Proof. One can check that (P1) is true because all equilibria lie at the $x$-axis and the abscissas of all equilibria are the roots of $g(x)=0$ which are independent of $\alpha$. In order to check (P2), let $\theta$ denote the angle from the $x$-axis to the vector $(X(x,y,\alpha ),Y(x,y,\alpha ))$ of system (1.1) in counter-clockwise direction. Then

\[ \frac{\partial \theta}{\partial\alpha}=\frac{\partial}{\partial\alpha}(\arctan({Y}/{X})) =\frac{1}{X^2+Y^2} \left| \begin{array}{cc} X & Y \\ \dfrac{\partial X}{\partial\alpha} & \dfrac{\partial Y}{\partial\alpha} \\ \end{array} \right|. \]

For $\mathcal {L}_\alpha$, we have $X(x,y,\alpha )=y$ and $Y(x,y,\alpha )=-g(x)-f(x,y,\alpha )y$. Then, it implies that the vector $(X(x,y,\alpha ), Y(x,y,\alpha ))$ rotates counter-clockwise at a regular point $P_0(x,y)$ as $\alpha$ varies since

\[ \left| \begin{array}{cc} X(x,y,\alpha) & Y(x,y,\alpha) \\ \dfrac{\partial X}{\partial\alpha}(x,y,\alpha) & \dfrac{\partial Y}{\partial\alpha}(x,y,\alpha) \\ \end{array} \right| = \left| \begin{array}{cc} y & -g(x)-f(x,y,\alpha)y \\ 0 & -\dfrac{\partial f(x,y, \alpha )}{\partial \alpha}y \\ \end{array} \right|={-}\frac{\partial f(x,y, \alpha )}{\partial \alpha}y^2<0 \]

at the regular point $P_0(x,y)$ for all $\alpha \in I$ except in the curves $\Omega (x,y):=\frac {\partial f(x,y, \alpha )}{\partial \alpha }y^2=0$. Thus, condition (P2) holds.

Proof. Assume that system (2.6) with hypothesis ($H_1$) has a limit cycle $\Gamma$, as shown in figure 2. For compound structure and fractional multiplicity, we need to consider a small annulus surrounding $\Gamma$. Since the function $g$ is piecewise analytic on $(a_1,x_1)\cup (x_1,x_2)\cup (x_2,x_3)\cup \ldots \cup (x_n,a_2)$, and functions $f(x,y,\alpha )$, $\partial f(x,y,\alpha )/\partial y$ are piecewise analytic on $D_1\cup D_2\cup \ldots \cup D_{n+1}$, there is a small $\varepsilon >0$ such that each normal at any $P\in \Gamma$ restricted to the open annular neighbourhood $S(\Gamma,\varepsilon )$ of $\Gamma$ with the radius $\varepsilon$ is non-contact. Thus, we can set up curvilinear coordinates in the annulus $S(\Gamma,\varepsilon )$. Each point $\underline {P}$ in $S(\Gamma,\varepsilon )$ has a corresponding point $P\in \Gamma$ such that $\underline {P}$ lies on the normal at $P$.

Figure 2. The curvilinear coordinates in the annulus $S(\Gamma,\varepsilon )$.

Note that a point on limit cycle $\Gamma$ can be written as

\[ (x,y)=(\varphi(s), \psi(s)), \]

where $s$ is the arclength (parameter) from $P$ to a specified point in the clockwise direction. Let $p$ be the length from $P$ to $\underline {P}$ positively in the outward direction. Thus, as shown in [Reference Zhang, Ding, Huang and Dong30, Chapter 4.2], we can represent the point $\underline {P}: (x,y)$ in the curvilinear coordinates $s$ and $p$ in each region $D_j$ ($j=1,2,\ldots,n+1$) as

\[ x=\varphi(s)-p\psi'(s), \quad y=\psi(s)+p\varphi'(s), \]

where

\begin{align*} & \varphi'(s)=\frac{{\rm d}x}{{\rm d}s}=\frac{X_0}{\sqrt{{X_0}^2+{Y_0}^2}}, \quad \psi'(s)=\frac{{\rm d}y}{{\rm d}s}=\frac{Y_0}{\sqrt{{X_0}^2+{Y_0}^2}}, \\ & X_0=X(\varphi(s),\psi(s),\alpha), \quad Y_0=Y(\varphi(s),\psi(s),\alpha). \end{align*}

Therefore, we have

\[ \frac{{\rm d}y}{{\rm d}x}=\frac{\psi'(s)+\frac{dp}{{\rm d}s}\varphi'(s)+p\varphi''(s)}{\varphi'(s)-\frac{dp}{{\rm d}s}\psi'(s)-p\psi''(s)}= \frac{Y(\varphi(s)-p\psi'(s),\psi(s)+p\varphi'(s),\alpha)}{X(\varphi(s)-p\psi'(s),\psi(s)+p\varphi'(s),\alpha)}, \]

implying that

(2.12)\begin{equation} \frac{dp}{{\rm d}s}=\frac{Y\varphi'-X\psi'-p(X\varphi''+Y\psi''(s))}{X\varphi'+Y\psi'}=F(s,p,\alpha),\end{equation}

where

\begin{align*} & \varphi''(s)=\frac{-Y_0}{\sqrt{{X_0}^2+{Y_0}^2}}\left({X_0}^2\frac{\partial Y}{\partial x}|_{p=0}+X_0Y_0\left(\frac{\partial Y}{\partial y}|_{p=0}-\frac{\partial X}{\partial x}|_{p=0}\right)-{Y_0}^2\frac{\partial X}{\partial y}|_{p=0}\right)\!, \\ & \psi''(s)=\frac{X_0}{\sqrt{{X_0}^2+{Y_0}^2}}\left({X_0}^2\frac{\partial Y}{\partial x}|_{p=0}+X_0Y_0 \left(\frac{\partial Y}{\partial y}|_{p=0}-\frac{\partial X}{\partial x}|_{p=0}\right)-{Y_0}^2\frac{\partial X}{\partial y}|_{p=0}\right)\!. \end{align*}

Having the above curvilinear coordinates, we first consider the case that $\Gamma$ does not intersect any switching line. Let $P_0:(0,p_0) \in \overline {P\underline {P}}$ and $p_0$ be the length from $P$ to $P_0$ positively in the outward direction, as shown in figure 2. It follows from (2.12) that the Poincaré map $\mathcal {P}$ satisfies

\[ \mathcal{P}(p_0)=p(l,p_0)=p_0+\int_0^lF(s,p(s,p_0),\alpha){\rm d}s, \]

where $l$ is the total arc length of $\Gamma$.

