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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2024 (2024), No. 57, pp. 1--23.\newline
ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu \newline
DOI: 10.58997/ejde.2024.57}
\thanks{\copyright 2024. This work is licensed under a CC BY 4.0 license.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2024/57\hfil Planar sector-wise linear systems]
{Global dynamics of a special class of planar sector-wise linear systems}


\author[Q.-Q. Han, S.-M. Huan \hfil EJDE-2024/57\hfilneg]
{Qian-Qian Han, Song-Mei Huan}

\address{Qian-Qian Han (corresponding author)\newline
School of Mathematics and Statistics, 
North China University of Water Resources and Electric Power,
Zhengzhou, Henan 450046, China}
\email{hanqianqian@ncwu.edu.cn}

\address{Song-Mei Huan \newline
School of Mathematics and Statistics,
Huazhong University of Science and Technology,
Hubei Key Laboratory of Engineering Modeling and Scientific Computing,
Wuhan, Hubei 430074, China}
\email{smhuan@hust.edu.cn}


\thanks{Submitted April 2, 2024. Published September 30, 2024.}
\subjclass[2020]{34C23, 34C60, 34C37}
\keywords{Sector-wise linear systems; sliding cycle bifurcation; 
\hfill\break\indent  sliding homoclinic bifurcation; heteroclinic cycle; non-regular point}

\begin{abstract}
In this article, we  study the global dynamics of a special class of 
planar sector-wise linear differential systems with two subsystems being 
the same except for their equilibriums. 
Taking the position of one equilibrium as the bifurcation parameter, we 
provide a complete analysis about sliding cycle bifurcation and sliding 
homoclinic bifurcation. Moreover, we obtain the existence of all important 
separatrix orbits and their dependence on the bifurcation parameter, 
including sliding heteroclinic orbit, heteroclinic cycle, limit cycle, 
sliding homoclinic cycle and sliding cycle.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}\label{sec1}

Piecewise smooth dynamical systems arise in applications, 
see \cite{Andronov66,Barnet85,Bazykin98,Bernardo08,Brogliato99,
Coombes12,Dercole07,Filippov88,Henry72,Jeffrey20,Krivan11,Kunze2000,
Maggio00,Novaes15,Ponce18,Stoker50,Tonnelier03}. As the simplest 
piecewise smooth dynamical system, planar piecewise linear systems 
(hereafter referred as PWLSs) with two pieces separated by a straight 
line has been deeply studied in recent years 
\cite{Euzebio15,Freire98,Freire12,Freire14a,Freire14b,Gu2017,Han2010,
Huan21a,Huan21b,oneparameter2003,Llibre17}, mainly due to their use in 
analyzing the dynamics of complex systems locally, the explicit solvability 
of each linear subsystem and their own applications in modeling real systems.

The discontinuity of vector field appeared in piecewise smooth systems 
induces many dynamics that are more complicated and a lot of DIBs having 
new bifurcation mechanisms, see 
\cite{Guardia2011,oneparameter2003,simpson2022,xiong2022}. 
Even in planar PWLSs, we can see some kind of sliding cycles, for example, 
sliding homoclinic cycles \cite{Giannakopoulos2001,Giannakopoulos2002} 
and even double homoclinic loops, which can be found in some engineering 
models such as Valve oscillators \cite{Lefschetz1965}, 
Coulomb friction \cite{Stoker50} and so on.

In recent years, the main research focus about planar PWLSs with a straight 
line separation has been transformed from the study of the existence and 
bifurcation of crossing limit cycles 
(see \cite{Braga13,Du2008, Euzebio15,Giannakopoulos2001,Han2010,
 Huan12b, Huan13, Huan14a, Llibre12,Llibre13,
Llibre15a, Llibre15b,Llibre17,Llibre18,Ponce18}) into the study of some critical 
sliding orbits and their bifurcations. For example, appearance of 
crossing-sliding bifurcation and a double tangency, pseudo-heteroclinic 
bifurcations, pseudo-homoclinic bifurcation, pseudo-Hopf bifurcation and 
so on (\cite{pseudohopf, guo2016, oneparameter2003}).

In addition to this, the perturbation to discontinuity boundary as an 
important common phenomenon should also be taken into consideration, 
since which can lead to many new interesting phenomena and affect both the 
number and type of crossing limit cycles, see 
\cite{Cardin16,Huan12a,Huan19,Huan21a,Llibre19,Novaes15,Zhao19a,Zhao19b,Zhao21}.

Therefore, in this article, we study a special family of planar PWLSs 
with two zones under two considerations, i.e., the discontinuity boundary 
is given by two rays starting from the same point (which are called planar 
sector-wise linear systems in \cite{Huan21b}) and investigate the global 
qualitative dynamics of such systems. Particularly, we have obtained 
explicit dependence on system parameters of the existence, stability and 
number of all kinds of special sliding points for these planar sector-wise 
linear systems in \cite{han2023}. Notice that the discontinuity boundary 
of planar sector-wise linear systems can be written as 
$\Sigma_{\pi/2}$ (see \cite[Theorem 1]{Huan19}), i.e.,
$$
\Sigma_{\pi/2}
=\{(x,y):x\geq0 \text{~and~} y=0\}\cup\{(x,y):x=0\text{~and~}y\geq0\}.
$$
So, here we consider the  planar PWLSs
\begin{equation}\label{eq1}
\dot{\mathbf{x}}=\begin{cases}
\mathbf{F}^+(\mathbf{x})=(F^+_1(\mathbf{x}),F^+_2(\mathbf{x}))^T=\mathbf{A}
 \cdot (\mathbf{x}-\mathbf{x}_e^{+}), & \text{if } 
  \mathbf{x} \in \mathcal{R}_{\pi/2}, \\
\mathbf{F}^-(\mathbf{x})=(F^-_1(\mathbf{x}),F^-_2(\mathbf{x}))^T=\mathbf{A} 
\cdot (\mathbf{x}-\mathbf{x}_e^{-}), & \text{if } 
 \mathbf{x}\in \mathcal{R}_{3\pi/2},
\end{cases}
\end{equation}
where $\mathbf{x}=(x,y)^T\in \mathbb{R}^2$,
$\mathbf{x}^{\pm}_e=(x^{\pm}_e,y^{\pm}_e)^T\in \mathbb{R}^2$, 
$x_e^{+}>0$, $y_e^{+}=-y_e^{-}>0$, 
$\mathbf{A}=[a_{ij}]$ are $2\times2$ real invertible matrices, and
$$
\mathcal{R}_{\pi/2}=\{(x,y):x>0\text{ and } y>0\},\quad 
\mathcal{R}_{3\pi/2}=\{(x,y):xy=0,\text{ or }x<0 \text{ or} y<0\}.
$$
We will investigate the global dynamics of system \eqref{eq1} as the value 
of $x_e^{-}$ changes when both $\mathbf{A}$ have two different real 
eigenvalues and $a_{12}a_{21}\neq0$.

For convenience, we refer to the systems 
$\dot{\mathbf{x}}=\mathbf{A}\cdot(\mathbf{x}-\mathbf{x}^+_e),
\mathbf{x}\in \mathcal{R}_{\pi/2} $ and 
$\dot{\mathbf{x}}=\mathbf{A}\cdot(\mathbf{x}-\mathbf{x}^-_e),
\mathbf{x}\in \mathcal{R}_{3\pi/2} $ as the $\oplus$-system and the 
$\ominus$-system, respectively.

The remainder of this article is organized as follows. 
Some preliminaries are given in Section 2. The main results can be found 
in Section 3. The proofs of the main results and some examples can be 
found in Section 4 and section 5. Our conclusions are provided in Section 6.

\section{Preliminaries}\label{sec2}

Let $\Sigma:H(x,y)=0$ be the discontinuity boundary of the 
 planar PWLSs with two zones
 \begin{equation}\label{eq2}
\mathbf{\dot{x}}= \begin{cases}
 \mathbf{F^+(x)}=(F_1^+(\mathbf{x}),F_2^+(\mathbf{x}))^T,
  & \text{if } \mathbf x\in\Sigma^+=\{(x,y):H(x,y)>0\}, \\
 \mathbf{F^-(x)}=(F_1^-(\mathbf{x}),F_2^-(\mathbf{x}))^T, & \text{if }
  \mathbf x\in\Sigma^-=\{(x,y):H(x,y)<0\}.
 \end{cases}
\end{equation}
It is often considered that the curve $H(x,y)=0$ is smooth, although it
maybe piecewise smooth. Precisely, the points on $\Sigma$ can be divided 
into two kinds: \emph{regular points} of the discontinuity boundary 
(at which the curve have tangent lines) and \emph{non-regular points} of 
the discontinuity boundary.

In this article, the discontinuity boundary we considered in system 
\eqref{eq1} has a unique non-regular point $(0,0)$. So, based on the 
existent definitions about various peculiar sliding points 
(see \cite{Giannakopoulos2001,oneparameter2003,simpson2022,xiong2022}), 
we first introduce some kinds of regular points of $\Sigma$ for system 
\eqref{eq2}, and then study the non-regular point of $\Sigma$ for system 
\eqref{eq1} separately.

\begin{definition}\label{def1} \rm
A regular point $\mathbf{p}\in\Sigma$ is called \emph{a crossing point}
if
$$
\langle\mathbf{F^+(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle
\cdot\langle\mathbf{F^-(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle>0,
$$
where $H_{\mathbf x}(\mathbf{p})$ is the transpose of the gradient of the 
function $H(x,y)$ at $\mathbf{p}$. Denoted by $\Sigma_c$ the set of all 
crossing points. Then $\Sigma_c$ is the \emph{crossing set} of 
system \eqref{eq2}.
\end{definition}

\begin{definition}\label{def2} \rm
A regular point $\mathbf{p}\in\Sigma$ is called \emph{a sliding point} if
$$
\langle\mathbf{F^+(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle
\cdot\langle\mathbf{F^-(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle<0.
$$
We denoted by $\Sigma_s$ the set of all sliding points. 
Then $\Sigma_s$ is the \emph{sliding set} of system \eqref{eq2}.
Moreover, $\Sigma_s$ is \emph{attractive (repulsive)} if
$$
\langle\mathbf{F^+(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle<0(>0),
\quad  \langle\mathbf{F^-(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle>0(<0),
\quad\textbf p\in\Sigma_s.
$$
In addition to this, the \emph{sliding vector field} $\mathbf{F_s}$ on 
$\Sigma_s$ is often defined by using the Filippov convex method 
(\cite{Filippov88,Kunze2000}) to be
\begin{equation}\label{slidingfield}
\mathbf{F_s}(\mathbf{p})=\frac{\langle\mathbf{F^+(\mathbf{p})},
H_{\mathbf x}(\mathbf{p})\rangle\mathbf{F^-}(\mathbf{p})
-\langle\mathbf{F^-(\mathbf{p})},H_{\mathbf x}
(\mathbf{p})\rangle\mathbf{F^+}(\mathbf{p})}
{\langle\mathbf{F^+(\mathbf{p})-F^-(\mathbf{p})},H_{\mathbf x}
(\mathbf{p})\rangle},\quad\mathbf{p}\in\Sigma_s,
\end{equation}
and a point $\mathbf{p}\in\Sigma$ satisfying 
$\mathbf{F_s}(\mathbf{p})=\mathbf0$ is called a \emph{pseudo-equilibrium}.
\end{definition}

