Linearization and Perturbations of Piecewise Smooth Vector Fields with a Boundary Equilibrium

Abstract

In this paper we study the linearization and perturbations of planar piecewise smooth vector fields that consist of two smooth vector fields separated by the straight line $y=0$ and sharing the origin as a non-degenerate equilibrium. In the sense of $\Sigma$-equivalence, we provide a sufficient condition for piecewise linearization near the origin, generalizing the classical linearization theorem to piecewise smooth vector fields. This condition is hard to be weakened because there exist vector fields that are not piecewise linearizable when this condition is not satisfied. Then a necessary and sufficient condition for local $\Sigma$-structural stability is established when the origin is still an equilibrium of both smooth vector fields under perturbations. In the opposition to this case, we prove that for any piecewise smooth vector field studied in this paper there are perturbations with crossing limit cycles bifurcating from the origin. Moreover, besides the fold-fold type given in previous publications we find some new types of singularities, such as types of center-center, center-saddle and saddle-saddle, to birth any finitely or infinitely many crossing limit cycles.

Appendix

Here we give the Proof of Lemma 3.2 by using the method introduced in [4, 5, 17].

Proof of item (1)

Because $\Omega _{ff}\subset \Omega _1\subset \Omega _0$, $Z\in \Omega _{ff}$ satisfies (1.3) by the definition of $\Omega _0$. Using the change $(x, y)\rightarrow (-x, y)$, we only need to consider the case

$$\begin{aligned} X_{2x}(0, 0)>0, \quad Y_{2x}(0, 0)>0. \end{aligned}$$

(6.1)

Hence, $\Sigma \cap {{\mathcal {U}}}_0$ is separated into two crossing sets by O, and the direction of X and Y on the right (resp. left) crossing set is upward (resp. downward) as it is seen in Lemma 3.1. Recalling [10, Theorem B] and [19, Theorem 1.2], we obtain that O is a stable pseudo-focus if $\ell <0$ and an unstable pseudo-focus if $\ell >0$ for $Z\in \Omega _{ff}$ satisfying (6.1), see Fig. 4. For $Z_{ff}\in \Omega _{ff}$ it is a linear vector field, and O is a stable focus as shown in (FF-1) of Fig. 2 if $\alpha =-1$ and an unstable focus as shown in (FF-2) of Fig. 2 if $\alpha =1$.

Fig. 4

Local phase portraits of $Z\in \Omega _{ff}$ satisfying (6.1) near O

We next prove this lemma for the case $\ell <0$ and $\alpha =-1$. The case $\ell >0$ and $\alpha =1$ can be treated similarly. Consider two sufficiently small neighborhoods $U\subset {{\mathcal {U}}}_0$ and $V\subset {{\mathcal {U}}}_0$ of O as shown in Fig. 5, where ${{\mathcal {U}}}_0$ is given in Lemma 3.1, U is surrounded by the closed line segment $\overline{CA}\subset \Sigma$ and the orbital arc of Z from A to C after passing through B, V is surrounded by the closed line segment $\overline{C_1A_1}\subset \Sigma$ and the orbital arc of $Z_{ff}$ from $A_1$ to $C_1$ after passing through $B_1$. Here overline denotes the closure. We need to construct a homeomorphism H from U to V implying the $\Sigma$-equivalence between Z with $\ell <0$ and $Z_{ff}$ with $\alpha =-1$.

Fig. 5

The homeomorphism H between $Z\in \Omega _{ff}$ with $\ell <0$ and $Z_{ff}$ with $\alpha =-1$

For $Z\in \Omega _{ff}$ satisfying (6.1), O is an anticlockwise rotary equilibrium of focus type of X and Y. Thus, given $P\in \overline{OA}$, there exist a first time $t_1=t_1(P)\ge 0$ such that $\Phi ^+(t_1, P)\in \overline{OB}$, and a first time $t_2=t_2(\Phi ^+(t_1, P))\ge 0$ such that $\Phi ^-\left( t_2, \Phi ^+(t_1, P)\right) \in \overline{OC}$, where $\Phi ^+$ and $\Phi ^-$ denote the flows of X and Y respectively. This means that we can define a Poincaré map ${\mathcal {P}}: \overline{OA}\rightarrow \overline{OC}$ by

$$\begin{aligned} {\mathcal {P}}(P)=\Phi ^-\left( t_2, \Phi ^+(t_1, P)\right) . \end{aligned}$$

(6.2)

In particular, ${\mathcal {P}}(O)=O$ and ${\mathcal {P}}(A)=C$, since A and C lie in the same orbit. Let $(x_P, 0)$ and $({\mathcal {P}}_1(x_P), {\mathcal {P}}_2(x_P))$ be the coordinates of P and ${\mathcal {P}}(P)$ respectively. Then ${\mathcal {P}}_2(x_P)=0$ and ${\mathcal {P}}_1(x_P)$ is given by

$$\begin{aligned} {\mathcal {P}}_1(x_P)=e^{\ell \pi } x_P+{\mathcal {O}}(x^2_P) \end{aligned}$$

from [19, Theorem 1.1, Theorem 1.2].