Second, consider the case that $\Gamma$ intersects only one switching line $x=x_j$. It is clear that $\Gamma$ is divided by $x=x_j$ into two parts: left part and right part. Consider $P_0:(0,p_0)$ to be the initial point of the Poincaré map on $x=x_j$, and let $(l_1,p_1)$ (resp. $(l_2,p_2)$) denote the first intersection point of the positive-(resp. negative-) half orbit with $x=x_j$. Without loss of generality, we can assume that the segment $\overline {PP_0}$ on the switching line $x=x_j$ is vertical to $\Gamma$. Otherwise, a rotation can achieve this because the switching line $x=x_j$ is transversal to $\Gamma$. Then, we can obtain the two half Poincaré maps

\[ \mathcal{P}_1(p_0)=p_1(l,p_0)=p_0+\int_0^{l_1}F(s,p(s,p_0),\alpha){\rm d}s \]

and

\[ \mathcal{P}_2(p_0)=p_2(l,p_0)=p_0-\int_{l_1}^lF(s,p(s,p_0),\alpha){\rm d}s, \]

as shown in figure 3. Notice that the denominator of the right-hand side of (2.12) does not equal zero since $X^2(\varphi (s),\phi (s),\alpha )+Y^2(\varphi (s),\phi (s),\alpha )\neq 0$. Moreover, the vector field of system (2.6) is analytic for $x< x_j$ and $x>x_j$ in $S(\Gamma,\varepsilon )$. Therefore,

\[ p_1(l,p_0)=p_0+\sum_{i=1}^{\infty}a_i{p_0}^i \text{ and } p_2(l,p_0)=p_0+\sum_{i=1}^{\infty}b_i{p_0}^i. \]

Thus, we obtain the following successive function

(2.13)\begin{equation} \varrho(p_0)= \mathcal{P}_1(p_0)- \mathcal{P}_2(p_0)=\sum_{i=1}^{\infty}(a_i-b_i){p_0}^i. \end{equation}

The coefficients $a_i-b_i$ in the above series have the two cases:

1. (i) $a_i-b_i=0$ for each positive integer $i\in \mathbb {Z}_+$;

2. (ii) there exists a positive integer $i_1$ such that $a_{i_1}-b_{i_1}\ne 0$ and $a_j-b_j= 0$ for all $1\le j< i_1$.

In case (i) we have a periodic annulus and no limit cycles exist. In case (ii), $\Gamma$ is a limit cycle of multiplicity $i_1\in \mathbb {Z}_+$ and there are at most $i_1$ zeros of $\varrho (p_0)$ near the origin by the Malgrange preparation theorem ([Reference Chow and Hale6]). Consequently, limit cycle $\Gamma$ is not compound and has a multiplicity $i_1 \in \mathbb {Z}_+$ if $\Gamma$ intersects only one switching line.

Figure 3. The successive function $\varrho (p_0)= \mathcal {P}_1(p_0)- \mathcal {P}_2(p_0)$.

In addition, if the crossing limit cycle $\Gamma$ intersects two or more switching lines, we can prove our result similarly as we do for the case of only one switching line. Actually, if $\Gamma$ intersects $n$ ($n>1$) switching lines, a small neighbourhood of the crossing limit cycle $\Gamma$ only possibly contains the crossing points of the switching lines. Thus, we can present the Poincaré map in $2n$ pieces, each of which is a map from a witching line to the next similar to (2.13). Since $\Gamma$ intersects transversally the switching lines, each piece of the Poincaré map is well defined and close to the corresponding segment of the orbit $\Gamma$ in the corresponding zone $D_i$ between the executive switching lines. In contrast, if $\Gamma$ is a grazing limit cycle, there are at most two grazing points, at which $\Gamma$ intersects two switching lines at the $x$-axis. Therefore, any small vicinity of $\Gamma$ (where $\Gamma$ is not included) only possibly contains the crossing points of the switching lines and thus the successive function can be constructed similarly as we did above for the case that $\Gamma$ intersects only one switching line.

Proof. By lemma 2.2, $k$ is a positive integer. In the following, we discuss the distance between an orbit without perturbation and the orbit under perturbation near a limit cycle. From the distance we can further investigate zeros of successive function and the relation between variational exponent and multiplicity of limit cycles.

Without loss of generality, we only discuss the case that the limit cycle $\Gamma$ is externally stable as $\alpha =\alpha _0$, i.e. those points $P_0, P_1$ and $P$ defined by the Poincaré map and the successive function $\varrho$ (see (2.9) and figure 1) are ranked by $0< x_1< x_0$. Otherwise, $\Gamma$ is externally unstable, for which we make a time-rescaling $t\to -t$, so that the limit cycle of the rescaled system is externally stable.