When $\mathbf{p}\in \Sigma_{s}$ is a pseudo-equilibrium, $\mathbf{p}$ may 
show properties of a node, a saddle or a focus according to the stability 
of $\mathbf{p}$ inside $\Sigma_s$ and the attractivity of $\Sigma_s$, 
which induces the following definitions.

\begin{definition}\label{def3} \rm
Let $\mathbf{p}\in \Sigma_{s}$ be an isolated pseudo-equilibrium. 
Then $\mathbf{p}$ will be an \emph{unstable (a stable) pseudo-node} of 
system \eqref{eq2} if $\Sigma_s$ is repulsive (attractive) and $\mathbf{p}$ 
is unstable (stable) inside $\Sigma_s$, a \emph{pseudo-saddle} 
if one of the followings is true: (i) $\Sigma_s$ is attractive and 
$\mathbf{p}$ is unstable inside $\Sigma_s$; 
(ii) $\Sigma_s$ is repulsive and $\mathbf{p}$ is stable inside $\Sigma_s$. 
Moreover, $\mathbf{p}$ will be a pseudo-saddle-node if $\mathbf{p}$ is 
half-stable inside $\Sigma_s$.
\end{definition}

\begin{definition}\label{def4} \rm
The point $\mathbf{p}\in \bar{\Sigma}$ is called \emph{a sliding boundary 
point} if
$$
\langle\mathbf{F^+(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle
\cdot\langle\mathbf{F^-(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle=0.
$$
The set of all sliding boundary points, denoted by $\Sigma_{sb}$, is called 
the \emph{sliding boundary set} of system \eqref{eq1}.
\end{definition}

\begin{definition}\label{def5} \rm
Let $\mathbf{p}\in\Sigma_{sb}$, we call $\mathbf{p}$ a \emph{boundary 
equilibrium} of system \eqref{eq1} if
$$
\mathbf{F^+}(\mathbf{p})=\mathbf{0}\text{ or }\mathbf{F^-}(\mathbf{p})
=\mathbf{0}.
$$
More specifically, $\mathbf{p}$ is called a \emph{double boundary 
equilibrium} when $\mathbf{F^+}(\mathbf{p})=\mathbf{F^-}(\mathbf{p})
=\mathbf{0}$, and a \emph{unilateral boundary equilibrium} induced by the 
$\ominus-$system ($\oplus-$system) if
\begin{gather*}
\mathbf{F^-}(\mathbf{p})=\mathbf{0},\quad
\mathbf{F^+}(\mathbf{p})\neq\mathbf{0}, \\
(\mathbf{F^+}(\mathbf{p})=\mathbf{0},\quad
\mathbf{F^-}(\mathbf{p})\neq\mathbf{0}).
\end{gather*}
\end{definition}

Let $\mathbf{p}_0=(0,0)$ and 
$\mathbf{F_s}(\mathbf{p}),\mathbf{p}\in \Sigma_s$ be the unique non-regular 
point of the discontinuity boundary $\Sigma$ and the sliding vector field 
of system \eqref{eq1}, respectively. Notice that we did not discuss the 
definition of the sliding field $\mathbf{F_s}$ at $\mathbf{p}_0$. 
In fact, we even do not care about whether or not this point is a sliding 
point, which we think is not necessary at least now. However, there is no 
doubt that the dynamics around $\mathbf{p}_0$ play an important role in 
studying the global dynamics. Therefore, we need to consider 
$\mathbf{F_s}$ around this special point.

Suppose that the sliding field $\mathbf{F_s}$ is well defined on 
$I_b^y\cup I_a^x$ with
$$
I_b^y=\{(0,y):y\in(0,b)\},\quad I_a^x=\{(x,0):x\in(0,a)\}.
$$
Then we call $\mathbf{p}_0$ a \emph{non-regular boundary saddle} of 
system \eqref{eq1} if there exist $a>0,~b>0$ such that
$$
\langle \mathbf{F_s}(\mathbf{p}_1),(0,1)\rangle
\cdot\langle \mathbf{F_s}(\mathbf{p}_2),(1,0)\rangle<0,\quad \forall
\mathbf{p}_1\in I_b^y,~\forall \mathbf{p}_2\in I_a^x,
$$
and a \emph{non-regular boundary source (sink)} if there exist 
$a>0$ and $b>0$ such that
\[
\langle \mathbf{F_s}(\mathbf{p}_1),(0,1)\rangle 
\cdot \langle \mathbf{F_s}(\mathbf{p}_2),(1,0)\rangle>0, \quad
\langle\mathbf{F_s}(\mathbf{p}_1),(0,1)\rangle>0 \quad (<0),
\]
for all $\mathbf{p}_1\in I_b^y$ and all $\mathbf{p}_2\in I_a^x$.


In addition to this, according to \cite{oneparameter2003,pi2013a,pi2013b}, 
we obtain that an orbit which has a common segment with the sliding set is 
called a \emph{sliding orbit}, and a closed orbit with sliding motion is 
called a \emph{sliding cycle}. Furthermore, a \emph{sliding-zero cycle} 
is a closed orbit intersecting $\Sigma$ at an end point of the sliding region.
A \emph{sliding heteroclinic orbit (or cycle)} 
is a hereroclinic one with sliding motion. 
A \emph{sliding homoclinic orbit (or cycle)} is a homoclinic one with 
sliding motion.

Moreover, for a piecewise smooth dynamical system with a parameter $\mu$, 
a \emph{sliding cycle bifurcation} happens at $\mu=\mu_0$ means that 
there is a sliding-zero cycle when $\mu=\mu_0$, while for 
$0<|\mu-\mu_0|\ll 1$ the sliding-zero cycle disappears, and instead 
there appear a limit cycle and a sliding cycle when $\mu$ crosses $\mu_0$ 
in opposite directions, respectively. And a \emph{sliding homoclinic 
bifurcation} of the system at $\mu=\mu_0$ means the system has a sliding 
homoclinic cycle when $\mu=\mu_0$, while for $0<|\mu-\mu_0|\ll 1$ the 
sliding homoclinic cycle disappears, instead there appears a sliding cycle 
when $\mu$ varies in one direction and neither sliding homoclinic orbits 
nor sliding cycles when $\mu$ varies in another direction.

Finally,  to give our main results, we introduce some notation for 
the planar linear system
\begin{equation} \label{eq3}
\dot{\mathbf{x}}=\mathbf{A}\cdot (\mathbf{x}-\mathbf{x}^\pm_e),
\end{equation}
where $\det(\mathbf{A})<0$ (i.e., $\mathbf{A}$ has two different real 
eigenvalues having opposite signs: $\lambda_1>0>\lambda_2$) and 
$[tr(\mathbf{A})]^2>4 \det(\mathbf{A})$. Then
\[
\lambda_1=\frac{tr(\mathbf{A})+\sqrt{[tr(\mathbf{A})]^2
-4 \det(\mathbf{A})}}{2},\quad
\lambda_2=\frac{tr(\mathbf{A})-\sqrt{[tr(\mathbf{A})]^2
-4 \det(\mathbf{A})}}{2}.
\]
Moreover, denote the invariant manifolds of $\mathbf{x}^\pm_e$ by 
$l_1^{\pm}, \ l_2^{\pm}$. Let $k_1^{\pm},k_2^{\pm}$ be the slopes of 
$l_1^{\pm},l_2^{\pm}$, $\mathbf{y}_{m1}^{\pm}=(0,y_{m1}^{\pm})$, 
$\mathbf{y}_{m2}^{\pm}=(0,y_{m2}^{\pm})$ and 
$\mathbf{x}_{m1}^{\pm}=(x_{m1}^{\pm}, 0)$, 
$\mathbf{x}_{m2}^{\pm}=(x_{m2}^{\pm}, 0)$ be the intersections of 
$l_1^{\pm}$ and $l_2^{\pm}$ with the $y$-axis and the $x$-axis, respectively.
In addition to this, let $\mathbf{y}^\pm_t=(0,y^\pm_t)$ and 
$\mathbf{x}^\pm_t=(x^\pm_t,0)$ be the contact points at which the 
trajectories of system \eqref{eq3} being tangent to the $y$-axis and the
 $x$-axis, respectively. Then by simple calculations, it follows that
\begin{gather}\label{k}
k_1^{\pm}=\frac{\lambda_1-a_{11}}{a_{12}},\quad 
k_2^{\pm} =\frac{\lambda_2-a_{11}}{a_{12}}, \\
\label{qiedian}
x^\pm_t=x^\pm_e+\frac{a_{22}}{a_{21}}y^\pm_e,\quad 
y^\pm_t=y^\pm_e+\frac{a_{11}}{a_{12}}x^\pm_e,
\end{gather}
\begin{equation}\label{fundation1}
\begin{gathered}
x_{m1}^{\pm}=x_e^{\pm}-\frac{a_{12}}{\lambda_1-a_{11}}y_e^{\pm}
 =x_t^{\pm}-\frac{\lambda_1}{a_{21}}y_e^{\pm}, \\
x_{m2}^{\pm}=x_e^{\pm}-\frac{a_{12}}{\lambda_2-a_{11}}y_e^{\pm}
 =x_t^{\pm}-\frac{\lambda_2}{a_{21}}y_e^{\pm},\\
y_{m1}^{\pm}=y_e^{\pm}-\frac{\lambda_1-a_{11}}{a_{12}}x_e^{\pm}
 =y_t^{\pm}-\frac{\lambda_1}{a_{12}}x_e^{\pm}, \\
y_{m2}^{\pm}=y_e^{\pm}-\frac{\lambda_2-a_{11}}{a_{12}}x_e^{\pm}
 =y_t^{\pm}-\frac{\lambda_2}{a_{12}}x_e^{\pm}.
\end{gathered}
\end{equation}

\begin{remark}\label{invariant manifold} \rm
It should be noted that, by system \eqref{eq1} with $\lambda_1>0>\lambda_2$,
 we obtain
\begin{gather*}
F_1{(0,y_{m1}^{\pm})}=-\lambda_1x_e^{\pm}, \quad
F_1{(0,y_{m2}^{\pm})}=-\lambda_2x_e^{\pm}, \\
F_2{(x_{m1}^{\pm},0)}=-\lambda_1y_e^{\pm}, \quad
F_2{(x_{m2}^{\pm},0)}=-\lambda_2y_e^{\pm},
\end{gather*}
which means $l_1^{\pm}$ and $l_2^{\pm}$ are the unstable and stable 
invariant manifolds of $\mathbf{x}_e^{\pm}$, respectively.
\end{remark}

\section{Main results}

In this paper, we give the qualitative analysis of system \eqref{eq1} 
under the following set of conditions (H):
\begin{equation}\label{H}
\begin{gathered}
 \det(\mathbf{A})=a_{11}a_{22}-a_{12}a_{21}<0,\\
a_{12}a_{21}\neq 0, \quad  a_{21}>0, \quad a_{22}>0,\\
x_e^{+}>0, \quad y_e^{+}=-y_e^{-}>0,
 \end{gathered}
\end{equation}
for  which, we have the following considerations.

First, in this article we  want to investigate the dynamics of 
system \eqref{eq1} related to sliding homoclinic/heteroclinic orbits, 
so we consider the case  $\det(\mathbf{A})<0$. The eigenvalues of 
$\mathbf{A}$ will  be denoted by $\lambda_1>0>\lambda_2$.
Moreover, since the discontinuity boundary of system \eqref{eq1} is given 
by the positive $x$-axis and the positive $y$-axis, we use 
$a_{21}a_{12}\neq 0$ to guarantee that the section return maps can be 
well defined.

