determined by the equation $(b/2+d/2)+(b/2-d/2)\psi(\theta)+Q(\psi(\theta))=0$, then it is the union of $k\leq n$ lines $\theta=\theta_{i},\,i=1,\ldots,k$. The slow flow is given by the solutions of the reduced problem represented by

[TABLE]

So, for $r=0$ the dynamics on each connected component of the slow manifold is given by $\operatorname{sgn}(\dot{x})$ and (b) is proved. ∎

Example 3. Consider the PSVF $X_{N}=(X_{+},X_{-})$ with $X_{+}=(1,-1)$, $X_{-}=(2,-1)$ and the continuous combination

[TABLE]

The corresponding slow–fast system is

[TABLE]

with $\lambda=\varphi(\cot\theta).$ Since $1-\lambda-2{\lambda}^{2}+{\lambda}^{3}=0$ has two solutions $\lambda_{1}\in(-1,0)$ and $\lambda_{2}\in(0,1)$ the slow manifold has two components $\theta=\theta_{1}\in(0,\pi/2)$ and $\theta=\theta_{2}\in(\pi/2,\pi).$  Finally $1/2-\lambda/2+{\lambda}^{2}>0$ for $\lambda\in[-1,1]$ implies that the slow flow is equivalent to $\dot{x}=1.$ See figure (3).

4.2. Saddle

We say that $X_{N}=(X_{+},X_{-})$ is the normal form of the saddle PSVF defined on $\mathbb{R}^{2}$ if $X_{+}(x,y)=(x,-1-y)$, $X_{-}(x,y)=(x,1-y)$, $h(x,y)=y$ and $\Sigma=h^{-1}(0)$. In this case we have $\Sigma=\Sigma^{s}$.

Proposition 4.2**.**

Let $X_{N}=(X_{+},X_{-})$ be the normal form of the saddle PSVF and the continuous combination

[TABLE]

• (a)

$\Sigma=\Sigma^{sl}_{n}$* and there exists a real continuous function $\Delta=\Delta(\lambda)$ such that if $\Delta$ has $k\leq n$ real roots $\lambda_{i}\in[-1,1]$, $i=1,\ldots,\,k$ then the singular perturbation problem (2) in the Theorem 2.1 is such that the slow manifold is the union of $k$ lines $\theta=\theta_{i},\,i=1,\ldots,k$.*

• (b)

The slow flow on $\theta=\theta_{i}$ follows the positive direction of $x$-axis if $x>-P(\psi(\theta_{i}))$ and follows the negative direction of $x$-axis if $x<-P(\psi(\theta_{i}))$.

Proof.

Let $\widetilde{X}$ be the continuous combination of $X_{N}$ given by the statement of the proposition. Then $\widetilde{X}=\left(P(\lambda)+x,-\lambda+Q(\lambda)-y\right).$ Using our definition, the nonlinear sliding region $\Sigma_{n}^{sl}$ in $\Sigma$ is given by the solutions of $(\widetilde{X}.h)(p)=0,\,\lambda\in[-1,1]$. So, let us consider the continuous function $\Delta=\Delta(\lambda)=-\lambda+Q(\lambda)$. Since $\Delta(1)=-1$ and $\Delta(-1)=1$, the intermediate value theorem guarantees that $\Sigma=\Sigma^{sl}_{n}$. Now, we consider the polar blow up coordinates given by $y=r\cos\theta$ and $\varepsilon=r\sin\theta$, with $r\geq 0$ and $\theta\in[0,\pi]$. Using these coordinates the parameter value $\varepsilon=0$ is represented by $r=0$ and the blow up induces the vector field given by

[TABLE]

The slow manifold is $\mathcal{S}=\{(\theta_{i},x)\in(0,\pi)\times\mathbb{R}:-\psi(\theta_{i})+Q(\psi(\theta_{i}))=0\}$. The slow flow is given by the solutions of the reduced problem represented by

[TABLE]

Then the reduced flow on $\theta=\theta_{i}$ follows the positive direction of $x$-axis if $x>-P(\psi(\theta_{i}))$ and follows the negative direction of $x$-axis if $x<-P(\psi(\theta_{i}))$. ∎

Corollary 4.1**.**

In the conditions of the Proposition 4.2, let $q$ be a normally hyperbolic singular point of the reduced problem (9) with $r=0$. Then there exists $\varepsilon_{0}>0$ such that for $0<\varepsilon<\varepsilon_{0}$, $X_{\varepsilon}$ has a saddle point or a repelling node point near $q$.

Proof.

