The Regularized Visible Fold Revisited

Abstract

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon \rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $\epsilon \rightarrow 0$, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.

Appendix A: Proof of Theorem 1.3(a)

The analysis in the chart ${\bar{y}}=-1$ (see (1.16)) plays little role and is similar to how we deal with ${\bar{y}}=1$. The details are therefore omitted. In the following we therefore consider ${\bar{\epsilon }}=1$ and ${\bar{y}}=1$ only.

1.1 A.1: Chart ${\bar{\epsilon }}=1$

In this chart, we obtain the following equations

$$\begin{aligned} {\dot{x}}&=\epsilon \phi (y_2,\epsilon )(1+f(x,\epsilon y_2)), \nonumber \\ {\dot{y}}_2&= \phi (y_2,\epsilon )(2x+\epsilon y_2 g(x,\epsilon y_2)) + 1-\phi (y_2,\epsilon ), \end{aligned}$$

(A.1)

and ${\dot{r}}_2=0$, using (1.11) and (1.1) in (1.12). This is a slow-fast system with x slow and $y_2$ fast. Setting $\epsilon =0$ gives the layer problem where x is a parameter and

$$\begin{aligned} {\dot{y}}_2&=\phi (y_2,0)2x+1-\phi (y_2,0), \end{aligned}$$

and hence a normally hyperbolic and attracting, but noncompact, critical manifold:

$$\begin{aligned} S =\{(x,y_2)\in U_\xi \vert \phi (y_2,0) =\frac{1}{1-2x},x<0\}. \end{aligned}$$

(A.2)

Fixing $J=[-\xi ,-\nu ]$, we can apply Fenichel’s theory to $S\cap \{x\in J\}$ and conclude the existence of the invariant manifold $S_\epsilon$ in Theorem 1.3(a). The fact that $Z\vert _{S_\epsilon }$ is a regular perturbation of the Filippov vector-field is standard and follows easily from the fact that the reduced problem on S:

$$\begin{aligned} x'&=\phi (y_2,0)(1+f(x,0)) = \frac{1}{1-2x}(1+f(x,0)), \end{aligned}$$

coincides with Filippov.

1.2 A.2: Chart ${\bar{y}}=1$

Upon inserting (1.16) into (1.12), using (1.1) and (1.11), we obtain the following equations

$$\begin{aligned} {\dot{r}}_1&=-r_1 F(r_1,x,\epsilon _1),\\ {\dot{x}}&= r_1 (1-\epsilon _1^k \phi _+(r_1,\epsilon _1))(1+f(x,r_1),\\ {\dot{\epsilon }}_1&=\epsilon _1 F(r_1,x,\epsilon _1), \end{aligned}$$

after dividing the righ-hand side by $\epsilon _1$ (as promised in our description of the blowup approach, see Sect. 1.4). In these equations, we have introduced the following function where

$$\begin{aligned} F(r_1,x,\epsilon _1) =- (1-\epsilon _1^k\phi _+(r_1,\epsilon _1))(2x+r_1 g(x,r_1))-\epsilon _1^k \phi _+(r_1,\epsilon _1). \end{aligned}$$

Remark A.1

Notice that within the invariant subset defined by $\epsilon _1=0$, we have $F_1(r_1,x,0)=-(2x+r_1g(x,r_1))$ and hence

$$\begin{aligned} {\dot{x}}&=y(1+f(x,y)),\\ {\dot{y}}&= -y (2x+yg(x,y)), \end{aligned}$$

using that $r_1=y$. But this is just $yZ_+$ (see (1.11)).

Now, focus first on $x\in J$. Then for $r_1\ge 0$ and $\epsilon _1\ge 0$ but sufficiently small we have $F>0$, and hence, the system is topological equivalent with the following version

$$\begin{aligned} {\dot{r}}_1&=-r_1, \nonumber \\ {\dot{x}}&= r_1 (1-\epsilon _1^k \phi _+(r_1,\epsilon _1))\frac{(1+f(x,r_1)}{F(r_1,x,\epsilon _1)},\nonumber \\ {\dot{\epsilon }}_1&=\epsilon _1, \end{aligned}$$

(A.3)

The set $r_1=\epsilon _1=0$ is therefore a line of equilibria having stable and unstable manifolds contained within $\epsilon _1=0$ and $r_1=0$, respectively. We can straighten out the individual unstable manifolds of points on $x\in J,\,r_1=\epsilon _1=0$ by a transformation of the form ${\tilde{x}}\mapsto x=m({\tilde{x}},r_1)$, $r_1\in [0,\xi ]$ with m smooth, with $m'_{{\tilde{x}}}>0$. Applying this transformation to (A.3) gives

$$\begin{aligned} {\dot{r}}_1&=-r_1,\\ {\dot{x}}&= \epsilon G(r_1,x,\epsilon _1),\\ {\dot{\epsilon }}_1&=\epsilon _1, \end{aligned}$$

dropping the tilde on x. Recall here that $\epsilon =r_1\epsilon _1$. Therefore if we consider an initial condition with $r_1(0)=\delta >0$ small, then

$$\begin{aligned} r_1(t)&=e^{-t}\delta , \nonumber \\ x(t)&=x(0)+{\mathcal {O}}(\epsilon t),\nonumber \\ \epsilon _1(t)&= e^{t}\epsilon _1(0). \end{aligned}$$

(A.4)

Now, we wish to extend the stable foliation of $S_\epsilon$ by the backward flow. For this, let $x\in J$, after possibly decreasing $\nu >0$ and $\xi$, and consider the leaf ${\mathcal {F}}_{x,\epsilon }$ of the Fenichel foliation of $S_\epsilon$. We therefore flow this set forward $t={\mathcal {O}}(\log \epsilon ^{-1})$, which is the time it takes for $r_1$ to go from ${\mathcal {O}}(1)$ to $\mathcal O(\epsilon )$. This gives a new x, $x'=x\cdot t$, say, and a new leaf ${\mathcal {F}}_{x',\epsilon }$. Notice $\phi _t \left( \mathcal F_{x,\epsilon }\right) \subset {\mathcal {F}}_{x',\epsilon }$, and hence, we extend ${\mathcal {F}}_{x,\epsilon }$ by flowing $\mathcal F_{x',\epsilon }$ backwards by time t. (In general, $\mathcal F_{x',\epsilon }$ will not be fully covered by the chart ${\bar{y}}=1$ and therefore we will have to work in separate charts.) We do this by using (A.4), which produces the extended leafs as images of Lipschitz mappings.