Notice that the first condition in \eqref{H} implies $a_{11}a_{22}\neq 0$. 
Moreover, the signs of $a_{12}$ and $a_{21}$ will be changed at the same 
time under the transformation $(x,y,t)\to (-x+2x_e^{+},y,t)$ with the 
signs of $a_{11}$ and $a_{22}$ unchanged. Similarly, the signs of 
$a_{11}$ and $a_{22}$ will be changed simultaneously under the
transformation $(x,y,t)\to (-x+2x_e^{+},y,-t)$ with the signs of 
$a_{12}$ and $a_{21}$ unchanged. Therefore, when we need to analyze 
the section return maps defined on the discontinuity boundary, 
we only need to discuss the case when $a_{21}>0, \ a_{22}>0$. 
More precisely, we only consider the following two cases:
\begin{gather*}
\Omega_1=\{\mathbf{A}: a_{21}>0, a_{22}>0, a_{12}a_{11}>0\}, \\
\Omega_2=\{\mathbf{A}: a_{21}>0, a_{22}>0, a_{12}a_{11}<0\}.
\end{gather*}

Finally, because the global dynamics of piecewise smooth 
systems is so complex that we put $x_e^{+}>0, \ y_e^{+}=-y_e^{-}>0$ and
 only left $x_e^{-}$ as a bifurcation parameter.

To state our main results about the global dynamics of system \eqref{eq1} 
under \eqref{H} by means of providing the existence of all important 
separatrix in the state space, we first give the following propositions to 
clarify the existence of sliding set and pseudo-equilibrium points, 
and the properties of the unique non-regular $\mathbf{p}_0=(0,0)$.


\begin{proposition}\label{pro1}
Suppose that condition \eqref{H} holds and 
$\mathbf{A}\in\Omega_1\cup \Omega_2$. Let
$$
\mu_1:= \frac{a_{22}}{a_{21}}y_e^{+},\quad
\mu_2:= \frac{a_{12}}{a_{11}}y_e^{+}.
$$
Then the sliding set $\Sigma_{s}$ of system \eqref{eq1} is 
$\Sigma_{s}=\Sigma^{x}_{s}\cup \Sigma^{y}_{s}$ with
\begin{equation}\label{sx}
\begin{split}
\Sigma^{x}_{s}
&=\{(x,0): x>0, x\in (\min\{x_t^{-},x_t^{+}\},
 \max\{x_t^{-},x_t^{+}\})\},\\
&=\begin{cases}
\{(x,0): 0\leq x<x_t^{+}\}, & \text{if }  x_e^{-}<\mu_1, \\
\{(x,0): x_t^{-}<x<x_t^{+}\}, 
 &\text{if }  \mu_1\leq x_e^{-}<x_e^{+}+2\mu_1, \\
\emptyset, & \text{if }  x_e^{-}=x_e^{+}+2\mu_1, \\
\{(x,0): x_t^{+}<x<x_t^{-}\}, & \text{if }  x_e^{-}>x_e^{+}+2\mu_1,
\end{cases}
\end{split}
\end{equation}
and
\begin{equation}\label{sy}
\Sigma^{y}_{s}=\{(0,y): y>0, y\in (\min\{y_t^{-},y_t^{+}\}, 
\max\{y_t^{-},y_t^{+}\})\}.
\end{equation}
About $\Sigma^{y}_{s}$, we have the following statements:
\begin{itemize}
\item[(a1)]
When $\mathbf{A}\in \Omega_1$, we have
\begin{equation}\label{sy1}
\Sigma_{s}^{y}=\begin{cases}
\{(0,y): 0\leq y<y_t^{+}\}, &\text{if }  x_e^{-}<\mu_2, \\
\{(0,y): y_t^{-}<y<y_t^{+}\}, &\text{if } 
 \mu_2\leq x_e^{-}<x_e^{+}+2\mu_2, \\
\emptyset, &\text{if }  x_e^{-}=x_e^{+}+2\mu_2, \\
\{(0,y): y_t^{+}<y<y_t^{-}\}, &\text{if }  x_e^{-}>x_e^{+}+2\mu_2.
\end{cases}
\end{equation}

\item[(a2)]
When $\mathbf{A}\in \Omega_2$ with $x_e^{+}+\mu_2>0$, we have
\begin{equation}\label{sy2}
\Sigma_{s}^{y}=\begin{cases}
\{(0,y): 0\leq y<y_t^{-}\}, &\text{if }  x_e^{-}<\mu_2, \\
\emptyset, &\text{if }  x_e^{-}\geq \mu_2.
\end{cases}
\end{equation}

\item[(a3)]
When $\mathbf{A}\in \Omega_2$ with $x_e^{+}+\mu_2\leq 0$, we have
\begin{equation}\label{sy3}
\Sigma_{s}^{y}=\begin{cases}
\{(0,y): y_t^{+}<y<y_t^{-}\}, &\text{if }  x_e^{-}<x_e^{+}+2\mu_2, \\
\emptyset, &\text{if } x_e^{-}=x_e^{+}+2\mu_2, \\
\{(0,y): y_t^{-}<y<y_t^{+}\}, &\text{if }  x_e^{+}+2\mu_2<x_e^{-}\leq \mu_2, \\
\{(0,y): 0\leq y<y_t^{+}\}, &\text{if }  x_e^{-}>\mu_2.
\end{cases}
\end{equation}
\end{itemize}
\end{proposition}

\begin{proof}
Note that the discontinuity boundary of system \eqref{eq1} is composed of 
the origin, the positive $x$-axis and the positive $y$-axis. 
So according to Definition \ref{def2}, when $\mathbf x=(x,0)$ is on the 
positive $x$-axis, we have
$$
\langle\mathbf{F^+(\mathbf{p})},H_{\mathbf x}(\mathbf{p})
\rangle\cdot\langle\mathbf{F^-(\mathbf{p})},H_{\mathbf x}
(\mathbf{p})\rangle=F_2^{+}(x,0)F_2^{-}(x,0)=a_{21}^2(x-x_t^{+})(x-x_t^{-}),
$$
and when $\mathbf x=(0,y)$ is on the positive $y$-axis, we have
$$
\langle\mathbf{F^+(\mathbf{p})},H_{\mathbf x}(\mathbf{p})\rangle
\cdot\langle\mathbf{F^-(\mathbf{p})},
H_{\mathbf x}(\mathbf{p})\rangle=F_1^{+}(0,y)F_1^{-}(0,y)
=a_{12}^2(y-y_t^{+})(y-y_t^{-}).
$$
Moreover, when $\mathbf{A}\in \Omega_1\cup \Omega_2$, we have 
$x_t^{+}>0$ and $0<\mu_1<2\mu_1+x_e^{+}$ under the condition \eqref{H}. 
And by easy computations, it follows that
\begin{equation}\label{miu1j}
x_e^{-}=\mu_1\Leftrightarrow x_t^{-}=0,\quad
 x_e^{-}=x_e^{+}+2\mu_1\Leftrightarrow x_t^{-}=x_t^{+}.
\end{equation}
It is obvious that $x_t^{-}$ increases with respect to $x_e^{-}$ 
by \eqref{qiedian}. Then \eqref{sx} and \eqref{sy} can be obtained directly.

In addition, by simple calculations, we obtain
\begin{equation}\label{miu2j}
x_e^{-}=\mu_2\Leftrightarrow y_t^{-}=0, \quad
x_e^{-}=x_e^{+}+2\mu_2 \Leftrightarrow y_t^{-}=y_t^{+}.
\end{equation}
On the one hand, when $\mathbf{A}\in \Omega_1$, by \eqref{qiedian}, 
we have $y_t^{+}>0$, and $y_t^{-}$ is a linear monotone increasing 
function with respect to $x_e^{-}$. Then \eqref{sy1} can be obtained easily. 
On the other hand, when $\mathbf{A}\in \Omega_2$, by \eqref{qiedian}, 
it follows that
\begin{equation}\label{cyt}
y_t^{+}\begin{cases}
<0, &\text{if }  x_e^{+}+\mu_2>0, \\
\geq 0, &\text{if }  x_e^{+}+\mu_2\leq 0.
\end{cases}
\end{equation}
And $y_t^{-}$ is a linear monotone decreasing function with respect to 
$x_e^{-}$. Then \eqref{sy2} and \eqref{sy3} can be proved directly.
\end{proof}

Based on the results about sliding sets given in Proposition \ref{pro1}, 
we can now write out the sliding vector field of system \eqref{eq1} 
defined by using the Filippov convex method as follows:
\begin{equation}\label{slidingfield}
\mathbf{F}_s(\mathbf{p})=\begin{cases}
 \frac{1}{F_2^+(\mathbf{p})- F_2^-(\mathbf{p})}
 \begin{pmatrix}
 F_2^+(\mathbf{p})F_1^-(\mathbf{p})- F_2^-(\mathbf{p})F_1^+(\mathbf{p}) \\
 0 
 \end{pmatrix}
, &\text{if } \mathbf{p}\in\Sigma_s^x;\\[4pt]
 \frac{1}{F_1^+(\mathbf{p})- F_1^-(\mathbf{p})}
 \begin{pmatrix}
 0 \\
 F_1^+(\mathbf{p})F_2^-(\mathbf{p})- F_1^-(\mathbf{p})F_2^+(\mathbf{p})
\end{pmatrix}, & \text{if }\mathbf{p}\in\Sigma_s^y.
 \end{cases}
\end{equation}
Then the existence, number and stability of pseudo-equilibrium points 
(i.e., the equilibrium points of $\mathbf{F}_s$) of system \eqref{eq1} 
are provided in the next proposition.

\begin{proposition}\label{pro2}
Suppose that condition \eqref{H} holds and 
$\mathbf{A}\in\Omega_1\cup \Omega_2$. There exists at most one 
pseudo-equilibrium point in system \eqref{eq1}. More precisely, 
about the pseudo-equilibrium point in $\Sigma_{s}^{x}$, we have:
\begin{itemize}
  \item[(a1)] If $x_e^-\geq -x_e^+$ and $x_e^-\neq x_e^{+}+2\mu_1$, 
  there exists a unique pseudo-equilibrium point $(x^\ast,0)$, which is a 
  stable (an unstable) pseudo-node when 
  $x_e^-<x_e^{+}+2\mu_1 \ (x_e^->x_e^{+}+2\mu_1)$;
  