Similarly, denoting the flows of $X_{ff}$ and $Y_{ff}$ by $\Psi ^+$ and $\Psi ^-$ respectively, we can define a Poincaré map ${\mathcal {Q}}: \overline{OA_1}\rightarrow \overline{OC_1}$ by

$$\begin{aligned} {\mathcal {Q}}(P)=\Psi ^-\left( s_2, \Psi ^+(s_1, P)\right) , \end{aligned}$$

(6.3)

which satisfies ${\mathcal {Q}}(O)=O$ and ${\mathcal {Q}}(A_1)=C_1$, where $s_1=s_1(P)\ge 0$ is the first time such that $\Psi ^+(s_1, P)\in \overline{OB_1}$, and $s_2=s_2(\Psi ^+(s_1, P))\ge 0$ is the first time such that $\Psi ^-\left( s_2, \Psi ^+(s_1, P)\right) \in \overline{OC_1}$. Let $({\mathcal {Q}}_1(x_P), {\mathcal {Q}}_2(x_P))$ be the coordinates of ${\mathcal {Q}}(P)$. Then ${\mathcal {Q}}_2(x_P)=0$ and a straightway calculation yields

$$\begin{aligned} {\mathcal {Q}}_1(x_P)=e^{-2\pi }x_P. \end{aligned}$$

Since we are considering the case of $\ell <0$, according to the linearization and conjugacy theory of smooth map [21], U and V can be chosen to ensure that there exists a homeomorphism $h: [0, x_A]\rightarrow [0, x_{A_1}]$ satisfying

$$\begin{aligned} h(0)=0, \qquad h(x_A)=x_{A_1}, \qquad h({\mathcal {P}}_1(x_P))={\mathcal {Q}}_1(h(x_P)), \end{aligned}$$

(6.4)

where $x_A$ and $x_{A_1}$ are the first coordinates of A and $A_1$ respectively. Consequently, we define a homeomorphism $H_0: \overline{OA}\rightarrow \overline{OA_1}$ by

$$\begin{aligned} H_0(P)=H_0(x_P, 0)=(h(x_P), 0) \qquad \textrm{for} \quad P\in \overline{OA}. \end{aligned}$$

(6.5)

Clearly, it follows from (6.4) that $H_0(O)=O$, $H_0(A)=A_1$ and $H_0(C)=C_1$.

Given $P\in \overline{OB}$, there exists a first time $t_3=t_3(P)\le 0$ such that $\Phi ^+(t_3, P)\in \overline{OA}$, since O is an anticlockwise rotary equilibrium of focus type of X. Then $H_0(\Phi ^+(t_3, P))\in \overline{OA_1}$ and there exists a first time $s_3=s_3(H_0(\Phi ^+(t_3, P)))\ge 0$ such that $\Psi ^+(s_3, H_0(\Phi ^+(t_3, P)))\in \overline{OB_1}$ because O is an anticlockwise rotary focus of $X_{ff}$. By the arc length parametrization we can identify the orbital arc of X from $\Phi ^+(t_3, P)$ to P with the one of $X_{ff}$ from $H_0(\Phi ^+(t_3, P))$ to $\Psi ^+(s_3, H_0(\Phi ^+(t_3, P)))$. Therefore, in this way we can define a homeomorphism $H^+: \overline{\Sigma ^+\cap U}\rightarrow \overline{\Sigma ^+\cap V}$ that maps $\overline{BA}$ onto $\overline{B_1A_1}$, maps the orbits of X in $\overline{\Sigma ^+\cap U}$ onto the orbits of $X_{ff}$ in $\overline{\Sigma ^+\cap V}$ and satisfies

$$\begin{aligned} \left. H^+\right| _{\overline{OA}}=H_0. \end{aligned}$$

(6.6)