We observe that any point $q$ on the slow manifold $\mathcal{S}$ is normally hyperbolic if

[TABLE]

Let us assume that, for every normally hyperbolic $q\in\mathcal{S}$, $(\partial\alpha/\partial\theta)(q,0)$ has $k^{s}$ eigenvalues with negative real part and $k^{u}$ eigenvalues with positive real part for the fast system. Lemma 14 in [2] implies that $X_{\varepsilon}$ has a singular point $q_{\varepsilon}$ with approaches $q$ when $\varepsilon$ is near to zero and $q_{\varepsilon}$ has a $(j^{s}+k^{s})$-dimensional local stable manifold $W_{\varepsilon}^{s}$ and $(j^{u}+k^{u})$-dimensional local unstable manifold $W_{\varepsilon}^{u}$, where $j^{s}$ and $j^{u}$ are the dimensions of the local stable and unstable manifold, respectively. In this case, $j^{u}=1$ and $j^{s}=0$. If $k^{u}=0$ and $k^{s}=1$, then $q_{\varepsilon}$ is a saddle or if $k^{u}=1$ and $k^{s}=0$, then $q_{\varepsilon}$ is a repelling node. ∎

Example 4. Consider the PSVF $X_{N}=(X_{+},X_{-})$ with $X_{+}=(x,-1-y)$, $X_{-}=(x,1-y)$ and the continuous combination

[TABLE]

The corresponding slow–fast system is

[TABLE]

with $\lambda=\varphi(\cot\theta).$ Since ${\lambda}^{2}-\lambda-1=0$ has two solutions $\lambda_{0}\in(-1,0)$ and $\lambda_{1}>1$ the slow manifold has only one component $\theta=\theta_{0}\in(\pi/2,\pi).$  The slow flow $\dot{x}=x+{\lambda_{0}}^{2}-1$ has one repelling equilibrium point $x_{0}\approx 0.6...$ See figure (3).

4.3. Fold

We say that $X_{N}=(X_{+},X_{-})$ is the normal form of the fold PSVF defined on $\mathbb{R}^{2}$ if $X_{+}(x,y)=(1,x)$, $X_{-}(x,y)=(1,1),$ $h(x,y)=y$ and $\Sigma=h^{-1}(0)$. In this case $\Sigma^{s}=\{(x,0)\in\Sigma:x<0\}$.

Proposition 4.3**.**

Let $X_{N}=(X_{1},\,X_{2})$ be the normal form of the fold PSVF defined on $\mathbb{R}^{2}$ and the continuous combination

[TABLE]

• (a)

The singular perturbation problem (2) in the Theorem 2.1 for this case is such that the slow manifold is a graphic $x=x(\theta)$ which is zero for $\theta=0$ and goes to $-\infty$ when $\theta$ goes to $\pi$.

• (b)

The slow flow is determined by $\dot{x}=1+P(\psi(\theta)).$

Proof.

Let $\widetilde{X}$ be the continuous combination of $X$ given by the statement of the proposition. Then $p=(x,0)\in\Sigma^{sl}_{n}$ if there exists $\lambda\in[-1,1]$ such that $(\widetilde{X}.h)(p)=0$, so solving this we obtain

[TABLE]

The singular perturbation problem (2) in the Theorem 2.1 for this case is

[TABLE]

The slow manifold is the graphic of a function which is 0 for $\theta=0$ and goes to $-\infty$ when $\theta$ goes to $\pi$. The reduced flow is determined by $\dot{x}>0$. The fast flow satisfies that that $\dot{\theta}>0$, for $0<\theta<M(x)$ and $\dot{\theta}<0$, for $M(x)<\theta<\pi$, with $M(x)$ given implicitly by $1/2-\psi(\theta)/2+Q(\psi(\theta))+(1/2+\psi(\theta)/2)x=0$. ∎

Example 5. Consider the PSVF $X_{N}=(X_{+},X_{-})$ with $X_{+}=(1,x)$, $X_{-}=(1,1)$ and the continuous combination

[TABLE]

The corresponding slow–fast system is

[TABLE]

with $\lambda=\varphi(\cot\theta).$ The equation $-1/2-\lambda/2+{\lambda}^{2}+(\lambda/2+1/2)x=0$ defines the slow manifold implicitly $x=x(\theta)$ as a smooth function satisfying that $x(0)=0$ and $\lim_{x\rightarrow\pi}x(\theta)=-\infty.$ The slow flow is locally equivalent to $\dot{x}=1.$ Moreover there exists one non normally hyperbolic point on the slow manifold. See figure (4).