  \item[(a2)] If $x_e^-<-x_e^+$ or $x_e^-=x_e^{+}+2\mu_1$, there exists 
  no pseudo-equilibrium point in $\Sigma_{s}^{x}$.
\end{itemize}
About the pseudo-equilibrium point in $\Sigma_{s}^{y}$, we have:
\begin{itemize}
  \item[(b1)] If $x_e^-\leq -x_e^+$ and $x_e^-\neq x_e^{+}+2\mu_2$, there exists a unique pseudo-equilibrium point $(0,y^\ast)$, which is a stable (an unstable) pseudo-node when $a_{11}[x_e^--(x_e^{+}+2\mu_2)]<0 \ (a_{11}[x_e^--(x_e^{+}+2\mu_2)]>0)$;
  \item[(b2)] If $x_e^->-x_e^+$ or $x_e^-=x_e^{+}+2\mu_2$, there exists no pseudo-equilibrium point in $\Sigma_{s}^{y}$.
\end{itemize}
\end{proposition}

\begin{proof}
By the definition of a pseudo-equilibrium given in Definition \ref{def2} 
and \eqref{slidingfield}, we know that $\mathbf{p}_{ex}=(x^\ast,0)$ is a 
pseudo-equilibrium of system \eqref{eq1} if and only if
$\mathbf{p}_{ex}\in \Sigma_{s}^{x}$ and
\begin{equation}\label{prop-eq1}
\begin{aligned}
&F_2^+(x^\ast,0)F_1^-(x^\ast,0)- F_2^-(x^\ast,0)F_1^+(x^\ast,0)\\
&=-2\det(\mathbf{A})\cdot y_e^+\cdot(x^\ast-\frac{x_e^-+x_e^+}{2})
=0.
\end{aligned}
\end{equation}
Similarly, $\mathbf{p}_{ey}=(0,y^\ast)$ is a pseudo-equilibrium of 
system \eqref{eq1} if and only if $\mathbf{p}_{ey}\in \Sigma_{s}^{y}$ and
\begin{equation}\label{prop-eq2}
\begin{split}
&F_1^+(0,y^\ast)F_2^-(0,y^\ast)- F_1^-(0,y^\ast)F_2^+(0,y^\ast)\\
&=-\det(\mathbf{A}) \left[(x_e^+-x_e^-)y^\ast+(x_e^++x_e^-)y_e^+\right]=0.
\end{split}
\end{equation}
Then we obtain $x^\ast=\frac{x_e^-+x_e^+}{2}$ and 
$y^\ast=\frac{x_e^-+x_e^+}{x_e^--x_e^+}y_e^+$ directly from \eqref{prop-eq1} 
and \eqref{prop-eq2}. In addition to this, we have
$$
x^{\ast}=\frac{x_t^{+}+x_t^{-}}{2}=\frac{x_{m1}^{+}+x_{m1}^{-}}{2}
=\frac{x_{m2}^{+}+x_{m2}^{-}}{2},
$$
which means that $\mathbf{p}_{ex}$ is the midpoint of $\mathbf{x}_e^{\pm}$, 
$\mathbf{x}_t^{\pm}$, $\mathbf{x}_{m1}^{\pm}$ and $\mathbf{x}_{m2}^{\pm}$ 
at the same time. And
$$
\lim_{x_e^{-}\to -\infty}y^{\ast}=y_e^{+}, \quad
\lim_{x_e^{-}\to (x_e^{+})^{-}}y^{\ast} =-\infty,\quad
\lim_{x_e^{-}\to (x_e^{+})^{+}}y^{\ast}=+\infty,\quad
\lim_{x_e^{-}\to +\infty}y^{\ast}=y_e^{+},
$$
this implies that $y^{\ast}$ decreases with respect to $x_e^{-}$, 
which is illustrated in Figure \ref{fig1}.

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1}
\end{center}
\caption{The changing of $y^{\ast}$ with respect to $x_e^{-}$}
\label{fig1}
\end{figure}

On the one hand, it is easy to see that $x^{\ast}\geq 0$ when 
$x_e^{-}\geq -x_e^{+}$, and $x^{\ast}=0$ if and only if
$x_e^{-}=-x_e^{+}$. Since $x_e^{-}=-x_e^{+}<0<\mu_1$, which means that 
$\mathbf{p}_{ex}=(0,0)\in \Sigma_{s}^{x}$ is a pseudo-equilibrium point 
by \eqref{sx}. Furthermore, to prove whether 
$\mathbf{p}_{ex}=(x^{\ast},0)\in \Sigma_{s}^{x}$ as $x^{\ast}>0$, we need 
the following calculations:
$$
x^{\ast}-x_t^{-}=\frac{1}{2}(x_e^{+}+2\mu_1-x_e^{-}), \quad
x^{\ast}-x_t^{+}=\frac{1}{2}(x_e^{-}-2\mu_1-x_e^{+}),
$$
which implies that
\begin{gather*}
x_t^{-}<x^{\ast}<x_t^{+},   \quad\text{if } x_e^{-}<x_e^{+}+2\mu_1;\\
x_t^{+}<x^{\ast}<x_t^{-},   \quad\text{if } x_e^{-}>x_e^{+}+2\mu_1.
\end{gather*}
Then by \eqref{sx}, we obtain 
$\mathbf{p}_{ex}=(x^{\ast},0)\in \Sigma_{s}^{x}$, that is
$\mathbf{p}_{ex}=(x^{\ast},0)$ is a pseudo-equilibrium point of 
system \eqref{eq1} only as $x_e^{-}>-x_e^{+}$ and 
$x_e^{-}\neq x_e^{+}+2\mu_1$.

On the other hand, according to Figure \ref{fig1}, it is easy to see that 
$y^{\ast}>0$ when $x_e^{-}\in(-\infty,-x_e^{+})\cup (x_e^{+},+\infty)$, 
and $y^{\ast}=0$ if and only if $x_e^{-}=-x_e^{+}$. Firstly, when 
$x_e^{-}=-x_e^{+}$, simple computation shows that 
$x_e^{-}-\mu_2=-x_e^{+}-\mu_2=-\frac{a_{12}}{a_{11}}y_t^{+}$. 
Then we obtain that $\mathbf{p}_{ey}=(0,0)\in \Sigma_{s}^{y}$ as long as 
$x_e^{+}+\mu_2\neq 0$ by \eqref{sy1}-\eqref{sy3}. 
Furthermore, to prove whether $\mathbf{p}_{ey}=(0,y^{\ast})\in \Sigma_{s}^{y}$
as $x_e^{-}\in(-\infty,-x_e^{+})\cup (x_e^{+},+\infty)$, we need the 
following statements:
\begin{gather*}
y^{\ast}-y_t^{-}=\frac{x_e^{-}}{x_e^{-}-x_e^{+}}
 \frac{a_{11}}{a_{12}} (x_e^{+}+2\mu_2-x_e^{-}),\\
y^{\ast}-y_t^{+}=\frac{x_e^{+}}{x_e^{-}-x_e^{+}}
 \frac{a_{11}}{a_{12}} (x_e^{+}+2\mu_2-x_e^{-}), \\
x_e^{+}+2\mu_2>0, \quad\text{if } \mathbf{A}\in\Omega_1, \\
x_e^{+}+2\mu_2<0, \quad\text{if } \mathbf{A}\in\Omega_2,
\end{gather*}
which implies that as $\mathbf{A}\in\Omega_1$ and 
$x_e^{-}<x_e^{+}$ ($\mathbf{A}\in\Omega_2$ and $x_e^{-}>x_e^{+}$), 
we have $x_e^{-}<x_e^{+}+2\mu_2$ ($x_e^{-}>x_e^{+}+2\mu_2$), then we 
obtain that
$$
y_t^{-}<y^{\ast}<y_t^{+} \quad  (y^{\ast}>\max\{y_t^{-},y_t^{+}\}).
$$
As $\mathbf{A}\in\Omega_1$ and $x_e^{-}>x_e^{+}$ ($\mathbf{A}\in\Omega_2$ and 
$x_e^{-}<-x_e^{+}$), we obtain that
\begin{gather*}
y^{\ast}<\min\{y_t^{-},y_t^{+}\}  (y_t^{-}<y^{\ast}<y_t^{+}),  
 \quad\text{if } x_e^{-}>x_e^{+}+2\mu_2; \\
y^{\ast}>\max\{y_t^{-},y_t^{+}\}  (y_t^{+}<y^{\ast}<y_t^{-}),  
 \quad\text{if } x_e^{-}<x_e^{+}+2\mu_2.
\end{gather*}
Then by \eqref{sy1}-\eqref{sy3}, it follows that 
$\mathbf{p}_{ey}=(0,y^{\ast})\in \Sigma_{s}^{y}$, that is 
$\mathbf{p}_{ey}=(0,y^{\ast})$ is a pseudo-equilibrium point of system 
\eqref{eq1} only as $x_e^{-}<-x_e^{+}$ and $x_e^{-}\neq x_e^{+}+2\mu_2$.

Finally, we discuss the stability of $\mathbf{p}_{ex}$ and $\mathbf{p}_{ey}$.
Let $f(x)=2\det(\mathbf{A})\cdot y_e^+\cdot(x-\frac{x_e^-+x_e^+}{2})$ and 
$g(x)=\det(\mathbf{A})\cdot\left[(x_e^+-x_e^-)y+(x_e^++x_e^-)y_e^+\right]$. 
Then $x^\ast$ and $y^\ast$ are the isolated root of the equation $f(x)=0$ 
and $g(x)=0$, respectively. We first discuss the stability of 
$\mathbf{p}_{ex}$.

By Definition \ref{def3}, on the one hand, 
 $\Sigma_{s}^{x}=\{(x,0): F_2^{+}(x,0)\cdot F_2^{-}(x,0)<0\}$. 
So the sliding set $\Sigma_{s}^{x}$ is escaping if $F_2^{-}(x,0)<0$ 
and attracting if $F_2^{-}(x,0)>0$. On the other hand, the sliding 
field \eqref{slidingfield} at $(x,0)\in\Sigma_{s}^{x}$ is 
\begin{equation}\label{pf-sx}
\mathbf{F_{sx}}(x,0)= \begin{pmatrix}
 F_{sx}^1(x,0) \\
 0 
 \end{pmatrix},\quad 
 F_{sx}^1(x,0)=\frac{f(x)}{F_2^-(x,0)- F_2^+(x,0)}.
\end{equation}
Then from \eqref{pf-sx} and the fact that $F_2^{+}(x,0)\cdot F_2^{-}(x,0)<0$, 
we have
\begin{equation}\label{sign}
\operatorname{sign}(F_{sx}^1(x,0))=\operatorname{sign}(F_2^{-}(x,0) f(x)).
\end{equation}
And it is easy to see $f'(x^{\ast})=2\det(\mathbf{A})\cdot y_e^+<0$. 
So we must have $f(x)>0$ on $(x^{\ast}-\varepsilon,x^{\ast})$ and 
$f(x)<0$ on $(x^{\ast},x^{\ast}+\varepsilon)$ for some small $\varepsilon>0$.
From \eqref{sign}, if the sliding set is attracting (escaping), 
i.e., $F_2^{-}(x,0)>0(<0)$, $\mathbf{p}_{ex}$ will be stable (unstable). 
That is $\mathbf{p}_{ex}$ will always be a pseudo-node of system \eqref{eq1} 
with $\mathbf{A}\in \Omega_1\cup \Omega_2$, which is a stable 
(an unstable) pseudo-node when $x_e^-<x_e^{+}+2\mu_1 (x_e^->x_e^{+}+2\mu_1)$. 
Similarly, we can also get the stability of $\mathbf{p}_{ey}$.
The proof is complete.
\end{proof}

Next, we give the properties of the non-regular point $\mathbf{p}_0$. 

\begin{proposition}\label{pro3}
Suppose that  condition \eqref{H} holds and 
$\mathbf{A}\in\Omega_1\cup \Omega_2$. We have the following statements 
about the non-regular point $\mathbf{p}_0$.