Given $P\in \overline{OC}$, there exists a first time $t_4=t_4(P)\le 0$ such that $\Phi ^-(t_4, P)\in \overline{OB}$. Then $H^+(\Phi ^-(t_4, P))\in \overline{OB_1}$ from the definition of $H^+$, and there exists a first time $s_4=s_4(H^+(\Phi ^-(t_4, P)))\ge 0$ such that $\Psi ^-\left( s_4, H^+(\Phi ^-(t_4, P))\right) \in \overline{OC_1}$. Similarly we can identify the orbital arc of Y from $\Phi ^-(t_4, P)$ to P with the one of $Y_{ff}$ from $H^+(\Phi ^-(t_4, P))$ to $\Psi ^-\left( s_4, H^+(\Phi ^-(t_4, P))\right)$, and thus define a homeomorphism $H^-: \overline{\Sigma ^-\cap U}\rightarrow \overline{\Sigma ^-\cap V}$ that maps $\overline{BC}$ onto $\overline{B_1C_1}$, maps the orbits of Y in $\overline{\Sigma ^-\cap U}$ onto the orbits of $Y_{ff}$ in $\overline{\Sigma ^-\cap V}$ and satisfies

$$\begin{aligned} \left. H^-\right| _{\overline{OB}}=\left. H^+\right| _{\overline{OB}}. \end{aligned}$$

(6.7)

Moreover, for any $P\in \overline{OC}$ we have

$$\begin{aligned} \begin{aligned} H^-(P)&=\Psi ^-\left( s_4, H^+(\Phi ^-(t_4, P))\right) =\Psi ^-\left( s_4, \Psi ^+(s_3, H_0(\Phi ^+(t_3, \Phi ^-(t_4, P))))\right) \\&={\mathcal {Q}}(H_0(\Phi ^+(t_3, \Phi ^-(t_4, P))))=H_0({\mathcal {P}}(\Phi ^+(t_3, \Phi ^-(t_4, P))))\\&=H_0(P) \end{aligned} \end{aligned}$$

by (6.2), (6.3), (6.4), (6.5) and the constructions of $H^\pm$. This implies that

$$\begin{aligned} \left. H^-\right| _{\overline{OC}}=\left. H_0\right| _{\overline{OC}}. \end{aligned}$$

(6.8)

Let

$$\begin{aligned} H(P)=\left\{ \begin{aligned}&H^+(P) \qquad{} & {} \textrm{for}\quad P\in (\Sigma ^+\cup \Sigma )\cap U,\\&H^-(P) \qquad{} & {} \textrm{for}\quad P\in (\Sigma ^-\cup \Sigma )\cap U. \end{aligned} \right. \end{aligned}$$

Then H is a homeomorphism from U to V because $H^+$ (resp. $H^-$) is a homeomorphism from $(\Sigma ^+\cup \Sigma )\cap U$ (resp. $(\Sigma ^-\cup \Sigma )\cap U$) to $(\Sigma ^+\cup \Sigma )\cap V$ (resp. $(\Sigma ^-\cup \Sigma )\cap V$) and $\left. H^+\right| _{\overline{BC}}=\left. H^-\right| _{\overline{BC}}$ by (6.6), (6.7) and (6.8). Furthermore, the construction of H ensures that H maps the orbits of $Z\in \Omega _{ff}$ with $\ell <0$ in U onto the orbits of $Z_{ff}$ with $\alpha =-1$ in V, preserving the direction of time and the switching line $\Sigma$. We eventually conclude that $Z\in \Omega _{ff}$ with $\ell <0$ and $Z_{ff}$ with $\alpha =-1$ are locally $\Sigma$-equivalent near O. $\square$

Proof of item (2)

By $(x, y)\rightarrow (x, -y)$ and $(x, y)\rightarrow (-x, y)$ we only need to consider $Z\in \Omega _{fn}$ satisfying (6.1) and

$$\begin{aligned} \lambda ^+_1,\lambda ^+_2\in {\mathbb {C}}\setminus {\mathbb {R}}, \qquad \lambda ^-_1,\lambda ^-_2\in {\mathbb {R}}, \qquad \lambda ^-_1\lambda ^-_2>0. \end{aligned}$$

In this case, O is an equilibrium of focus type of X and a node of Y by [34, Theorems 4.2, 4.3, 5.1]. Thus, recalling the dynamics on $\Sigma$ given in Lemma 3.1, we get two different types of the local phase portraits of Z near O depending on the sign of $\lambda ^-_1+\lambda ^-_2$, namely the stability of O when it is regarded as an equilibrium of Y, see Fig. 6. In Fig. 6a, the strong unstable manifold $m^u_s$ lies in the left side of the weak unstable manifold $m^u_w$, while in Fig. 6b, the strong stable manifold $m^s_s$ lies in the right side of the weak stable manifold $m^s_w$. Here we use the assumption of $\lambda ^-_1\ne \lambda ^-_2$ for all vector fields in $\Omega _1$. Regarding the vector field $Z_{fn}$, we easily verify that its phase portrait is the one either as shown in (FN-1) of Fig. 2 if $\beta =1$, or as shown in (FN-2) of Fig. 2 if $\beta =-1$.