5. Nonlinear regularization of Codimension One PSVF on $\mathbb{R}^{2}$

The codimension one singularities of the PSVF are those which generically occur in one parameter families of PSVF. The classification of codimension one local bifurcations was achieved by [8, 14, 23]. In what follows we discuss about the normal forms $X_{N}$ of codimension one PSVF defined on $\mathbb{R}^{2}$. Saddle-node, elliptical fold, hyperbolic fold and parabolic fold are considered. For each one of these normal forms we consider a continuous combination $\widetilde{X}$ given by (5) and analyze the dynamics of their nonlinear regularizations using singular perturbation techniques.

5.1. Saddle-node

We say that $X_{N}=(X_{1},X_{2})$ is the normal form of the saddle-node PSVF defined on $\mathbb{R}^{2}$ if $X_{+}(x,y)=(-x^{2},-1)$, $X_{-}(x,y)=(0,1)$, $h(x,y)=y$ and $\Sigma=h^{-1}(0)$. In this case $\Sigma^{s}=\Sigma$.

Proposition 5.1**.**

Let $X_{N}=(X_{+},\,X_{-})$ be the normal form of the saddle-node PSVF and the continuous combination

[TABLE]

• (a)

There exists a real continuous function $\Delta=\Delta(\lambda)$ such that if $\Delta$ has $k\leq m$ real roots $\lambda_{i}\in[-1,1]$, $i=1,\ldots,\,k$ then $\Sigma=\Sigma^{sl}_{n}$.

• (b)

The singular perturbation problem (2) in the Theorem 2.1 is such that the slow manifold is the union of $k$ lines $\theta=\theta_{i},\,i=1,\ldots,k\leq m$. Moreover the slow flow on each $\theta=\theta_{i}$ is determined by $\dot{x}=\operatorname{sgn}\left(P(\psi(\theta))-(x^{2}/2)(1+\psi(\theta))\right)$.

• (c)

For each $i=1,\ldots,k\leq m$, the number of singular points of the slow flow on $\theta=\theta_{i}$ is at most two whenever $P(\psi(\theta_{i}))\geq 0$.

Proof.

The items (a) and (b) follow using the same ideas as in the proofs of the previous theorems. For the item (c), we consider the singular perturbation problem (2)

[TABLE]

The there are two singular points at $(\theta,x)=\left(\theta_{i},\pm\sqrt{2P(\psi(\theta_{i}))/(1+\psi(\theta_{i}))}\right)$, if $P(\psi(\theta_{i}))\geq 0$, for each $i=1,\ldots,k$. ∎

Example 6. Consider the PSVF $X_{N}=(X_{+},X_{-})$ with $X_{+}=(-x^{2},-1)$, $X_{-}=(0,1)$ and the continuous combination

[TABLE]

The corresponding slow–fast system is

[TABLE]

with $\lambda=\varphi(\cot\theta).$ The equation $-1-\lambda+{\lambda}^{2}=0$ has only one solution $\lambda_{0}\in(-1,0)$ and consequently the slow manifold is $\theta=\theta_{0}$ with $\theta_{0}\in(\pi/2,\pi).$ Since $-1+{\lambda_{0}}^{2}+(-1/2-\lambda_{0}/2)x^{2}<0$ the slow flow is locally equivalent to $\dot{x}=-1.$ See figure (4).

5.2. Elliptical fold

We say that $X_{N}=(X_{+},X_{-})$ is the normal form of the elliptical fold PSVF defined on $\mathbb{R}^{2}$ if $X_{+}(x,y)=(1,-x)$, $X_{-}(x,y)=(-1,-x)$, $h(x,y)=y$ and $\Sigma=h^{-1}(0)$.

Proposition 5.2**.**

Let $X_{N}=(X_{+}\,X_{-})$ be the normal form of the elliptical fold PSVF and the continuous combination

[TABLE]

• (a)

The singular perturbation problem (2) in the Theorem 2.1, for this case, is such that the slow manifold is a graphic $x=Q(\psi(\theta)$ which joins $(0,0)$ with $(\pi,0)$. Moreover, it is possible that the curve $x=Q(\psi(\theta)$ has points $(\theta_{0},0)$ such that $Q(\psi(\theta_{0})=0$, $\theta_{0}\in(0,\pi)$.

• (b)

The reduced flow is determined by $\dot{x}=\operatorname{sgn}(\psi(\theta_{0})+P(\psi(\theta_{0}))$, $\theta_{0}\in(0,\pi)$.