\noindent(a) When $\mathbf{A}\in \Omega_1$, about the non-regular point 
  $\mathbf{p}_0$, we have
  \begin{itemize}
  \item[(a1)] As $x_e^-=\mu_1$ or $x_e^-=\mu_2$, $\mathbf{p}_0$ is a 
   boundary equilibrium point;
  \item[(a2)] As $\min\{\mu_1,\mu_2\}<x_e^{-}<\max\{\mu_1,\mu_2\}$,
    $\mathbf{p}_0$ is a sliding boundary point;
  \item[(a3)] As $x_e^{-}>\max\{\mu_1,\mu_2\}$, $\mathbf{p}_0$ is a crossing 
   point;
  \item[(a4)] When $a_{12}>0$ ($a_{12}<0$), $\mathbf{p}_0$ is a non-regular 
  boundary saddle (a non-regular boundary sink), a stable pseudo-node 
  (a pseudo-saddle-node), a non-regular boundary saddle (a non-regular 
   boundary source) for $x_e^-<-x_e^+$, $x_e^-=-x_e^+$ and 
   $-x_e^{+}<x_e^{-}<\min\{\mu_1,\mu_2\}$, respectively.
  \end{itemize}
  
\noindent(b) When $\mathbf{A}\in \Omega_2$, about the non-regular point 
 $\mathbf{p}_0$, we have
  \begin{itemize}
  \item[(b1)] As $x_e^-=\mu_1$ or $x_e^-=\mu_2$, $\mathbf{p}_0$ is a 
  boundary equilibrium point;
  \item[(b2)] As $x_e^{-}>\mu_1$, $\mathbf{p}_0$ is a crossing point;
  \item[(b3)] When $x_e^{+}+\mu_2>0$ ($x_e^{+}+\mu_2<0$), $\mathbf{p}_0$ 
  is a non-regular boundary sink (a sliding boundary point), 
  a pseudo-saddle-node (a stable pseudo-node), a non-regular boundary 
  source (a non-regular boundary saddle), a sliding boundary point 
  (a non-regular boundary saddle)  for $x_e^-<\min\{-x_e^+,\mu_2\}$, 
  $x_e^-=-x_e^+$, $\min\{-x_e^+,\mu_2\}<x_e^{-}<\max\{-x_e^+,\mu_2\}$ and
   $\max\{-x_e^+,\mu_2\}<x_e^{-}<\mu_1$, respectively.
\end{itemize}
\end{proposition}

\begin{proof}
To discuss the non-regular point $\mathbf{p}_0$, we need the following 
statements:
$$
\mu_2-\mu_1=-\frac{\det(\mathbf{A})}{a_{11}a_{21}}y_e^{+},\quad
\mu_2-(-x_e^{+})=\frac{a_{12}}{a_{11}}y_t^{+}, \quad
\mu_1-(-x_e^{+})=x_t^{+}>0.
$$
Then it is easy to see that when $\mathbf{A}\in \Omega_1$, we obtain
\begin{gather*}
-x_e^{+}<0<\mu_1<\mu_2, \quad\text{for }a_{12}>0;\\
-x_e^{+}<0<\mu_2<\mu_1, \quad\text{for } a_{12}<0.
\end{gather*}
And when $\mathbf{A}\in \Omega_2$, we obtain
\begin{gather}\label{size121}
-x_e^{+}<\mu_2<0<\mu_1, \quad\text{for } x_e^{+}+\mu_2>0; \\
\label{size122}
\mu_2<-x_e^{+}<0<\mu_1, \quad\text{for } x_e^{+}+\mu_2\leq 0.
\end{gather}
And for $x_e^-=-x_e^+$, it is easy to see 
$\mathbf{p}_{ex}=\mathbf{p}_{ey}=\mathbf{p}_0$. 
Thus by Proposition \ref{pro1} and Proposition \ref{pro2}, the statements
 in this proposition can be obtained directly.
\end{proof}

Before we give the global qualitative dynamics of system \eqref{eq1}, 
we denote 
\begin{equation}\label{miu1}
\begin{gathered}
\mu_3:= x_e^{+}+\frac{a_{22}-a_{11}}{a_{21}}y_e^{+},\quad
\mu_4:= x_e^{+}+\frac{a_{22}(\lambda_2-a_{11})-a_{12}a_{21}}{a_{21}(\lambda_2-a_{11})}y_e^{+}, \\
\mu_5:= x_e^{+}-\frac{2a_{12}}{\lambda_2-a_{11}} y_e^{+},\quad
\mu_6:= \frac{(\lambda_2-a_{11})x_e^{+}-2a_{12}y_e^{+}}{\lambda_1-a_{11}}, \\
\mu_7:= -\frac{a_{11}x_e^{+}+2a_{12}y_e^{+}}{\lambda_1-a_{11}}, \quad
\mu_8:= \frac{2a_{12}y_e^{+}-(\lambda_2-a_{11})}{a_{11}}, \\
\mu_9:= -\frac{a_{12}}{\lambda_1-a_{11}}y_e^{+}.
\end{gathered}
\end{equation}
And by simple computations, it follows that
\begin{equation}\label{miu2}
\begin{gathered}
x_e^{-}=\mu_3\Leftrightarrow x_{m2}^{+}=x_{m1}^{-},  \quad
  x_e^{-}=\mu_4\Leftrightarrow x_{m2}^{+}=x_t^{-}, \\
x_e^{-}=\mu_5\Leftrightarrow x_{m2}^{+}=x_{m2}^{-}, \quad
x_e^{-}=\mu_6\Leftrightarrow y_{m1}^{-}=y_{m2}^{+}, \\
  x_e^{-}=\mu_7\Leftrightarrow y_{m1}^{-}=y_t^{+},\quad
x_e^{-}=\mu_8\Leftrightarrow y_t^{-}=y_{m2}^{+}, \\
x_e^{-}=\mu_9\Leftrightarrow y_{m1}^{-}=0.
\end{gathered}
\end{equation}

Secondly, to make our analysis of the dynamics more concrete, without loss of generality, we assume that
\begin{equation}\label{assumption}
x_{m1}^{+}>0\Leftrightarrow 0<y_e^{+}<\frac{\lambda_1-a_{11}}{a_{12}}x_e^{+}.
\end{equation}
Since the discussion with $x_{m1}^{+}\leq 0$ is similar, so we omit it.

\begin{theorem}\label{th1}
Suppose that  condition \eqref{H} holds and 
$\mathbf{A}\in \Omega_1\cup \Omega_2$. We have
$$
\max\{2\mu_1+x_e^{+},\mu_3\}<\mu_4<\mu_5
$$
and the following statements hold.
\begin{itemize}
  \item[(a)] When $x_e^{-}=\mu_3$, there exists one heteroclinic cycle;
  
  \item[(b)] When $\min\{2\mu_1+x_e^{+},\mu_3\}<x_e^{-}<\max\{2\mu_1+x_e^{+},
   \mu_3\}$, there exists one limit cycle, which is repulsive (attractive) 
   if $\mu_3<2\mu_1+x_e^{+}$ ($\mu_3>2\mu_1+x_e^{+}$);
   
  \item[(c)] When $x_e^{-}=\mu_4$ and $x_e^{-}=\mu_5$, there both exist 
  two sliding heteroclinic orbits.
\end{itemize}
\end{theorem}

\begin{theorem}\label{th2}
Suppose that condition \eqref{H} holds and $\mathbf{A}\in \Omega_1$, the 
following statements are true.

\noindent(a) When $a_{11}>0$, we have
  $$
  \mu_6<\mu_7<0<\mu_1<\mu_3<2\mu_1+x_e^{+}<\mu_4<\mu_5
  $$
  and the following statements hold.
  \begin{itemize}
  \item[(a1)] When $x_e^{-}<\min\{-x_e^{+}, \mu_6\}$, there exists one 
   sliding heteroclinic orbit;
   
  \item[(a2)] When $x_e^{-}=\mu_6$, there exist one heteroclinic orbit 
   and one sliding heteroclinic orbit;
   
  \item[(a3)] When $x_e^{-}=-x_e^{+}$, if $x_e^{-}<\mu_6$, there exists 
  one sliding heteroclinic orbit; Or there exist two sliding heteroclinic 
  orbits;
  
  \item[(a4)] When $\min\{-x_e^{+}, \mu_6\}<x_e^{-}<\mu_3$, there exist 
  two sliding heteroclinic orbits.
  \end{itemize}
  
\noindent(b) When $a_{11}<0$, we have
  $$
  -x_e^{+}<0<\mu_9<\min\{2\mu_1+x_e^{+},\mu_3\}
  < \max\{2\mu_1+x_e^{+},\mu_3\}<\mu_4<\mu_5
  $$
  and the following statements.
  \begin{itemize}
  \item[(b1)] When $x_e^{-}<-x_e^{+}$, there exists one sliding orbit 
   containing $\mathbf{p}_{ey}$, $\mathbf{p}_0$ and $\mathbf{x}_e^{+}$;
  \item[(b2)] When $-x_e^{+}\leq x_e^{-}<\mu_9$, there exists one sliding 
   heteroclinic orbit;
  \item[(b3)] When $\mu_9\leq x_e^{-}<\min\{2\mu_1+x_e^{+},\mu_3\}$, 
   there exist two sliding heteroclinic orbits.
  \end{itemize}
\end{theorem}

\begin{theorem}\label{th3}
Suppose that condition \eqref{H} holds and $\mathbf{A}\in \Omega_2$, 
the following statements are true.

\noindent(a) When $a_{11}<0$ and $0<y_e^{+}<-\frac{a_{11}}{a_{12}}x_e^{+}$,
   we have
  $$
  \mu_8<\min\{\mu_6, \mu_2\}<0<\mu_1<\max\{2\mu_1+x_e^{+}, \mu_3\}
   <\mu_4<\mu_5
  $$
  and the following statements hold.
  \begin{itemize}
  \item[(a1)] When $x_e^{-}<\min\{\mu_8, -x_e^{+}\}$, there exists one 
  sliding orbit containing $\mathbf{x}_e^{+}$, $\mathbf{p}_0$ and 
  $\mathbf{p}_{ey}$;
  
  \item[(a2)] When $x_e^{-}=\mu_8$, there exists one sliding cycle;
  
  \item[(a3)] When $x_e^{-}=\mu_6$, there exists one sliding heteroclinic 
  orbit;
  
  \item[(a4)] If $\mu_8<-x_e^{+}$, we obtain
  \begin{itemize}
  \item[(a41)] When $\mu_8<x_e^{-}<-x_e^{+}$, there exists one sliding 
  cycle;
  \item[(a42)] When $x_e^{-}=-x_e^{+}$, there exist one sliding homoclinic 
  cycle and one sliding heteroclinic orbit;
  \end{itemize}
  
  \item[(a5)] When $\min\{-x_e^{+},\mu_6, \mu_2\}<x_e^{-}
   <\min\{2\mu_1+x_e^{+}, \mu_3\}$, there exist two sliding heteroclinic 
   orbits and at most one sliding cycle.
  \end{itemize}
  
\noindent(b) When $a_{11}<0$ and $-\frac{a_{11}}{a_{12}}x_e^{+}<y_e^{+}
  <\frac{\lambda_1-a_{11}}{a_{12}}x_e^{+}$, we have
  $$
  \mu_8<\mu_6<\mu_7, \quad \mu_8<2\mu_2+x_e^{+}<\mu_2<-x_e^{+}
   <0<\max\{2\mu_1+x_e^{+}, \mu_3\}<\mu_4<\mu_5
  $$
  and the following statements hold.
  \begin{itemize}
  \item[(b1)] When $x_e^{-}=\mu_8$, there exists one sliding heteroclinic 
   orbit;
  \item[(b2)] When $x_e^{-}=\mu_6$, there exists one heteroclinic orbit;
  \item[(b3)] If $2\mu_2+x_e^{+}<\mu_6<\mu_2$, when 
   $\mu_8<x_e^{-}<2\mu_2+x_e^{+}$, there exists one sliding homoclinic 
    cycle;
  \item[(b4)] If $\mu_6<2\mu_2+x_e^{+}$, when 
   $\mu_8<x_e^{-}<2\mu_2+x_e^{+}$, there exist a sliding cycle 
   bifurcation and a sliding homoclinic bifurcation;
  \item[(b5)] When $2\mu_2+x_e^{+}<x_e^{-}<\min\{2\mu_1+x_e^{+}, \mu_3\}$,
   there exist two sliding heteroclinic orbits.
  \end{itemize}
  \end{theorem}

There is no other separatrix of system \eqref{eq1} except for the 
separatrixes given in the above theorems.