Fig. 6

Local phase portraits of $Z\in \Omega _{fn}$ satisfying (6.1) and $\lambda ^+_1,\lambda ^+_2\in {\mathbb {C}}\setminus {\mathbb {R}}, \lambda ^-_1,\lambda ^-_2\in {\mathbb {R}}, \lambda ^-_1\lambda ^-_2>0$ near O

We only consider $\lambda ^-_1+\lambda ^-_2>0$ and $\beta =1$ because the case of $\lambda ^-_1+\lambda ^-_2<0$ and $\beta =-1$ is similar. Consider two sufficiently small neighborhoods $U\subset {{\mathcal {U}}}_0$ and $V\subset {{\mathcal {U}}}_0$ of O as shown in Fig. 7, where ${{\mathcal {U}}}_0$ is given in Lemma 3.1, U is surrounded by orbital arc $\widehat{AB}$ of X from A to B, and arc $\widehat{BA}$ on which Y is transverse to it, V is surrounded by orbital arc $\widehat{A_2B_2}$ of $X_{fn}$ from $A_2$ to $B_2$, and arc $\widehat{B_2A_2}$ on which the vector field $Y_{fn}$ is transverse to it. We need to construct a homeomorphism H from U to V providing the $\Sigma$-equivalence between $Z\in \Omega _{fn}$ with $\lambda ^-_1+\lambda ^-_2>0$ and $Z_{fn}$ with $\beta =1$.

Fig. 7

The homeomorphism H between $Z\in \Omega _{fn}$ with $\lambda ^-_1+\lambda ^-_2>0$ and $Z_{fn}$ with $\beta =1$

By the arc length parametrization there exists a homeomorphism $H_0: \overline{OA}\rightarrow \overline{OA_2}$ such that $H_0(O)=O$ and $H_0(A)=A_2$. Since O is an anticlockwise rotary equilibrium of focus type of X, the forward orbit of X starting from $P\in \overline{OA}$ evolves in $\overline{\Sigma ^+\cap U}$ until it reaches $\overline{OB}$ at a point Q. Then $H_0(P)\in \overline{OA_2}$. Since O is an anticlockwise rotary center of $X_{fn}$, the forward orbit of $X_{fn}$ starting from $H_0(P)$ evolves in $\overline{\Sigma ^+\cap V}$ until it reaches $\overline{OB_2}$ at a point $Q_2$. By the arc length parametrization we can identify the orbital arc of X from P to Q with the one of $X_{fn}$ from $H_0(P)$ to $Q_2$. In this way we can define a homeomorphism $H_f: \overline{\Sigma ^+\cap U}\rightarrow \overline{\Sigma ^+\cap V}$ that maps $\overline{BA}$ onto $\overline{B_2A_2}$, maps the orbits of X in $\overline{\Sigma ^+\cap U}$ onto the orbits of $X_{fn}$ in $\overline{\Sigma ^+\cap V}$ and satisfies

$$\begin{aligned} \left. H_f\right| _{\overline{OA}}=H_0. \end{aligned}$$

(6.9)

Consider the region $R_{BOC}$ surrounded by $\overline{OB}$, $\widehat{BC}$ and the strong unstable manifold $\widehat{OC}$, and the corresponding region $R_{B_2OC_2}$ surrounded by $\overline{OB_2}$, $\widehat{B_2C_2}$ and the strong unstable manifold $\widehat{OC_2}$. Given $P\in \overline{OB}$, there exists a unique point $Q\in \widehat{BC}$ such that the backward orbit of Y starting from Q evolves in $\overline{R_{BOC}}$ until it reaches or tends to $\overline{OB}$ at P, since $\widehat{OC}$ is the strong unstable manifold of the node O for Y and we are assuming that the vector field Y on $\widehat{BA}$ is transverse to $\widehat{BA}$. Analogously, there exists a unique point $Q_2\in \widehat{B_2C_2}$ such that the backward orbit of $Y_{fn}$ starting from $Q_2$ evolves in $\overline{R_{B_2OC_2}}$ until it reaches or tends to $\overline{OB_2}$ at $H_f(P)$. Therefore, by the arc length parametrization again we can identify the orbital arc of Y from P to Q with the one of $Y_{fn}$ from $H_f(P)$ to $Q_2$, and then define a homeomorphism $H^1_n: \overline{R_{BOC}}\rightarrow \overline{R_{B_2OC_2}}$ that maps the orbits of Y in $\overline{R_{BOC}}$ onto the orbits of $Y_{fn}$ in $\overline{R_{B_2OC_2}}$ and satisfies

$$\begin{aligned} \left. H^1_n\right| _{\overline{OB}}=\left. H_f\right| _{\overline{OB}}. \end{aligned}$$