We always let $Q:(x_Q, y_Q)$ be a general point $Q$. Notice that the closed orbit $\Gamma$ has two intersection points $P_0$ and $Q_0$ with the $x$-axis. Moreover, the orbit $\widehat {PQ_1P_1}$ starts from $P:(x_0,0)$, passes through $Q_1:(x_{Q_1},0)$ and returns the positive $x$-axis at $P_1:(x_1,0)$ such that $x_{Q_1}<0< x_{P_1}< x_P$, as shown in figure 1. We consider the perturbed system (2.6)$|_{\alpha =\alpha _0+\varepsilon }$ and let $\widehat {Q_3Q_1Q_2}$ be the orbit of system (2.6)$|_{\alpha =\alpha _0+\varepsilon }$ crossing $Q_1$ and having two intersection points $Q_2$ and $Q_3$ with the positive $x$-axis. For simplicity, let

\[ x_{12}:=\min\{x_1,x_{Q_2}\} ~~{\rm and} ~~x_{03}:=\min\{x_0,x_{Q_3}\}. \]

For $x\in (x_{Q_1}, x_{12})$, let

(2.14)\begin{equation} z_1:={\tilde y}_{\varepsilon}(x)-{\tilde y}_0(x),\end{equation}

where $y={\tilde y}_0(x)$ and $y={\tilde y}_{\varepsilon }(x)$ represent the orbit segments $\widehat {Q_1P_1}$ and $\widehat {Q_1Q_2}$, respectively. For $x\in (x_{Q_1}, x_{03})$, let

(2.15)\begin{equation} z_2:={\hat y}_{\varepsilon}(x)-{\hat y}_0(x),\end{equation}

where $y={\hat y}_0(x)$ and $y={\hat y}_{\varepsilon }(x)$ represent the orbit segments $\widehat {P Q_1}$ and $\widehat {Q_3 Q_1}$, respectively. Moreover, let

\[ P_2= \left\{ \begin{array}{ll} (x_{12}, \tilde y_0(x_{12}) & \mbox{ if } x_1\geq x_{Q_2}, \\ (x_{12}, \tilde y_{\varepsilon}(x_{12}) & \mbox{ if } x_1< x_{Q_2}, \end{array} \right. \ \ P_3= \left\{ \begin{array}{ll} (x_{03}, \hat y_{\varepsilon}(x_{12}) & \mbox{ if } x_0\leq x_{Q_3}, \\ (x_{03}, \hat y_0(x_{12}) & \mbox{ if } x_0> x_{Q_3}. \end{array} \right. \]

By the mean value theorem, we see from the equations ${\tilde y}_{\varepsilon }(x_{Q_1})={\tilde y}_0(x_{Q_1})=0$ that

(2.16)\begin{align} z_1(x) & = z_1(x)-z_1(x_{Q_1}) \nonumber\\ & =\left\{\tilde y_\varepsilon(\tau)-\tilde y_0(\tau)\right\}\mid^{\tau=x}_{\tau=x_{Q_1}} \nonumber\\ & =\int_{x_{Q_1}}^{x} \Bigg(\frac{-g(\tau )-f(\tau,\tilde y_\varepsilon(\tau), \alpha+\varepsilon) \tilde y_\varepsilon(\tau)}{\tilde y_\varepsilon(\tau)}- \nonumber\\ & \quad\frac{-g(\tau )-f(\tau,\tilde y_0(\tau), \alpha )\tilde y_0(\tau)}{\tilde y_0(\tau)} ~ \Bigg) {\rm d}\tau \nonumber \\ & =\int_{x_{Q_1}}^{x} \Bigg( \frac{-g(\tau )}{\tilde y_\varepsilon(\tau)} + \frac{g(\tau )}{\tilde y_0(\tau)} -f(\tau,\tilde y_\varepsilon(\tau), \alpha+\varepsilon )+f(\tau,\tilde y_0(\tau), \alpha+\varepsilon ) \nonumber\\ & \qquad - f(\tau,\tilde y_0(\tau), \alpha+\varepsilon ) +f(\tau,\tilde y_0(\tau), \alpha ) ~ \Bigg) {\rm d}\tau \nonumber\\ & =\int_{x_{Q_1}}^{x} \Bigg(\frac{-g(\tau )}{\tilde y_\varepsilon(\tau)} + \frac{g(\tau )}{\tilde y_0(\tau)} -\frac{\partial f(\tau,\tilde y^*(\tau), \alpha+\varepsilon )}{\partial y}z_1(\tau) \nonumber\\ & \quad -\frac{\partial f(\tau,\tilde y_0(\tau), \alpha+\varepsilon_1 )}{\partial \alpha}\varepsilon~ \Bigg) {\rm d}\tau \nonumber\\ & =h_1(x)+\int^x_{x_{Q_1}}z_1(\tau)h_2(\tau){\rm d}\tau, \end{align}

where $\tilde y^*$ lies between $\tilde y_0$ and $\tilde y_\varepsilon$, $\varepsilon _1$ lies between $0$ and $\varepsilon$, and

\begin{align*} h_1(x)& :={-}\varepsilon\int_{x_{Q_1}}^{x}\frac{\partial f(\tau,\tilde y_0(\tau), \alpha+\varepsilon_1 )}{\partial \alpha}{\rm d}\tau, \ \ h_2(x):=\frac{g(x )}{\tilde y_0(x)\tilde y_\varepsilon(x)}\\ & \quad-\frac{\partial f(\tau,\tilde y^*(\tau), \alpha+\varepsilon )}{\partial y}. \end{align*}

By (2.16),

\[ h_2(x)z_1(x)=h_2(x)h_1(x)+h_2(x)\int^x_{x_{Q_1}}z_1(\tau)h_2(\tau){\rm d}\tau, \]

indicating that the function $h_3(x):=\int ^x_{x_{Q_1}}z_1(\tau )h_2(\tau ){\rm d}\tau$ satisfies

(2.17)\begin{equation} \frac{dh_3(x)}{{\rm d}x}=z_1(x)h_2(x)=h_2(x)h_3(x)+h_1(x)h_2(x).\end{equation}

By the variation of constant formula, we get from (2.17) that

(2.18)\begin{equation} h_3(x)=\int^x_{x_{Q_1}} ~h_1(\tau)h_2(\tau)\exp{\left\{\int^x_{\tau}h_2(\eta){\rm d}\eta\right\}}{\rm d}\tau.\end{equation}

Since $h_1(x_{Q_1})=0$, we see from (2.7), (2.16) and (2.18) that for all $x\in (x_{Q_1},x_{12})$