Example 7. Consider the PSVF $X_{N}=(X_{+},X_{-})$ with $X_{+}=(1,-x)$, $X_{-}=(-1,-x)$ and the continuous combination

[TABLE]

The corresponding slow–fast system is

[TABLE]

with $\lambda=\varphi(\cot\theta).$ The equation $-1-x+{\lambda}^{2}=0$ defines implicitly the slow manifold as a smooth curve connecting $\theta=0$ and $\theta=\pi$. The slow flow is determined by $\dot{x}=-1+\lambda+{\lambda}^{2}$. Thus $\dot{x}>0$ if $\lambda\in(\lambda_{0},1)$ and $\dot{x}<0$ if $\lambda\in(-1,\lambda_{0})$, where $\lambda_{0}=\varphi(\cot\theta_{0}),\theta_{0}\in(0,\pi/2).$ See figure (5).

5.3. Hyperbolic fold

We say that $X_{N}=(X_{+},X_{-})$ is the normal form of the hyperbolic fold PSVF defined on $\mathbb{R}^{2}$ if $X_{+}(x,y)=(1,x)$, $X_{-}(x,y)=(1,-2x)$, $h(x,y)=y$ and $\Sigma=h^{-1}(0)$.

Proposition 5.3**.**

Let $X_{N}^{-}$ be the normal form of the hyperbolic fold and the continuous combination

[TABLE]

The singular perturbation problem (2) in the Theorem 2.1 is such that the reduced flow has only two singular points, $(0,0)$ and $(\pi,0)$, and the slow manifold is the curve $\frac{x}{2}(1-3\psi(\theta))=Q(\psi(\theta))$ that joins the two folds and tends to $\pm\infty$ if $\theta$ tens to $\theta_{0}$, where $\psi(\theta_{0})=1/3$. Moreover, the reduced flow is singular.

Example 8. Consider the PSVF $X_{N}=(X_{+},X_{-})$ with $X_{+}=(1,x)$, $X_{-}=(1,-2x)$ and the continuous combination

[TABLE]

The corresponding slow–fast system is

[TABLE]

with $\lambda=\varphi(\cot\theta).$ The equation $-1+{\lambda}^{2}+(-1/2+3\lambda/2)x=0$ defines implicitly the slow manifold as a pair of smooth curves with an asymptote on $\theta_{0}$ with $\varphi(\cot\theta_{0})=1/3.$ The slow flow is determined by $\dot{x}={\lambda}^{2}$ , thus it is locally equivalent to $\dot{x}=1.$ See figure (5).

5.4. Parabolic fold

We say that $X_{N}=(X_{+},X_{-})$ is the normal form of the parabolic fold PSVF defined on $\mathbb{R}^{2}$ if $X_{+}(x,y)=(-1,-x)$, $X_{-}(x,y)=(1,2x)$, $h(x,y)=y$ and $\Sigma=h^{-1}(0)$.

Proposition 5.4**.**

Let $X_{N}=(X_{+},X_{-})$ be the normal form of the parabolic fold PSVF and the continuous combination

[TABLE]

The singular perturbation problem (2) is

[TABLE]

The slow manifold a graphic $x=x(\theta)$ given implicitly by $x(3\psi(\theta)-1)=2Q(\psi(\theta))$, satisfying that $x(0)=x(\pi)=0$ and tends to $\pm\infty$ if $\theta$ tends to $\theta_{0}$ with $\psi(\theta_{0})=1/3$. The reduced flow is given by $\dot{x}=\operatorname{sgn}(-\psi(\theta_{0})+P(\psi(\theta_{0})))$.

Example 9. Consider the PSVF $X_{N}=(X_{+},X_{-})$ with $X_{+}=(-1,-x)$, $X_{-}=(1,2x)$ and the continuous combination

[TABLE]

The corresponding slow–fast system is

[TABLE]

with $\lambda=\varphi(\cot\theta).$ The equation $-1+{\lambda}^{2}+(1/2-3\lambda/2)x=0$ defines implicitly the slow manifold as a smooth curve with an asymptote on $\theta_{0}$ with $\varphi(\cot\theta_{0})=1/3.$ The slow flow is determined by $\dot{x}=-1-\lambda+{\lambda}^{2}$. Thus $\dot{x}<0$ if $\lambda\in(\lambda_{0},1)$ and $\dot{x}>0$ if $\lambda\in(-1,\lambda_{0})$, where $\lambda_{0}=\varphi(\cot\theta_{0}),\theta_{0}\in(\pi/2,\pi).$ See figure (6).

6. Some Examples of PSVF’s on $\mathbb{R}^{3}$

In this section we present some examples of nonlinear regularization of PSVF’s on $\mathbb{R}^{3}$. In the previous section, the continuous combinations of $X_{+}$ and $X_{-}$ had nonlinear terms depending only on $\lambda$. Indeed, in all the examples we considered $\widetilde{X}$ of the kind (5).