\section{Proofs of main results}\label{sec5}

\begin{proof}[Proof of Theorem \ref{th1}]
By \eqref{qiedian} and \eqref{fundation1}, it is easy to see that
$x_{m1}^{-}$ $x_{m2}^{-}$ and $x_t^{-}$ increase with respect to $x_e^{-}$. 
And by \eqref{assumption} and simple calculation, it follows that for 
$\mathbf{A}\in \Omega_1\cup \Omega_2$, we have
\begin{equation}\label{oder1}
x_{m2}^{-}<x_t^{-}<x_{m1}^{-},\ 0<x_{m1}^{+}<x_t^{+}<x_{m2}^{+}.
\end{equation}
Then $\mu_3<\mu_4<\mu_5$ and $2\mu_1+x_e^{+}<\mu_4$ can be obtained 
directly since \eqref{miu1j} and \eqref{miu2}, which means 
$\max\{\mu_3,2\mu_1+x_e^{+}\}<\mu_4<\mu_5$.

As $x_e^{-}=\mu_3$, we know $x_{m1}^{\mp}=x_{m2}^{\pm}$, it is also easy 
to see $x_{m1}^{+}<\min\{x_t^{+},x_t^{-}\}<\max\{x_t^{+},x_t^{-}\}<x_{m2}^{+}$
by \eqref{assumption}. So there is a heteroclinic cycle connecting 
$\mathbf{x}_e^{+}$ with $\mathbf{x}_e^{-}$.

In addition, by simple calculation, it follows that
$$
\mu_3-(2\mu_1+x_e^{+})=-\frac{y_e^{+}}{a_{21}}(a_{11}+a_{22}).
$$
Then for $\mathbf{A}\in \Omega_1\cup\Omega_2$, we obtain that
\begin{equation}\label{comp1}
\begin{gathered}
\mu_3>2\mu_1+x_e^{+}, \quad\text{if } a_{11}+a_{22}<0; \\
\mu_3<2\mu_1+x_e^{+},  \quad\text{if } a_{11}+a_{22}>0.
\end{gathered}
\end{equation}
For $\mu_3<x_e^{-}<2\mu_1+x_e^{+}$, we have 
$x_{m1}^{+}<x_{m2}^{-}<x_t^{-}<x_t^{+}<x_{m2}^{+}<x_{m1}^{-}$. 
And by Proposition \ref{pro2}, $\mathbf{p}_{ex}\in \Sigma_{s}^{x}$ is stable. 
Then there must exist a repulsive limit cycle $\mathcal{L}_1$. 
To prove this, we need to construct the Poincar$\acute{e}$ map as follows:
\[
\mathcal{P}_1:  D_{\mathcal{P}_1}\mapsto D_{\mathcal{P}_1}, \quad
\mathcal{P}_1(x_0,0)=(x_1,0),
\]
where $D_{\mathcal{P}_1}=\{(x,0): x\in[x_t^{+},x_{m2}^{+}]\}$, and 
$(x_1,0)$ is the first arriving point at $D_{\mathcal{P}_1}$ of the orbit
of system \eqref{eq1} with the initial point $(x_0,0)\in D_{\mathcal{P}_1}$. 
Since $x_t^{-}<x_t^{+}<x_{m2}^{+}$, there must exist 
$(x^{\star}_1,0)$, $(x^{\star}_2,0)\in D_{\mathcal{P}_1}$ and 
$x_t^{+}<x^{\star}_2<x^{\star}_1<x_{m2}^{+}$, such that
$$
\mathcal{P}_1(x^{\star}_1,0)=(x_{m2}^{+},0), \quad
\mathcal{P}_1(x^{\star}_2,0)=(x_t^{+},0).
$$
This means there must be $(x^{\star},0)\in D_{\mathcal{P}_1}$, such that
$$
\mathcal{P}_1(x^{\star},0)=(x^{\star},0),
$$
which implies the existence of the repulsive limit cycle $\mathcal{L}_1$. 
Similarly, for $\mu_3<x_e^{-}<2\mu_1+x_e^{+}$, we can prove the existence 
of an attractive limit cycle $\mathcal{L}_2$.

As $x_e^{-}>\max\{2\mu_1+x_e^{+},\mu_3\}$, we have $x_t^{+}<x_t^{-}$ and 
$x_{m1}^{-}>x_{m2}^{+}$. Then according to Proposition \ref{pro1} and 
Proposition \ref{pro2}, it follows that 
$\Sigma_{s}^{x}=\{(x,0): x_t^{+}<x<x_t^{-}\}$ is repulsive and 
$\mathbf{p}_{ex}$ is unstable. So there is no other special dynamics 
expect for two sliding heteroclinic orbits $\mathcal{L}_3$ and 
$\mathcal{L}_4$, both of which connect $\mathbf{x}_e^{\pm}$ and 
$\mathbf{p}_{ex}$ as $x_e^{-}=\mu_4$ and $x_e^{-}=\mu_5$, respectively. 
The proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{th2}]
(a) In this case, since $a_{11}>0$ and \eqref{assumption}, we obtain that
$$
y_{m2}^{+}>y_t^{+}>y_e^{+}>0>y_{m1}^{+}, \quad
0<x_{m1}^{+}<x_e^{+}<x_t^{+}<x_{m2}^{+}.
$$
By \eqref{fundation1}, it is obvious that $y_{m1}^{-}$ decreases with 
respect to $x_e^{-}$. So we obtain $\mu_6<\mu_7<0$ directly because 
of \eqref{miu2}. Moreover, when $x_e^{-}=\mu_5$, we have 
$x_{m1}^{\mp}=x_{m2}^{\pm}$, which means $x_t^{-}>0$. So we obtain that 
$0<\mu_1<\mu_3$. Since \eqref{comp1}, it is easy to see 
$\mu_3<2\mu_1+x_e^{+}$. Then by Theorem \ref{th1}, we obtain that
$$
\mu_6<\mu_7<0<\mu_1<\mu_3<2\mu_1+x_e^{+}<\mu_4<\mu_5.
$$
In addition, by simple calculation, it follows that
\begin{equation}\label{k2-xe}
\mu_6-(-x_e^{+})=\frac{(a_{22}-a_{11})x_e^{+}-2a_{12}y_e^{+}}
{\lambda_1-a_{11}}, \quad
\mu_7-(-x_e^{+})=\frac{\lambda_2-a_{11}-2a_{12}y_e^{+}}{\lambda_1-a_{11}},
\end{equation}
the signs of which are not sure. Then $-x_e^{+}<\mu_6<\mu_7<0$, 
$\mu_6<-x_e^{+}<\mu_7<0$ or $\mu_6<\mu_7<-x_e^{+}<0$ could happen.

Firstly, according to Proposition \ref{pro1}, Proposition \ref{pro2} 
and Remark \ref{invariant manifold}, it is obvious to see that there is 
a sliding heteroclinic orbit $\mathcal{M}_1$ from $\mathbf{x}_e^{+}$ to 
$\mathbf{p}_{ey}$ for $x_e^{-}\leq -x_e^{+}$, and a sliding 
heteroclinic orbit $\mathcal{M}_2$ from $\mathbf{x}_e^{+}$ to 
$\mathbf{p}_{ex}$ for $x_e^{-}>-x_e^{+}$. Moreover, a heteroclinic orbit 
$\mathcal{M}_3$ from $\mathbf{x}_e^{-}$ to $\mathbf{x}_e^{+}$ appears only 
as $x_e^{-}=\mu_6$ obviously. For $\mu_6<x_e^{-}<\mu_7$, the orbit of 
system \eqref{eq1} starting from $(0,y_{m1}^{-})$ will arrive at the 
sliding set $\Sigma_{s}$. And for $\mu_7\leq x_e^{-}<\mu_3$, the invariant 
manifold $l_1^{-}$ of $\mathbf{x}_e^{-}$ will intersects with $\Sigma_{s}$. 
So there is a sliding heteroclinic orbit $\mathcal{M}_4$ from 
$\mathbf{x}_e^{-}$ to the unique pseudo-equilibrium point $\mathbf{p}_{ex}$ 
or $\mathbf{p}_{ey}$ for $\mu_6<x_e^{-}<\mu_3$.

Thus we obtain that $\mathcal{M}_1$ exists for 
$x_e^{-}<\min\{-x_e^{+},\mu_6\}$. When $\min\{-x_e^{+},\mu_6\}<x_e^{-}<\mu_3$,
there exist two sliding heteroclinic orbits $\mathcal{M}_2$ and 
$\mathcal{M}_4$. As $x_e^{-}=\mu_6$, there exists the heteroclinic orbit 
$\mathcal{M}_3$ and the sliding heteroclinic orbit $\mathcal{M}_4$.

(b) Since $a_{11}<0$,  we obtain that
$$
x_{m2}^{-}<x_t^{-}<x_{m1}^{-}<x_e^{-}, \quad
0<x_e^{+}<x_{m1}^{+}<x_t^{+}<x_{m2}^{+}.
$$
By \eqref{fundation1} and \eqref{miu2}, we know $x_{m1}^{-}=y_{m1}^{-}=0$ 
as $x_e^{-}=\mu_9$, then $0<\mu_9<\min\{2\mu_1+x_e^{+},\mu_3\}$ can 
be obtained directly. And by Theorem \ref{th1}, we have
$$
-x_e^{+}<0<\mu_9<\min\{2\mu_1+x_e^{+},\mu_3\}
< \max\{2\mu_1+x_e^{+},\mu_3\}<\mu_4<\mu_5.
$$

According to Propositions \ref{pro1}, \ref{pro2} and  \ref{pro3}, 
it is easy to see there exists a sliding orbit connecting $\mathbf{x}_e^{+}$, 
$\mathbf{p}_{ey}$ and the non-regular boundary sink $\mathbf{p}_0$ 
for $x_e^{-}<-x_e^{+}$, which will become a sliding heteroclinic orbit 
from $\mathbf{x}_e^{+}$, $\mathbf{p}_{ex}$ for 
$-x_e^{+}\leq x_e^{-}<\min\{2\mu_1+x_e^{+},\mu_3\}$. 
Once $x_e^{-}=\mu_9$, there appears a sliding heteroclinic orbit from 
$\mathbf{x}_e^{-}$ to $\mathbf{p}_{ex}$, which will persist until 
$x_e^{-}=\min\{2\mu_1+x_e^{+},\mu_3\}$.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th3}]
When $\mathbf{A}\in \Omega_2$, it is not difficult to verify that the 
case of matrix $\mathbf{A}$ with $a_{21}>0$, $a_{22}>0$, $a_{12}<0$, 
$a_{11}>0$ is invalid since $\det(\mathbf{A})=a_{11}a_{22}-a_{12}a_{21}>0$, 
which contradicts that $\det(\mathbf{A})<0$ in the condition \eqref{H}.
 So we just need to give the discussion of system \eqref{eq1} with 
$\mathbf{A}\in \Omega_2$ and $a_{11}<0$ under  assumption \eqref{assumption} 
as follows.