(2.19)\begin{equation} \begin{aligned} z_1(x) & = h_1(x)+\int^x_{x_{Q_1}}h_1(\tau)h_2(\tau)\exp{\left\{\int^x_{\tau}h_2(\eta){\rm d}\eta\right\}}{\rm d}\tau \\ & = h_1(x)-\int^x_{x_{Q_1}}h_1(\tau) ~ d\Bigg(\exp{\left\{\int^x_{\tau}h_2(\eta){\rm d}\eta\right\}} \Bigg) \\ & = h_1(x_{Q_1})\exp\left\{\int^x_{x_{Q_1}}h_2(\eta){\rm d}\eta\right\}+\int_{x_{Q_1}}^xh_1'(\tau)\exp\left\{\int^x_{\tau}h_2(\eta){\rm d}\eta\right\}{\rm d}\tau \\ & ={-}\varepsilon\int_{x_{Q_1}}^x\frac{\partial f(\tau,\tilde y_0(\tau), \alpha+\varepsilon_1 )}{\partial \alpha}\exp\left\{\int^x_{\tau}h_2(\eta){\rm d}\eta\right\}{\rm d}\tau<0. \end{aligned} \end{equation}

Similarly, for all $x\in (x_{Q_1},x_{03})$ we can obtain

(2.20)\begin{equation} z_2(x) ={-}\varepsilon\int_{x_{Q_1}}^x\frac{\partial f(\tau,\hat y_0(\tau), \alpha+\varepsilon_2 )}{\partial \alpha}\exp\left\{\int^x_{\tau}h_4(\eta){\rm d}\eta\right\}{\rm d}\tau<0, \end{equation}

where $\varepsilon _2\in (0, \varepsilon )$ and

\[ h_4(x):=\frac{g(x )}{\hat y_0(x)\hat y_\varepsilon(x)}-\frac{\partial f(\tau,\hat y^*(\tau), \alpha+\varepsilon )}{\partial y} \]

for $\hat y^*$ lying between $\hat y_0$ and $\hat y_\varepsilon$.

Construct an energy function

(2.21)\begin{equation} E(x,y)=\int_0^xg(s ){\rm d}s+\frac{y^2}{2}.\end{equation}

Then

(2.22)\begin{equation} \frac{{\rm d}E }{{\rm d}t}={-}f(x,y,\alpha )y^2\end{equation}

restricted on system (2.6). From (2.19) and (2.20) we see that the two points $P_2$ and $P_3$ lie on the two orbit segments $\widehat {Q_1P_1}$ and $\widehat {Q_3Q_1}$ respectively, where $x_{P_2}=x_{Q_2}$, $x_{P_3}=x_{P}$, $y_{P_2}>0$ and $y_{P_3}<0$, as shown in figure 1. From (2.19)–(2.22), and $\int _{\widehat {P_2P_1}}{\rm d}E=E(P_1)-E(P_2)$, we obtain

\begin{align*} \int^0_{ {-}z_1(x_{Q_2}) } ~\frac{ f(x,y,\alpha )y^2}{ g(x )+f(x,y,\alpha )y} {\rm d}y & =\frac{y_{P_1}^2}{2}+\int_0^{x_1} ~ g(s ){\rm d}s -\frac{y_{P_2}^2}{2}-\int_0^{x_{Q_2}}~ g(s ){\rm d}s\\ & = \int_{x_{Q_2}}^{x_1} ~ g(s ){\rm d}s-\frac{z_1^2(x_{Q_2})}{2}, \end{align*}

where $z_1(\cdot )$ is defined in (2.14). Thus, there exist $x^*\in [x_{Q_2},x_1]$ and $y^*\in [0, -z_1(x_{Q_2})]$ such that

\begin{align*} \int^0_{{-}z_1(x_{Q_2})} ~\frac{ f(x,y,\alpha )y^2}{ g(x )+f(x,y,\alpha )y} {\rm d}y & = \frac{ f(x,y^*,\alpha )(y^*)^2}{ g(x )+f(x,y^*,\alpha )y^*} z_1(x_{Q_2}) \\ & = (x_1-x_{Q_2})g(x^* ) -\frac{z_1^2(x_{Q_2})}{2} \end{align*}

by the mean value theorem for integrals. Thus,

(2.23)\begin{equation} x_1-x_{Q_2}=\frac{z_1^2(x_{Q_2})}{2g(x_0)}+o(\varepsilon^2) \end{equation}

because $g(x^*)=g(x_0)+O(\varepsilon )$. From (2.19)–(2.22) and the equality

\[ \int_{\widehat{Q_3P_3}}{\rm d}E=E(P_3)-E(Q_3), \]

we obtain

(2.24)\begin{align} -\int^0_{ z_2(x_0) } ~\frac{ f(x,y,\alpha )y^2}{ g(x )+f(x,y,\alpha )y} {\rm d}y& =\frac{y_{P_3}^2}{2}+\int_0^{x_0} ~ g(s ){\rm d}s -\frac{y_{Q_3}^2}{2}-\int_0^{x_{Q_3}}~ g(s ){\rm d}s\nonumber\\ & = \int_{x_{Q_3}}^{x_0} ~ g(s ){\rm d}s +\frac{{z_2}^2(x_0)}{2}, \end{align}

where $z_2(\cdot )$ is defined in (2.15). It follows from equality (2.24) that

(2.25)\begin{equation} x_{Q_3}-x_0=\frac{z_2^2(x_0)}{2g(x_0 )}+o(\varepsilon^2).\end{equation}

On the other hand, $\Gamma$ is a limit cycle of multiple $k$, i.e.

(2.26)\begin{equation} x_0-x_1={-}a_k{x_1}^k+o({x_1}^k),\end{equation}

as seen in the definition of a limit cycle of multiple $k$ in [Reference Zhang, Ding, Huang and Dong30, Chapter 4.2]. Moreover, $\Gamma$ is externally stable, i.e. $a_k<0$. It follows from (2.23)–(2.26) that

(2.27)\begin{equation} x_{Q_3}-x_{Q_2}= \frac{z_1^2(x_{Q_2})}{2g(x_0 )}+\frac{z_2^2(x_0)}{2g(x_0 )}+o(\varepsilon^2)-a_k {x_1}^k+o({x_1}^k).\end{equation}

Since $\dot {y}=-g(x)<0$ on both the positive $x$-axis and the negative $x$-axis near the origin, we see that $g(x_0)>0$.