(6.10)

Consider the region $R_{COA}$ surrounded by $\widehat{OC}$, $\widehat{CA}$ and $\overline{OA}$, and the corresponding region $R_{B_2OC_2}$ surrounded by $\widehat{OC_2}$, $\widehat{C_2A_2}$ and $\overline{OA_2}$. Regarding arcs $\widehat{CA}$ and $\widehat{C_2A_2}$, we obtain a homeomorphism $H^0_n: \widehat{CA}\rightarrow \widehat{C_2A_2}$ such that $H^0_n(C)=C_2$ and $H^0_n(A)=A_2$ by the arc length parametrization. Since the choice of U ensures that the vector field Y on $(\widehat{CA}\cup \overline{OA})\setminus O$ is transverse to $(\widehat{CA}\cup \overline{OA})\setminus O$, the backward orbit of Y starting from $P\in (\widehat{CA}\cup \overline{OA})\setminus O$ evolves in $\overline{R_{COA}}$ and finally tends to O. Let $P_2=H_0(P)$ if $P\in \overline{OA}$ and $P_2=H^0_n(P)$ if $P\in \widehat{CA}$. Then the backward orbit of $Y_{fn}$ starting from $P_2$ evolves in $\overline{R_{C_2OA_2}}$ and tends to O. Identify the orbital arc of Y from P to O with the orbital arc of $Y_{fn}$ from $P_2$ to O. In this way we can define a homeomorphism $H^2_n: \overline{R_{COA}}\rightarrow \overline{R_{C_2OA_2}}$ that maps the orbits of Y in $\overline{R_{COA}}$ onto the orbits of $Y_{fn}$ in $\overline{R_{C_2OA_2}}$ and satisfies

$$\begin{aligned} \left. H^2_n\right| _{\widehat{OC}}=\left. H^1_n\right| _{\widehat{OC}}, \qquad \left. H^2_n\right| _{\overline{OA}}=H_0, \qquad \left. H^2_n\right| _{\widehat{CA}}=H^0_n. \end{aligned}$$

(6.11)

Define

$$\begin{aligned} H_n(P)=\left\{ \begin{aligned}&H^1_n(P) \qquad{} & {} \textrm{for} \quad P\in \overline{R_{BOC}},\\&H^2_n(P) \qquad{} & {} \textrm{for} \quad P\in \overline{R_{COA}}. \end{aligned} \right. \end{aligned}$$

(6.12)

Since $H^1_n$ (resp. $H^2_n$) is a homeomorphism from $\overline{R_{BOC}}$ (resp. $\overline{R_{COA}}$) to $\overline{R_{B_2OC_2}}$ (resp. $\overline{R_{C_2OA_2}}$) and $\left. H^2_n\right| _{\widehat{OC}}=\left. H^1_n\right| _{\widehat{OC}}$ from (6.11), we get that $H_n$ is a homeomorphism from $\overline{\Sigma ^-\cap U}$ to $\overline{\Sigma ^-\cap V}$ that maps the orbits of Y in $\overline{\Sigma ^-\cup U}$ onto the orbits of $Y_{fn}$ in $\overline{\Sigma ^-\cup V}$.

Let

$$\begin{aligned} H(P)=\left\{ \begin{aligned}&H_f(P) \qquad{} & {} \textrm{for}\quad P\in (\Sigma ^+\cup \Sigma )\cap U,\\&H_n(P) \qquad{} & {} \textrm{for}\quad P\in (\Sigma ^-\cup \Sigma )\cap U. \end{aligned} \right. \end{aligned}$$

Since $H_f$ (resp. $H_n$) is a homeomorphism from $(\Sigma ^+\cup \Sigma )\cap U$ (resp. $(\Sigma ^-\cup \Sigma )\cap U$) to $(\Sigma ^+\cup \Sigma )\cap V$ (resp. $(\Sigma ^-\cup \Sigma )\cap V$) and $\left. H_n\right| _{\overline{BA}}=\left. H_f\right| _{\overline{BA}}$ from (6.9), (6.10), (6.11), and (6.12), $H: U\rightarrow V$ is a homeomorphism. Furthermore, the construction of H ensures that it maps the orbits of $Z\in \Omega _{fn}$ with $\lambda ^-_1+\lambda ^-_2>0$ in U onto the orbits of $Z_{fn}$ with $\beta =1$ in V, preserving the direction of time and the switching line $\Sigma$. This concludes the proof of item (2). $\square$