(a) When $a_{11}<0$ and $0<y_e^{+}<-\frac{a_{11}}{a_{12}}x_e^{+}$, we know that
$y_t^{+}<0$ by \eqref{cyt}. In addition, we obtain that
$$
y_{m2}^{+}>y_e^{+}>0>y_t^{+}>y_{m1}^{+}, \quad
0<x_{m1}^{+}<x_e^{+}<x_t^{+}<x_{m2}^{+}.
$$
Note that $y_{m1}^{-}>y_t^{-}$ and both of them decrease with respect to 
$x_e^{-}$ by \eqref{qiedian} and \eqref{fundation1}. 
Then as $x_e^{-}=\mu_8$, we must have $y_{m1}^{-}>y_t^{-}=y_{m2}^{+}>0$, 
which means $\mu_8<\mu_6<0$ and $\mu_8<\mu_2<0$ since \eqref{miu2j} 
and \eqref{miu2}. Moreover, by simple calculation, it follows that
\begin{equation}\label{miu2-miu6}
\begin{gathered}
\mu_8-(-x_e^{+})=\frac{2a_{11}y_e^{+}-(\lambda_2-2a_{11})x_e^{+}}{a_{11}},\\
\mu_2-\mu_6=\frac{(\lambda_1+a_{11})a_{12}y_e^{+}
  -a_{11}(\lambda_2-a_{11})x_e^{+}}{(\lambda_1-a_{11})a_{11}}.
\end{gathered}
\end{equation}
And by \eqref{k2-xe}, we know the signs of $\mu_8-(-x_e^{+})$, 
$\mu_2-\mu_6$ and $\mu_6-(-x_e^{+})$ are not sure. 
Then by \eqref{size121} and Theorem \ref{th1}, we know
$$
\mu_8<\min\{\mu_6,\mu_2\}<0<\mu_1<\max\{2\mu_1+x_e^{+}, \mu_3\}
<\mu_4<\mu_5.
$$
Meanwhile, we obtain that $-x_e^{+}<\mu_8<\min\{\mu_6,\mu_2\}$, 
$\mu_8<-x_e^{+}<\min\{\mu_6,\mu_2\}$ or $\mu_8<\mu_6<-x_e^{+}<\mu_2$ 
could take place.

By Propositions \ref{pro1}, \ref{pro2} and \ref{pro3},
and Remark \ref{invariant manifold}, for $x_e^{-}<\min\{\mu_8,-x_e^{+}\}$, 
there is a sliding orbit $\mathcal{J}_1$ containing $\mathbf{p}_{ey}$, 
$\mathbf{x}_e^{+}$ and $\mathbf{p}_0$, which is a non-regular boundary sink. 
As $x_e^{-}=\mu_8$, if $\mu_8<-x_e^{+}$, there is a sliding cycle 
$\mathcal{J}_2$ connecting $\mathbf{p}_{ey}$, $\mathbf{x}_e^{+}$ and 
$\mathbf{p}_0$, which is a non-regular boundary sink; 
If $\mu_8>-x_e^{+}$, there is a sliding cycle $\mathcal{J}_3$ connecting 
$\mathbf{p}_{ex}$, $\mathbf{x}_e^{+}$ and $\mathbf{p}_0$, which is a 
non-regular boundary source. Moreover, as $x_e^{-}=\mu_6$, there is a 
heteroclnic orbit $\mathcal{J}_4$ from $\mathbf{x}_e^{-}$ to 
$\mathbf{x}_e^{+}$.

As $\mu_8<x_e^{-}<\mu_3$, the orbits of $\oplus$-system starting from 
$(0,y_t^{-})$ will arrive at $\Sigma_{s}^{x}$ since $y_t^{+}<0$ and 
$\Sigma_{s}^{x}$ is attractive before $x_e^{-}=\mu_3$. 
Then on the one hand, if $\mu_8<-x_e^{+}$, for $\mu_8<x_e^{-}<-x_e^{+}$, 
$\mathcal{J}_2$ will become a sliding cycle $\mathcal{J}_{21}$ connecting
$\mathbf{p}_{ey}$ and the non-regular boundary sink $\mathbf{p}_0$. 
Then $\mathcal{J}_{21}$ will turn into a sliding homoclinic cycle 
$\mathcal{J}_{22}$ as $x_e^{-}=-x_e^{+}$, which containing the 
pseudo-saddle-node $\mathbf{p}_{ex}=\mathbf{p}_{ey}=\mathbf{p}_0$, 
meanwhile, a sliding heteroclinic orbit $\mathcal{J}_5$ from 
$\mathbf{x}_e^{+}$ to $\mathbf{p}_{ex}$ occurs, which will exist until 
$x_e^{-}=\min\{2\mu_1+x_e^{+}, \mu_3\}$. 
And $\mathcal{J}_{22}$ will turn into a sliding cycle $\mathcal{J}_{23}$ 
connecting $\mathbf{p}_{ex}$ and the non-regular boundary source 
$\mathbf{p}_0$ as long as the first arriving point on $\Sigma_{s}^{x}$ 
of the orbit of $\oplus$-system starting from $(0,y_t^{-})$ is on the right 
of $\mathbf{p}_{ex}$. However, if $\mu_8>-x_e^{+}$, $\mathcal{J}_3$ can 
only becomes $\mathcal{J}_{23}$ before $x_e^{-}=\mu_3$.

On the other hand, if $\mu_6<-x_e^{+}<\mu_2$, we have as 
$\mu_6<x_e^{-}<\mu_2$, $\mathcal{J}_4$ will firstly become a sliding
heteroclinic orbit $\mathcal{J}_{41}$ from 
$\mathbf{x}_e^{-}$ to $\mathbf{p}_{ey}$ for $x_e^{-}<-x_e^{+}$, then 
$\mathcal{J}_{41}$ will become a sliding heteroclinic orbit 
$\mathcal{J}_{42}$ from $\mathbf{x}_e^{-}$ to $\mathbf{p}_{ex}$ for 
$x_e^{-}\geq -x_e^{+}$, which will persist until 
$x_e^{-}=\min\{2\mu_1+x_e^{+}, \mu_3\}$; If $-x_e^{+}<\min\{\mu_6,\mu_2\}$, 
we obtain that $\mathcal{J}_4$ will only become $\mathcal{J}_{42}$.

(b) In this case, we know $y_t^{+}>0$ by \eqref{cyt}. 
In addition, we obtain that
$$
y_{m2}^{+}>y_e^{+}>y_t^{+}>0>y_{m1}^{+}, \quad y_{m1}^{-}>y_t^{-}<y_{m2}^{-}.
$$
Then since $y_t^{-}$ decreases with respect to $x_e^{-}$, we obtain 
$\mu_8<2\mu_2+x_e^{+}<\mu_2$ directly by \eqref{miu2j} and \eqref{miu2}. 
Note that $\mu_2<-x_e^{+}$ by \eqref{size122}. So by Theorem \ref{th1}, 
we obtain
$$
\mu_8<2\mu_2+x_e^{+}<\mu_2<-x_e^{+}<0<\max\{2\mu_1+x_e^{+}, 
\mu_3\}<\mu_4<\mu_5.
$$
By \eqref{miu2j} and \eqref{miu2}, it easy to see that $\mu_8<\mu_6<\mu_7$ 
and $2\mu_3+x_e^{+}<\mu_7$. However, the signs of 
$\mu_7-(-x_e^{+})$ and $\mu_2-\mu_6$ in this case are not sure 
by \eqref{k2-xe} and \eqref{miu2-miu6}, respectively. 
And by easy computation, we obtain that
$$
2\mu_2+x_e^{+}-\mu_6=\frac{\lambda_1-\lambda_2}{\lambda_1-a_{11}}x_e^{+}
+\frac{2a_{12}\lambda_1}{a_{11}(\lambda_1-a_{11})}y_e^{+},
$$
which implies $2\mu_2+x_e^{+}<\mu_6$ and $2\mu_2+x_e^{+}>\mu_6$ both may be 
true.

By Propositions \ref{pro1}, \ref{pro2} and \ref{pro3},
and Remark \ref{invariant manifold}, we see that there is no 
special separatrixes as $x_e^{-}<\mu_8$.
As $x_e^{-}=\mu_8$, there exists a sliding heteroclinic orbit from 
$\mathbf{p}_{ey}$ to $\mathbf{x}_e^{+}$. In addition, it is obvious 
that a heteroclinic orbit from $\mathbf{x}_e^{-}$ to $\mathbf{x}_e^{+}$ 
appears as $x_e^{-}=\mu_6$. Furthermore, as $x_e^{-}>\max\{\mu_6, \mu_2\}$, 
the orbit of $\oplus$-system starting from $\mathbf{y}_{m1}^{-}$ will 
arrive $\Sigma_{s}$. Thus there will exist a sliding heteroclinic orbit 
from $\mathbf{x}_e^{-}$ to $\mathbf{p}_{ey}$ if $x_e^{-}<-x_e^{+}$, 
or from $\mathbf{x}_e^{-}$ to $\mathbf{p}_{ex}$ if $x_e^{-}\geq -x_e^{+}$ 
until $x_e^{-}=\min\{2\mu_1+x_e^{+}, \mu_3\}$.

In addition to this, when $x_e^{-}<2\mu_2+x_e^{+}$, we have
 $y_t^{-}>y_t^{+}>0$, then let $\mathbf{y}^{1}=(0,y^{1})$ and 
$\mathbf{y}^2=(0,y^2)$ be the first arriving point on the positive 
$y$-axis of the forward orbits of $\circleddash$-system starting from 
$\mathbf{y}_t^{+}$ and $\mathbf{p}_{ey}$, respectively. 
And when $x_e^{-}<\mu_2$, denote by $\mathbf{y}^{-}=(0,y^{-})$ and 
$\mathbf{y}^{+}=(0,y^{+})$ the first arriving point on the positive 
$y$-axis of the forward and backward orbits of $\circleddash$-system and 
$\oplus$-system with initial condition $\mathbf{p}_0$, respectively. 
Then when $x_e^{-}<2\mu_2+x_e^{+}<\mu_2$, we must have 
$y_{m1}^{-}>y^{-}>y^{1}>y^2$. Since $\mathbf{y}^{+}=(0,y^{+})$ and 
$\mathbf{y}_{m2}^{+}=(0,y_{m2}^{+})$ will not change with the value of 
$x_e^{-}$, moreover, $y_{m2}^{+}>y^{+}$, so there must be 
$\xi_1<\xi_2, \ \xi_1<\mu_6, \ \xi_2<\mu_7$ such that
\begin{gather*}
y^{-}=y_{m2}^{+}, \quad\text{if } x_e^{-}=\xi_1\,; \\
y^{-}=y^{+}, \quad\text{if } x_e^{-}=\xi_2\,.
\end{gather*}
This mean there exist a sliding homoclinic cycle containing 
$\mathbf{x}_e^{+}$ called $\mathcal{H}_1$ and a sliding-zero cycle 
containing $\mathbf{p}_0$ called $\mathcal{H}_2$ as 
$x_e^{-}=\xi_1$ and $x_e^{-}=\xi_2$, respectively.