Consider the case that $k$ is odd, i.e. $\Gamma$ is stable in both internal and external neighbourhoods of $\Gamma$. In the external neighbourhood of $\Gamma$, we can obtain that $x_{Q_3}-x_{Q_2}>0$ by (2.27) and the inequalities $a_k<0$, $g(x_0)>0$ and $x_1>0$, implying that no limit cycles exist in the external neighbourhood of $\Gamma$ when $\alpha$ increases. In the internal neighbourhood of $\Gamma$, we can also obtain equality (2.27) and the inequalities $a_k<0$, $g(x_0)>0$ but $x_1<0$ from a similar discussion to the external neighbourhood of $\Gamma$. Since the solution of system (2.6) is Lipschitzian, the implicit function theorem is applicable. Thus, from (2.27) with $X_1:={x_1}^k$, we see that equality $x_{Q_3}-x_{Q_2}=0$ has a unique root $X_1= (z_1^2(x_{Q_2})+z_2^2(x_0))/(2a_kg(x_0 )) +o(\varepsilon ^{2})<0$, which is equivalent to

(2.28)\begin{equation} x_1=\left( \frac{z_1^2(x_{Q_2})+z_2^2(x_0)}{2a_kg(x_0 )} \right)^{1/k}+o(\varepsilon^{2/k})<0.\end{equation}

It implies that system (2.6)$|_{\alpha =\alpha _0+\varepsilon }$ produces a stable limit cycle $\Gamma _{\alpha }$ in the internal neighbourhood of $\Gamma$, where the bifurcated limit cycle $\Gamma _{\alpha }$ passes through the point $(x_1, 0)$. Moreover, the stable limit cycle expands inwards (or contracts) monotonically as $\alpha$ increases. From (2.8) and (2.28), we can obtain

\[ \Delta_\alpha(\varepsilon):=x_1-0=x_1, \]

implying that the variational exponent is $2/k$. This proves the results of statement (a).

In the case that $k$ is even, i.e. $\Gamma$ is semi-stable (externally stable but internally unstable), we obtain equality (2.27), $a_k<0$ and $g(x_0)>0$ in the neighbourhood of $\Gamma$, implying that $x_{Q_3}-x_{Q_2}>0$ near the origin and then system (2.6) has no limit cycles in a neighbourhood of $\Gamma$ as $\alpha$ increases.

In order to completely investigate limit cycles bifurcating from semi-stable $\Gamma$ for even $k$, we consider the case that $\alpha :=\alpha _0-\varepsilon$ decreases. By a similar calculation to (2.27), we obtain

(2.29)\begin{equation} x_{Q_3}-x_{Q_2}={-}\frac{z_1^2(x_{P_2})}{2g(x_0 )}-\frac{z_2^2(x_{Q_3})}{2g(x_0 )} +o(\varepsilon^2)-a_k{x_1}^k+o({x_1}^k),\end{equation}

where $g(x_0)>0$ and $a_k<0$. Let $k:=2n$ for $n\in \mathbb {Z}_+$ and $X_2:={x_1}^n$. By the implicit function theorem and (2.29), the equality $x_{Q_3}-x_{Q_2}=0$ has exactly two roots

$X_2= \pm \sqrt { -(z_1^2(x_{P_2})+z_2^2(x_{Q_3}))/(2a_kg(x_0 ))}+o(\varepsilon ^{2})$, which is equivalent to

(2.30)\begin{equation} x_1={\pm}\left(-\frac{z_1^2(x_{P_2})+z_2^2(x_{Q_3})}{2a_kg(x_0 )}\right)^{1/k}+o(\varepsilon^{2/k}).\end{equation}

It follows that two simple limit cycles $\Gamma _{\alpha }^{\pm }$ of system (2.6)$|_{\alpha =\alpha _0-\varepsilon }$ exist in a neighbourhood of $\Gamma$, and the outer limit cycle $\Gamma _{\alpha }^+$ is stable but the inner one $\Gamma _{\alpha }^-$ is unstable. Moreover, $\Gamma _{\alpha }^+$ expands outwards monotonically and $\Gamma _{\alpha }^-$ expands inwards monotonically as $\alpha$ decreases. From (2.8) and (2.30), we can obtain $\Delta _\alpha (\varepsilon ):=x_1-0=x_1$, implying that the variational exponent is $2/k$. This proves the results of statement (b).

In conclusion, for odd $k$ system (2.6) produces a stable (resp. unstable) limit cycle $\Gamma _{\alpha }$ when $\Gamma$ is stable (resp. unstable) in an internal (resp. external) neighbourhood of $\Gamma _{\alpha }$ as $\alpha$ increases. On the other hand, for even $k$ the externally stable and internally unstable (resp. externally stable and internally unstable) $\Gamma$ splits into exact two simple limit cycles $\Gamma _{\alpha }^{\pm }$. Moreover, the outer limit cycle $\Gamma _{\alpha }^+$ is stable (resp. unstable) and expands outwards, but the inner one $\Gamma _{\alpha }^-$ is unstable (resp. stable) and expands inwards as $\alpha$ decreases (resp. increases). However, the limit cycle $\Gamma$ disappears as $\alpha$ varies in the opposite direction. Furthermore, the variational exponent of the new limit cycle is $\varepsilon ^{2/k}$.