Proof of item (3)

Using the changes $(x, y)\rightarrow (x, -y)$ and $(x, y)\rightarrow (-x, y)$, we only need to consider $Z\in \Omega _{fs}$ satisfying (6.1) and

$$\begin{aligned} \lambda ^+_1, \lambda ^+_2\in {\mathbb {C}}\setminus {\mathbb {R}},\qquad \lambda ^-_1, \lambda ^-_2\in {\mathbb {R}}, \qquad \lambda ^-_1\lambda ^-_2<0. \end{aligned}$$

In this case, O is an equilibrium of focus type of X and a saddle of Y by [34, Theorems 4.2, 4.4, 5.1]. Reviewing the dynamics on $\Sigma$ given in Lemma 3.1, we depict the local phase portrait of Z near O as shown in Fig. 8. The phase portrait of the vector field $Z_{fs}$ is as shown in Fig. 2.

Fig. 8

Local phase portrait of $Z\in \Omega _{fs}$ satisfying (6.1) and $\lambda ^+_1, \lambda ^+_2\in {\mathbb {C}}\setminus {\mathbb {R}}, \lambda ^-_1, \lambda ^-_2\in {\mathbb {R}}, \lambda ^-_1\lambda ^-_2<0$ near O

Fig. 9

The homeomorphism H between $Z\in \Omega _{fs}$ and $Z_{fs}$

Consider two sufficiently small neighborhoods $U\subset {{\mathcal {U}}}_0$ and $V\subset {{\mathcal {U}}}_0$ of O as shown in Fig. 9, where $\widehat{AB}$ and $\widehat{A_3B_3}$ are the corresponding orbital arcs, $\widehat{BA}$ (resp. $\widehat{B_3A_3}$) is the arc where the vector field Y (resp. $Y_{fs}$) is transverse to it. As done in the proof of item (2), we can define a homeomorphism $H_f: \overline{\Sigma ^+\cap U}\rightarrow \overline{\Sigma ^+\cap V}$ that maps $\overline{BA}$ onto $\overline{B_3A_3}$, and maps the orbits of X in $\overline{\Sigma ^+\cap U}$ onto the orbits of $X_{fs}$ in $\overline{\Sigma ^+\cap V}$.

In order to complete this proof, next we construct a homeomorphism $H_s: \overline{\Sigma ^-\cap U}\rightarrow \overline{\Sigma ^-\cap V}$ that maps the orbits of Y in $\overline{\Sigma ^-\cap U}$ onto the orbits of $X_{fs}$ in $\overline{\Sigma ^-\cap V}$ and satisfies $\left. H_s\right| _{\overline{BA}}=\left. H_f\right| _{\overline{BA}}$. Let

$$\begin{aligned} \widehat{OD}=\left\{ (x, y)\in \overline{\Sigma ^-\cap U}: Y_2(x, y)=0\right\} , \qquad \overline{OD_3}=\left\{ (x, y)\in \overline{\Sigma ^-\cap V}: x=0\right\} , \end{aligned}$$

where $Y_2$ is the ordinate of Y. Then there exists a homeomorphism $H^0_s: \widehat{OD}\rightarrow \overline{OD_3}$ such that $H^0_s(O)=O$ and $H^0_s(D)=D_3$ by the arc length parametrization. Consider the region $R_{BOD}$ surrounded by $\overline{OB}$, $\widehat{BD}$ and $\widehat{OD}$, and the region $R_{B_3OD_3}$ surrounded by $\overline{OB_3}$, $\widehat{B_3D_3}$ and $\overline{OD_3}$. Given $P\in \overline{OB}\cup \widehat{OD}$, there exists a unique point $Q\in \widehat{BD}$ such that the backward orbit of Y starting from Q evolves in $\overline{R_{BOD}}$ until it either reaches $(\overline{OB}\cup \widehat{OD})\setminus O$ when $P\ne O$ or tends to O when $P=O$, since we require that the vector field Y on $\widehat{BD}$ is transverse to $\widehat{BD}$. Let $P_3=H_f(P)$ if $P\in \overline{OB}$ and $P_3=H^0_s(P)$ if $P\in \widehat{OD}$. We obtain a unique point $Q_3\in \widehat{B_3D_3}$ such that the backward orbit of $Y_{fs}$ starting from $Q_3$ evolves in $\overline{R_{B_3OD_3}}$ until it reaches or tends to $P_3$. The arc length parametrization allows to identify the orbital arc of Y from Q to P and the one of $Y_{fs}$ from $Q_3$ to $P_3$. In this way we can define a homeomorphism $H^1_s: \overline{R_{BOD}}\rightarrow \overline{R_{B_3OD_3}}$ that maps the orbits of Y in $\overline{R_{BOD}}$ onto the orbits of $Y_{fs}$ in $\overline{R_{B_3OD_3}}$ and satisfies

$$\begin{aligned} \left. H^1_s\right| _{\overline{OB}}=\left. H_f\right| _{\overline{OB}}, \qquad \left. H^1_s\right| _{\widehat{OD}}=H^0_s. \end{aligned}$$