Now, in order to obtain some generic situations, the examples of continuous combinations of discontinuous systems in $\mathbb{R}^{3}$ will have the nonlinear terms also depending on $p\in\Sigma$, with $\Sigma=\{(x,y,0)\in\mathbb{R}^{3}\}$.

Example 10. Consider the PSVF $X=(X_{+},X_{-})$ with $X_{+}=(0,0,-1)$, $X_{-}=(0,1,1)$ and the continuous combination

[TABLE]

where $P(\lambda,x,y,z)=x(\lambda^{2}-1)$, $Q(\lambda,x,y,z)=-y/2(\lambda^{2}-1)$ and $R(\lambda,x,y,z)=-\lambda(\lambda^{2}-1)$ are continuous polynomials satisfying the condition $P(\pm 1,x,y,z)=Q(\pm 1,x,y,z)=R(\pm 1,x,y,z)=0$. Thus we have

[TABLE]

The nonlinear sliding region, $\Sigma_{n}^{sl}$, is $\Sigma$ and for any $p\in\Sigma_{n}^{sl}$ the nonlinear sliding vector field is given by

[TABLE]

Now, let us consider the nonlinear regularization given by $X^{\varepsilon}=\widetilde{X}(\varphi(z/\varepsilon),x,y,z)$ and apply Theorem (2.1). We get the singular perturbation problem

[TABLE]

The slow manifold is $\theta=\pi/2$ and the corresponding slow system is determined by $\dot{x}=-x,\quad\dot{y}=1/2+y/2$, and it has a saddle point in $(0,-1)$. See figure (7).

Example 11. Consider the PSVF $X=(X_{+},X_{-})$ with $X_{+}=(1,0,x)$, $X_{-}=(0,0,1)$. The regular points in $\Sigma$ are $\{(x,y,0)\in\mathbb{R}^{3}:\,x\neq 0\}$ (sewing if $x>0$ and sliding if $x<0$). The singular points in $\Sigma$ are $\{(0,y,0)\in\mathbb{R}^{3}\}$.

Let us consider a continuous combination of $X_{+}$ and $X_{-}$ given by

[TABLE]

where

[TABLE]

Solving

[TABLE]

in the variable $\lambda$ we have that the nonlinear sliding region is

[TABLE]

The sliding vector field is given by $X^{\lambda}=(-x-1,y,0).$

Let $X^{\varepsilon}=\widetilde{X}(\varphi(z/\varepsilon),x,y,z)$ be the $\varphi$-nonlinear regularization. For any $p\in\Sigma^{sl}_{n}$, the associated singular perturbation problem is

[TABLE]

with $\psi(\theta)=\varphi(\cot\theta)$. The slow manifold is the curve implicitly defined by

[TABLE]

and the slow flow is determined by $\dot{x}=-x-1,\quad\dot{y}=y$. In this case, the slow flow has a saddle in $(-1,0)$. See figure (8).

Now, take the continuous combination $\widetilde{X}$ of $X_{+}$ and $X_{-}$ with

[TABLE]

By solving

[TABLE]

in the variable $\lambda$, we find that $\Sigma^{sl}_{n}=\Sigma\setminus\{x=0\}$; besides on $\{(x,y,0)\in\Sigma:\,x>0\}$ we have two nonlinear sliding vector field and on $\{(x,y,0)\in\Sigma:\,x<0\}$ only one.

The sliding vector field $X^{\lambda_{1}}$ is

[TABLE]

and the sliding vector field $X^{\lambda_{2}}$ is

[TABLE]

The $\varphi$-nonlinear regularization $X^{\varepsilon}=\widetilde{X}(\varphi(z/\varepsilon),x,y,z)$. For any $p\in\Sigma^{sl}_{n}$, the singular perturbation problem associated is

[TABLE]

where $\psi(\theta)=\varphi(\cot\theta)$. Then the slow manifold is the curve given by the plane $\theta=\pi/2$ and $\psi(\theta)=\frac{1-x}{1+x}$, $x>0$. The slow flow on $\theta=\pi/2$ is given by

[TABLE]

and for $\psi(\theta)=\dfrac{1-x}{1+x}$, the slow flow is

[TABLE]

The slow-fast system obtained blowing up the nonlinear regularization is shown in figure (9).

7. Acknowledgments

The first author is partially supported by CAPES, CNPq, FAPESP, FP7-PEOPLE-2012-IRSES 318999, and PHB 2009-0025-PC. The second author is partially supported by PNPD-CAPES scholarship. The third author is partially supported by FAPESP 2016/11471-2. All authors thank the hospitality of CRM (Centre de Recerca Matemàtica, Barcelona) during our visit in April 2016.