Proof. First, we prove property (PR3). Without loss of generality, assume that the origin of system (2.6) is a weak focus as $\alpha =\alpha _0$ and the weak focus is stable, as shown in figure 4. When the vector field of system (2.6) is analytic in a small neighbourhood of the origin, the conclusion holds directly by [Reference Perko23]. Since the origin is a weak focus, we obtain that

\[ \mathcal{P}(x)-x=a_kx^k+o(x^k) \]

on the $x$-axis, where $\mathcal {P}(x)$ is the Poincaré map, $k$ is a positive integer and ${x>0}$ is small. As proven in theorem 2.3, (2.27) still holds, where $P, P_1, Q_1, Q_2, Q_3$, $x_0,x_1, x_{Q_1}, x_{Q_2}, x_{Q_3}$ are defined similarly in figure 1 and the proof of theorem 2.3, as shown in figure 4. Then, $a_k<0$ since the weak focus is stable. For positive integer $k$ and $\alpha =\alpha _0-\varepsilon$, the equality $x_{Q_3}-x_{Q_2}=0$ has a unique positive zero

\[ x_1=\left(-\frac{z_1^2(x_{P_2})+z_2^2(x_{Q_3})}{2a_k g(x_0)} \right)^{1/k}+o(\varepsilon^{2/k})>0, \]

where $\varepsilon >0$ is small and $g(x_0)>0$. When $\alpha =\alpha _0+\varepsilon$, for even $k$ the equality $x_{Q_3}-x_{Q_2}=0$ has no positive zeros. Thus, system (2.6)$|_{\alpha =\alpha _0-\varepsilon }$ produces a new stable limit cycle in a small neighbourhood of the origin. Thus, property (PR3) still holds.

Figure 4. Poincaré map near $O$ in system (2.6).

Next, we prove property (PR4). Assume that system (2.6) exhibits a homoclinic loop $\Gamma _0$ as $\alpha =\alpha _0$. Without loss of generality, we consider that the saddle in the homoclinic loop is the origin and the homoclinic loop is stable. By the sign of the vector field $(y,-g(x)-f(x,y,\alpha )y)$ near the saddle, in a small neighbourhood of the origin one side of the stable manifold and one side of the unstable manifold of the saddle lie in the left-half plane, but the other sides of the two manifolds lie in the right-half plane. Assume that $\Gamma _0$ intersects the positive (or negative) $y$-axis and surrounds neither one stable manifold nor one unstable manifold of the origin other than those in $\Gamma _0$, as shown in figures 5a, b. Notice that $\dot x=y>0$ in the positive $y$-axis and $\dot x=y<0$ in the negative $y$-axis. However, in figures 5a, b the sign of $\dot x$ at the intersection point $\Gamma _0$ and the $y$-axis is opposite by the location of the stable and unstable manifolds of $\Gamma _0$. Thus, system (2.6) has no homoclinic loops which intersect the positive (or negative) $y$-axis and do not surround the other stable and unstable manifolds. In other words, if system (2.6) has a homoclinic loop $\Gamma _0$ which does not surround one stable and one unstable manifolds of $O$ other than those in $\Gamma _0$, then $\Gamma _0$ lies in the left-half (or right-half) plane. It is similar to prove that system (2.6) has no homoclinic loops which intersect the positive and negative $y$-axes, and surround the other two manifolds, as shown in figures 5c, d. Therefore, it is sufficient to prove that the homoclinic loop has one of the configurations shown in figure 6. Otherwise, we apply the transformation $(x,y)\to (-x,-y)$. The case of figure 6b can be treated by a similar mean to the case of figure 6a, where the only difference is that the Poincaré map need to be considered in the outer neighbourhood of the homoclinic loop for the case of figure 6b. Therefore, we only need to discuss the case of figure 6a. Now, consider $\delta _0:=-f(0,0, \alpha _0)\neq 0$. In other words, the sum of the two eigenvalues of the hyperbolic saddle is not equal to zero. Let

\[ \delta:={-}\frac{\partial f(\varphi(t),\phi(t), \alpha_0)}{\partial y}\phi(t)-f(\varphi(t),\phi(t), \alpha_0), \]

where $(x,y)=(\varphi (t),\phi (t))$ represents the homoclinic loop. Consider the perturbed system of (2.6) for $\alpha =\alpha _0+\varepsilon$ with sufficiently small $|\varepsilon |$. By Theorem 3.7 of [Reference Chow, Li and Wang7, Chapter 3], we have the following conclusions:

1. (a) There is exactly one limit cycle bifurcating from the homoclinic loop of system (2.6) when $\delta _0\varepsilon \Delta >0$ (resp. $<0$), which is stable for $\delta _0<0$ and unstable for $\delta _0>0$, where

   \[ \Delta:={-}\int_{-\infty}^{+\infty}e^{-\int_0^t \delta(s){\rm d}s} \frac{\partial f(\varphi(t),\phi(t),\alpha)}{\partial \alpha}|_{\alpha=\alpha_0} \phi^2(t){\rm d}t. \]

2. (b) There are no limit cycles in a small neighbourhood of the homoclinic loop of system (2.6) when $\delta _0\varepsilon \Delta <0$ (resp. $>0$).

Figure 5. Impossible cases of homoclinic loops for system (2.6).

Figure 6. Classification of homoclinic loops for system (2.6).

We consider the case $\delta _0=0$. In this case we cannot apply [Reference Chow, Li and Wang7, Theorem 3.7 of Chapter 3] directly, which only considers the case $\delta _0\ne 0$. We can obtain the Poincaré map again

\[ \mathcal{P}(x)-x=a_kx^k+o(x^k) \]

on the $x$-axis, defined in an interior neighbourhood of any homoclinic loop, where $k$ is a positive integer. As proven in theorem 2.3, (2.27) still holds, where $P$, $P_0$, $P_1$, $Q_1$, $Q_2$, $Q_3$, $x_0,x_1, x_{Q_2}, x_{Q_3}$ are defined similarly in the proof of theorem 2.3, as shown in figure 6. Then, $a_k>0$ since the homoclinic loop is stable. For $\alpha =\alpha _0+\varepsilon$ and positive integer $k$, the equality $x_{Q_3}-x_{Q_2}=0$ has a unique zero

\[ x_1=\tilde{x}_0- \left(\frac{z_1^2(x_{Q_2})+z_2^2(x_0)}{2a_kg(x_0 )} \right)^{1/k} +o(\varepsilon^{2/k})<\tilde{x}_0, \]

where $\tilde {x}_0$ is the abscissa of $P_0$ and $\varepsilon >0$ is small. When $\alpha =\alpha _0-\varepsilon$, for even $k$ and small $\varepsilon _1>0$ the equality $x_{Q_3}-x_{Q_2}=0$ has no zeros in $(\tilde {x}_0-\varepsilon _1, \tilde {x}_0)$. Thus, the proof is completed.