(6.13)

A similar argument to the last paragraph yields a homeomorphism $H^2_s: \overline{R_{DOA}}\rightarrow \overline{R_{D_3OA_3}}$ that maps the orbits of Y in $\overline{R_{DOA}}$ onto the orbits of $Y_{fs}$ in $\overline{R_{D_3OA_3}}$ and satisfies

$$\begin{aligned} \left. H^2_s\right| _{\overline{OA}}=\left. H_f\right| _{\overline{OA}}, \qquad \left. H^2_s\right| _{\widehat{OD}}=H^0_s. \end{aligned}$$

(6.14)

Joining the homeomorphisms $H^1_s$ and $H^2_s$ we construct $H_s$ as

$$\begin{aligned} H_s(P)=\left\{ \begin{aligned}&H^1_s(P) \qquad{} & {} \textrm{for} \quad P\in \overline{R_{BOD}},\\&H^2_s(P) \qquad{} & {} \textrm{for} \quad P\in \overline{R_{DOA}}. \end{aligned} \right. \end{aligned}$$

(6.15)

From (6.13) and (6.14) it follows that $\left. H^2_s\right| _{\widehat{OD}}=\left. H^1_s\right| _{\widehat{OD}}$, so that $H_s$ is a homeomorphism from $\overline{\Sigma ^-\cap U}$ to $\overline{\Sigma ^-\cap V}$ maps the orbits of Y in $\overline{\Sigma ^-\cap U}$ onto the orbits of $X_{fs}$ in $\overline{\Sigma ^-\cap V}$.

Let

$$\begin{aligned} H(P)=\left\{ \begin{aligned}&H_f(P) \qquad{} & {} \textrm{for}\quad P\in (\Sigma ^+\cup \Sigma )\cap U,\\&H_s(P) \qquad{} & {} \textrm{for}\quad P\in (\Sigma ^-\cup \Sigma )\cap U. \end{aligned} \right. \end{aligned}$$

Since $H_f$ (resp. $H_s$) is a homeomorphism from $(\Sigma ^+\cup \Sigma )\cap U$ (resp. $(\Sigma ^-\cup \Sigma )\cap U$) to $(\Sigma ^+\cup \Sigma )\cap V$ (resp. $(\Sigma ^-\cup \Sigma )\cap V$) and $\left. H_s\right| _{\overline{BA}}=\left. H_f\right| _{\overline{BA}}$ from (6.13), (6.14), and (6.15), $H: U\rightarrow V$ is a homeomorphism. Furthermore, the construction of H ensures that it maps the orbits of $Z\in \Omega _{fs}$ in U onto the orbits of $Z_{fs}$ in V, preserving the direction of time and the switching line $\Sigma$. This proves item (3). $\square$

Fig. 10

Local phase portraits of $Z\in \Omega _{nn}$ satisfying (6.1) near O

Proof of item (4)

For $Z\in \Omega _{nn}$ we know that O is a node of both X and Y with two different eigenvalues by [34, Theorem 4.3]. Moreover, using the change $(x, y)\rightarrow (-x, y)$ it is enough to consider $Z\in \Omega _{nn}$ satisfying (6.1). In this case, according to the dynamics on $\Sigma$ given in Lemma 3.1, we get four local phase portraits of Z near O as shown in Fig. 10, depending on the sign of $\lambda ^\pm _1+\lambda ^\pm _2$, namely the stability of O as an equilibrium of X and Y. However, we notice that the phase portrait (d) of Fig. 10 can be transformed into (b) of Fig. 10 by the change $(x, y)\rightarrow (-x, -y)$, so that there are essentially three different types of the local phase portraits of Z near O. Besides, a simple analysis implies that the phase portrait of $Z_{nn}$ is (NN-1) (resp. (NN-2) and (NN-3)) of Fig. 2 if $\gamma =\eta =1$ (resp. $\gamma =-\eta =1$ and $\gamma =\eta =-1$).