Proof. Without loss of generality, we can assume that the cusp is located at $(0,0)$. Since the vector field of system (2.6) at the cusp is analytic, in a small neighbourhood of $(0,0)$ system (2.6) can be rewritten as

(3.1)\begin{equation} \begin{aligned} \dot x & = y, \\ \dot y & = a_kx^k \big(1+p_1(x )\big) +b_nx^ny\big(1+p_2(x )\big)+y^2p_3(x,y ) \\ & \quad +(\alpha-\alpha_0)y^lp_4(x,y,\alpha ) \end{aligned} \end{equation}

by [Reference Dumortier, Llibre and Artés11, Chapter 3] or [Reference Zhang, Ding, Huang and Dong30, Chapter 2], where integer $k\ge 2$, integers $n,l$ are positive, and functions $p_1(x)$, $p_2(x)$, $p_3(x,y)$ and $p_4(x,y,\alpha )$ are analytic such that $p_1(0)=0$, $p_2(0)=0$ and $y^{l}p_4(x,y,\alpha )\geq 0$. By Theorem 3.5 of [Reference Dumortier, Llibre and Artés11, Chapter 3] or Theorem 7.3 of [Reference Zhang, Ding, Huang and Dong30, Chapter 2], the origin $(0,0)$ of system (3.1) is a cusp when one of the following statements holds:

1. (a) $k=2\,m$ ($m\ge 1$), $l\geq 2$ and $b_n=0$.

2. (b) $k=2\,m$ ($m\ge 1$), $n\geq m$, $l\geq 2$ and $b_n\neq 0$.

Therefore, according to the conditions of this theorem, we can define the Poincaré map on both sides of $\Gamma _0$ when $|\alpha -\alpha _0|\ll 1$. Moreover, system (3.1) is of the form of system (2.6), the remainder research is similar to the proof of theorem 2.3 and we omit it.

Proof. Let $\tilde b=b\xi$ and take $(a, \tilde b, \xi )$ as new parameter. With the transformation $(x,y)\to (x, y+\tilde bx+\xi x^3)$, system (4.1) can be changed into the following form

(4.2)\begin{equation} \dot x=y, \quad \dot y={-}x\left(1-\frac{1}{\sqrt{x^2+a^2}}\right)-(\tilde b+3\xi x^2)y. \end{equation}

We can check that $(y,-x(1-{1}/{\sqrt {x^2+a^2}})-(\tilde b+3\xi x^2)y)$ is a rotated vector field with respect to $\tilde b$ and $\xi$. Moreover, the parameter region $\mathcal {G}$ can be changed into

\begin{align*} & \mathcal{G}_1:=\Bigg\{(a,\tilde b,\xi)\in \mathbb{R}^+{\times}\mathbb{R}\times\mathbb{R}^+: 0< a<\frac{\sqrt{3}}{3}, (2\sqrt{3}a-4)\xi\\ & \qquad <\tilde b<\min\{\xi\varphi(a,\xi),(3a^2-3)\xi\}\Bigg\}. \end{align*}

By theorem 2.3, unstable limit cycles of system (4.2) expand and stable limit cycles contract as either $\tilde b$ or $\xi$ increases. Assume that system (4.2) has at least three limit cycles $\Gamma _1$, $\Gamma _2$ and $\Gamma _3$ surrounding $E_R$ only. Without loss of generality, we assume that $\Gamma _1$, $\Gamma _2$ and $\Gamma _3$ are innermost limit cycles and $\Gamma _i$ lies in the interior region surrounded by $\Gamma _{i+1}$ for $i=1,2$.

In the following we prove by reduction to absurdity that at most two small limit cycles surround $E_R$ only as the parameter belongs to region $\mathcal {G}_1$. First, we exhibit the idea of the proof. We assume that there may exist three small limit cycles surrounding $E_R$ only as the parameter belongs to region $\mathcal {G}_1$. Then we enlarge or lessen those rotated parameters $\tilde b$ and $\xi$ in order, and the parameters lie in a region at last for such case the number of limit cycles has been obtained which is smaller than three. Thus, it induces a contradiction.

By the instability of $E_R$, we can consider the case that $\Gamma _1$ and $\Gamma _3$ are stable and $\Gamma _2$ is unstable. Otherwise, we will vary the rotated parameters to obtain such case. Now we go on the following steps:

Step 1: Lessen $\xi =\xi _1<\xi _0$ till either $\Gamma _1$ and $\Gamma _2$ coincidence or $\Gamma _3$ becomes a homoclinic loop or $\Gamma _3$ becomes a semi-stable limit cycle for fixed $a=a_0$ and $\tilde b=\tilde b_0$. We claim that $(a,\tilde b, \xi )=(a_0,\tilde b_0, \xi _1)\in \mathcal {G}_1$. Otherwise, system (4.1) has at least three small limit cycles surrounding $E_R$ as parameters lie in the other region except $\mathcal {G}_1$ by the rotated properties of vector fields. By theorem 1 of [Reference Chen, Llibre and Tang4], this is a contradiction. We also claim that $\xi _1$ is not small. By the results of [Reference Sun and Liu27], system (4.1) has at most two small limit cycles surrounding $E_R$ for small $\xi _1$, also implying a contradiction.

Step 2: Enlarge $\tilde b=\tilde b_1>\tilde b_0$. We can get three limit cycles, which are still denoted by $\Gamma _1, \Gamma _2$ and $\Gamma _3$ by the rotated properties of vector fields. We continue to enlarge $\tilde b$ till $\Gamma _2$ and $\Gamma _3$ coincide. In other words, we get a stable $\Gamma _1$ and a semi-stable $\tilde \Gamma _{23}$. Moreover, $\tilde \Gamma _{23}$ is internally unstable and externally stable. We claim that $(a,\tilde b, \xi )=(a_0,\tilde b_1, \xi _1)\in \mathcal {G}_1$ and $\tilde b_1$ is not small. Otherwise, we can similarly obtain a contradiction as tep 1.