The homeomorphism between $Z\in \Omega _{nn}$ and $Z_{nn}$ can be constructed by a similar method to the proofs of foregoing proofs. In fact, consider the case of $\lambda ^+_1+\lambda ^+_2>0, \lambda ^-_1+\lambda ^-_2>0$ and $\gamma =\eta =1$ as an example. We can choose two sufficiently small neighborhoods $U\subset {{\mathcal {U}}}_0$ and $V\subset {{\mathcal {U}}}_0$ of O such that Z is transverse to the boundary of U and $Z_{nn}$ is transverse to the boundary of V. Then there is always a homeomorphism $H: \Sigma \cap U\rightarrow \Sigma \cap V$ satisfying $H(O)=O, H(\Sigma _l\cap U)=\Sigma _l\cap V$ and $H(\Sigma _r\cap U)=\Sigma _r\cap V$, where $\Sigma _l=\{(x, 0)\in {{\mathcal {U}}}: x<0\}$ and $\Sigma _r=\{(x, 0)\in {{\mathcal {U}}}: x>0\}$. Like the construction of $H_n$ in the proof of item (2), we are able to extend H for $\Sigma ^+\cap U$ and $\Sigma ^-\cap U$ respectively, and finally obtain a homeomorphism from U to V that provides the $\Sigma$-equivalence between $Z\in \Omega _{nn}$ with $\lambda ^+_1+\lambda ^+_2>0, \lambda ^-_1+\lambda ^-_2>0$ and $Z_{nn}$ with $\gamma =\eta =1$. That is, item (4) holds. $\square$

Proof of item (5)

Using the changes $(x, y)\rightarrow (x, -y)$ and $(x, y)\rightarrow (-x, y)$, we only need to consider $Z\in \Omega _{ns}$ satisfying (6.1), $\lambda ^+_1\lambda ^+_2>0$ and $\lambda ^-_1\lambda ^-_2<0$. In this case, O is a node of X and a saddle of Y by [34, Theorems 4.3, 4.4]. Combining with the dynamics on $\Sigma$ given in Lemma 3.1, we get two different types of the local phase portraits of Z near O as shown in Fig. 11, depending on the sign of $\lambda ^+_1+\lambda ^+_2$. Regarding $Z_{ns}$, its phase portrait is (NS-1) (resp. (NS-2)) of Fig. 2 if $\xi =1$ (resp. $\xi =-1$).

Fig. 11

Local phase portraits of $Z\in \Omega _{ns}$ satisfying (6.1), $\lambda ^+_1\lambda ^+_2>0$ and $\lambda ^-_1\lambda ^-_2<0$ near O

Consider two sufficiently small neighborhoods $U\subset {{\mathcal {U}}}_0$ and $V\subset {{\mathcal {U}}}_0$ of O such that Z is transverse to the boundary of U and $Z_{ns}$ is transverse to the boundary of V. For each one of the above two cases, we can define a homeomorphism H with $H(O)=O$ to identify $\Sigma \cap U$ with $\Sigma \cap V$ by the arc length parametrization. Then H can be extended for $\Sigma ^+\cap U$ (resp. $\Sigma ^-\cap U$) as the construction of $H_n$ (resp. $H_s$) in the proof of item (2) (resp. item (3)). That is, H is a homeomorphism from U to V that provides $\Sigma$-equivalence, and then item (5) holds. $\square$

Proof of item (6)

For $Z\in \Omega _{ss}$ we know that O is a saddle of both X and Y by [34, Theorem 4.4]. Using the change $(x, y)\rightarrow (-x, y)$ we only need to consider $Z\in \Omega _{ss}$ satisfying (6.1). Together with the dynamics on $\Sigma$ given in Lemma 3.1, this implies that the local phase portrait of Z near O is as shown in Fig. 12. Moreover, the phase portrait of $Z_{ss}$ is Fig. 2.

Fig. 12

Local phase portrait of $Z\in \Omega _{ss}$ satisfying (6.1) near O

Consider two sufficiently small neighborhoods $U\subset {{\mathcal {U}}}_0$ and $V\subset {{\mathcal {U}}}_0$ of O such that Z is transverse to the boundary of U and $Z_{ss}$ is transverse to the boundary of V . We can define a homeomorphism H with $H(O)=O$ to identify $\Sigma \cap U$ with $\Sigma \cap V$ by the arc length parametrization. Repeating the construction of $H_s$ in the proof of item (3), we extend H for $\Sigma ^+\cap U$ and $\Sigma ^-\cap U$ respectively, and finally obtain a homeomorphism from U to V that provides $\Sigma$-equivalence between $Z\in \Omega _{ss}$ and $Z_{ss}$. This proves item (6). $\square$