Step 3: Repeat step 1, i.e. lessen $\xi =\xi _2<\xi _1$ till either $\Gamma _1$ and $\Gamma _2$ coincide or $\Gamma _3$ becomes a homoclinic loop or $\Gamma _3$ becomes a semi-stable limit cycle for fixed $a=a_0$ and $\tilde b=\tilde b_1$. We also claim that $(a,\tilde b, \xi )=(a_0,\tilde b_1, \xi _2)\in \mathcal {G}_1$ and $\xi _2$ is not small. Otherwise, we can obtain the similar contradiction as step 1. Then repeat step 2, step 1, step 2,…, up to $n$ times.

On the other hand, by the proof of theorem 2.3, both the variations of $\tilde b$ and $\xi$ in the aforementioned steps are not sufficiently small (i.e. there exists a positive $d_0$ such that the variations are larger than $d_0$) because none of distances between any two of given $\Gamma _1, \Gamma _2$ and $\Gamma _3$ is sufficiently small. Thus, we stop the aforementioned process in finitely many steps when either $\xi$ is sufficiently close to $0$ or $\tilde b\ge \min \{\xi \varphi (a,\xi ),(3a^2-3)\xi \}$. Then there exists $n_0\in \mathbb {Z}_+$ such that the number of limit cycles is larger than $2$ as $(a,\tilde b, \xi )=(a_0,\tilde b_n, \xi _n)$ for $n>n_0$, contradicting the result ‘a unique limit cycle’ for $\tilde b=\min \{\xi \varphi (a,\xi ),(3a^2-3)\xi \}$ and the result ‘at most $2$ limit cycles’ for small $\xi$. This is a contradiction. Therefore, there are at most two small limit cycles only surrounding $E_R$ as the parameter belongs to region $\mathcal {G}$. By the instability of $E_0$, the interior small limit cycle is stable and the outer one is unstable respectively if there exist exactly two small limit cycles surrounding $E_R$.

Proof. Without loss of generality, we only discuss $\mathcal {G}_1$ since the remainder cases $\mathcal {G}_2$ , $\mathcal {G}_3$, $\mathcal {G}_4$ can be discussed similarly. With the transformation $(x,y)\to (x,y+F(x))$, system (5.1) is changed into the following discontinuous system

(5.2)\begin{equation} \dot x={-}y, \qquad \dot y= g(x)+f(x)y,\end{equation}

where

\[ f(x)= \left\{ \begin{array}{l} t_r,~~\mbox{if}~x> v, \\ t_c,\ \mbox{if}~-u\leq x\leq v, \\ t_l,~~\mbox{if}~x<{-}u. \end{array}\right. \]

We can check that the vector field $(-y, g(x)+f(x)y)$ of system (5.2) is rotated about $t_c, t_l$ and $t_r$.

Assume that system (5.2) has at least three limit cycles as parameters lie in the region $\mathcal {G}_1$ and $(t_c,t_r)=(t_c^0, t_r^0)$ for $h_1< t_c^0<-\varepsilon$ and $0< t_r^0<2$, and $\Gamma _1$, $\Gamma _2$ and $\Gamma _3$ are the three innermost limit cycles, where $\Gamma _2$ lies in the annular region surrounded by inner boundary $\Gamma _1$ and outer boundary $\Gamma _3$. Without loss of generality, we can let $\Gamma _1, \Gamma _3$ be unstable and $\Gamma _2$ be stable. By theorem 2.3, $\Gamma _1, \Gamma _3$ contract and $\Gamma _2$ expands when one of $t_c$, $t_l$, $t_r$ increases. Now we make a similar process to the proof of theorem 4.1 as follows.

Step 1: There is $t_c^1\in (t_c^0,0)$ such that $\Gamma _2, \Gamma _3$ coincide when the other parameters are fixed. We claim that $|t_c^1|$ is not small. Otherwise, system (5.2) has more than one limit cycle as $t_c=0$ by the rotated properties of vector fields. This contradicts proposition 5.2 and what we claimed is true.

Step 2: There is $t_r^1\in (0, t_r^0)$ such that $\Gamma _1, \Gamma _2$ coincide or $\Gamma _3$ becomes a semi-stable limit cycle when the other parameters are fixed. In this step, we claim that $t_r^1$ is not small. Otherwise, system (5.2) has more than one limit cycle as $t_r=0$. By [Reference Llibre, Ordóñez and Ponce17], system (5.2) has at most one limit cycle for $t_r=0$. This is a contradiction. We claim that $t_c^1$ keeps in $(h_1, -\varepsilon )$. Otherwise, assume that $t_c^1\leq h_1$. From proposition 5.3 system (5.2) has no limit cycles if $t_c^1\leq h_1$. This is also a contradiction. Then, we turn to step 1 again.

Finally, for arbitrary small $\varepsilon >0$ there exists $n_0\in \mathbb {Z}_+$ such that $|t_c^n|<\varepsilon$ or $t_r^n<\varepsilon$ as $n>n_0$ step by step. We can find that system (5.2) still has at least three limit cycles as $|t_c^n|$ or $t_r^n$ is small, which contradicts proposition 5.3. Therefore, in the parameter region $\mathcal {G}_1$ for $h_1< t_c<-\varepsilon$ system (5.1) has at most two limit cycles. Applying the rotated properties from theorem 2.3, the continuity of vector fields and propositions 5.1–5.3, there exists a function $t_c=\varphi _1(t_r)$ such that system (5.1) has exactly two limit cycles when $\varphi _1(t_r)< t_c<0$, where the inner limit cycle which only intersects $\Gamma _R$ is unstable and the outer one which intersects $\Gamma _L$ and $\Gamma _R$ is stable; system (5.1) has exactly one semi-stable limit cycle when $t_c=\varphi _1(t_r)$, where the limit cycle which intersects $\Gamma _L$ and $\Gamma _R$ is externally stable; and system (5.1) has no limit cycles when $t_c<\varphi _1(t_r)$. The proof is completed.