Apart from this, one the one hand, if $\mu_6<2\mu_2+x_e^{+}<\mu_2$, 
there must be $\xi_{11}<\xi_{12}<\xi_1$, such that
\begin{gather*}
y^{1}=y_{m2}^{+}, \quad\text{if } x_e^{-}=\xi_{11}; \\
y^2=y_{m2}^{+}, \quad\text{if } x_e^{-}=\xi_{12},
\end{gather*}
which mean there both appears a sliding heteroclinc orbit 
$\mathcal{H}_3$ from $\mathbf{p}_{ey}$ to $\mathbf{x}_e^{+}$ as 
$x_e^{-}=\xi_{11}$ and $x_e^{-}=\xi_{12}$, that is $\mathcal{H}_3$ 
and $\mathcal{H}_1$ will appear in turn before $x_e^{-}=\mu_6$; 
If $2\mu_2+x_e^{+}<\mu_6<\mu_2$, it is easy to see only $\mathcal{H}_1$ 
will appear before $x_e^{-}=\mu_6$; If $2\mu_2+x_e^{+}<\mu_2<\mu_6$, 
neither $\mathcal{H}_3$ nor $\mathcal{H}_1$ will appear before 
$x_e^{-}=\mu_6$ since $\mathbf{p}_0\in \Sigma_{s}$. 
On the other hand, because of the sign of $2\mu_2+x_e^{+}-\mu_6$ is not sure, 
so the size relationship of $\xi_2$ with $2\mu_2+x_e^{+}$ is uncertain. 
Then there may exist some special separatrixes, which will be discussed 
as follows:

If $y^{-}>y^{+}$ is true for $x_e^{-}<2\mu_2+x_e^{+}$, there must be a 
sliding cycle $\mathcal{H}_4$ containing $\mathbf{p}_0$ before $y^{-}=y^{+}$ 
since the orbit of $\oplus$-system starting from $\mathbf{y}^{-}$ will 
arrive at $\Sigma_{s}^{x}$. If $y^{-}>y^{+}$ is true for 
$2\mu_2+x_e^{+}<x_e^{-}<\mu_2$, $\mathcal{H}_4$ must exist, meanwhile, 
there must exist a repulsive limit cycle $\mathcal{H}_5$ before 
$x_e^{-}=\xi_2$. To prove this, we need constructing the Poincar\'e map as 
follows:
$$
\mathcal{P}_2: D_{\mathcal{P}_2}\mapsto D_{\mathcal{P}_2},\quad 
\mathcal{P}_2(0,y_0)=(0,y_1),
$$
where $D_{\mathcal{P}_2}=\{(0,y): y\in[y_t^{-}, y^{-}]\}$, and 
$(0,y_1)$ is the first arriving point at $D_{\mathcal{P}_2}$ of the 
forward orbit of system \eqref{eq1} starting from $(0,y_0)$. 
Then we must have $\mathcal{P}_2((0,y^{+}))=(0,y^{-})$. 
And there must be $(0,y^{\ast}_1)$ and $(0,y^{\ast}_2)$, which satisfy 
$y_t^{+}<y^{\ast}_2<y^{\ast}_1<y^{+}$, such that 
$\mathcal{P}_2(0,y^{\ast}_1)=(0,y^{\ast}_2)$ since 
$\mathbf{p}_{ey}$ is atable. So there must exist 
$y^{\ast}_1<y_{p}^{\ast}<y^{+}$, such that
$$
\mathcal{P}_2(0,y_{p}^{\ast})=(0,y_{p}^{\ast}).
$$
which implies the existence of the repulsive $\mathcal{H}_5$.

Whether $y^{-}=y^{+}$ is true for $x_e^{-}<2\mu_2+x_e^{+}$ or 
$2\mu_2+x_e^{+}<x_e^{-}<\mu_2$, it is obvious that there must arise a 
sliding-zero cycle $\mathcal{H}_2$.

If $y^{-}<y^{+}$ is true for $x_e^{-}<2\mu_2+x_e^{+}$, we obtain a 
attractive limit cycle $\mathcal{H}_6$ before $x_e^{-}=2\mu_2+x_e^{+}$. 
To prove this, we need constructing the Poincar$\acute{e}$ map as follows:
$$
\mathcal{P}_3:  D_{\mathcal{P}_3}\mapsto D_{\mathcal{P}_3},\quad
\mathcal{P}_3(0,y_0)=(0,y_1),
$$
where $D_{\mathcal{P}_3}=\{(0,y): y\in[0,y_t^{+}]\}$, and $(0,y_1)$ is 
the first arriving point at $D_{\mathcal{P}_3}$ of the forward orbit of 
system \eqref{eq1} starting from $(0,y_0)$. Thus there must be 
$(0,y^{\diamond}_1)$ and $(0,y^{\diamond}_2)$, 
$0<y^{\diamond}_1<y^{\diamond}_2<y_t^{+}$, such that
$$
\mathcal{P}_3(0,0)=(0,y^{\diamond}_1), \quad
\mathcal{P}_3((0,y^{+}_t))=(0,y^{\diamond}_2),
$$
since $y^{-}<y^{+}$ and $0<y_t^{+}<y_t^{-}$, respectively. 
So there must be $(0,y_{p}^{\ast})$, such that 
$y^{\diamond}_1<y_{p}^{\diamond}<y^{\diamond}_2$
$$
\mathcal{P}_3(0,y_{p}^{\diamond})=(0,y_{p}^{\diamond}).
$$
which implies the existence of the attractive 
$\mathcal{H}_6$. If $y^{-}<y^{+}$ is true for $2\mu_2+x_e^{+}<x_e^{-}<\mu_2$, 
there must exist a sliding heteroclinic orbit $\mathcal{H}_{71}$ from 
$\mathbf{x}_e^{+}$ to $\mathbf{p}_{ey}$ until $x_e^{-}=-x_e^{+}$ since 
$\Sigma_{s}^{y}$ is attractive. Then $\mathcal{H}_{71}$ turns into a 
sliding heteroclinic orbit $\mathcal{H}_{72}$ from $\mathbf{x}_e^{+}$ to 
$\mathbf{p}_{ex}$ until $x_e^{-}=\min\{2\mu_1+x_e^{+}, \mu_3\}$.

Thus according to the above analysis, we obtain $\mathcal{H}_4$, 
$\mathcal{H}_2$ and $\mathcal{H}_6$ will appears in turn if 
$y^{-}<y^{+}$ is true for $x_e^{-}<2\mu_2+x_e^{+}$, which implies that
there is a sliding cycle bifurcation as $x_e^{-}=\xi_2$. 
In addition, $\mathcal{H}_1$ and $\mathcal{H}_4$ will appear in turn 
if $y^{-}>y^{+}$ is true for $x_e^{-}<2\mu_2+x_e^{+}$, which implies that
there is a sliding homoclinic bifurcation as $x_e^{-}=\xi_1$.
The proof is complete.
\end{proof}

\section{Examples}\label{example}

In this section, three examples are provided to illustrate some 
separatrixes in the theorems in Section 3. It should be noted that the 
red and the blue curves in the following figures represent the orbits 
of the $\oplus$-system and the $\ominus$-system, respectively. 
And the arrows represents the directions of the orbits on the forward time.

\begin{example} \rm
Let
\begin{equation}\label{exm-1}
  A=\begin{pmatrix} -4  & 2 \\ 3 & 1 \end{pmatrix},
  \quad
  X^{-}_e=\begin{pmatrix} 5 \\ -3 \end{pmatrix},
  \quad
  X^{+}_e=\begin{pmatrix} 2 \\ 3 \end{pmatrix}.
\end{equation}
By simple calculations, we have
\begin{gather*}
x_t^{+}=3,  \quad x_{m1}^{+}=1, \quad x_{m2}^{+}=8,\quad x_t^{-}=4, 
\quad x_{m1}^{-}=6, \quad x_{m2}^{-}=-1,\\
2\mu_1+x_e^{+}=4<\mu_3=7.
\end{gather*}
Through numerical simulation, there exists an attractive limit cycle 
illustrated in Figure \ref{fig2}, which supported the conclusion about 
$\mathcal{L}_2$ in Theorem \ref{th1}.

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig2}
\end{center}
\caption{Existence of the attractive limit cycle of equation \eqref{exm-1}}
\label{fig2}
\end{figure}
\end{example}

\begin{example} \rm 
Let
 \begin{equation}\label{exm-2}
  A=\begin{pmatrix} -1  & 2 \\ 3 & 4 \end{pmatrix}, \quad
  X^{-}_e=\begin{pmatrix} 9 \\ -3 \end{pmatrix}, \quad
  X^{+}_e=\begin{pmatrix} 2 \\ 3 \end{pmatrix}.
\end{equation}
By simple calculations, we have
\begin{gather*}
x_t^{+}=6,  \quad x_{m1}^{+}=1, \quad x_{m2}^{+}=8,\quad x_t^{-}=5, 
\quad x_{m1}^{-}=10, \quad x_{m2}^{-}=3, \\
2\mu_1+x_e^{+}=10>\mu_3=7.
\end{gather*}
Through numerical simulation, there exists a repulsive limit cycle 
illustrated in Figure \ref{fig3}, which supported the conclusion about 
$\mathcal{L}_1$ in Theorem \ref{th1}.

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3}
\end{center}
\caption{Existence of the repulsive limit cycle of equation \eqref{exm-2}}
\label{fig3}
\end{figure}
\end{example}

\begin{example} \rm
Let
 \begin{equation}\label{exm-3}
  A=\begin{pmatrix} -4  & 2 \\ 3 & 1 \end{pmatrix},  \quad
  X^{-}_e=\begin{pmatrix} \mu \\ -10 \end{pmatrix}, \quad
  X^{+}_e=\begin{pmatrix} 4 \\ 10 \end{pmatrix}.
  \end{equation}
For $-9<\mu<-6$, we will show there exist a sliding homoclinic bifurcation 
and a sliding cycle bifurcation. By easy calculations, it follows that
$$
x_t^{+}=22/3,\quad y_t^{+}=2, \quad y_{m1}^{+}=-2, \quad y_{m2}^{+}=12.
$$
In addition, we obtain the following statements:

(1) As $\mu=-8.4$, we have
$$
x_t^{-}=-176/15,\quad y_t^{-}=6.8, \quad y_{m1}^{-}=15.2, \quad
 y_{m2}^{-}=-14.2.
$$
By numerical simulation, we have $y^{-}=y_{m2}^{+}$, and there exists 
a sliding homoclinic cycle illustrated in Figure \ref{fig4}, which is in 
agreement with the orbit $\mathcal{H}_1$ in Theorem \ref{th3}.

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig4}
\end{center}
\caption{Existence of the sliding homoclinic cycle of equation \eqref{exm-3}
 with $\mu=-8.4$}
\label{fig4}
\end{figure}

(2) As $\mu=-7$, we have
$$
x_t^{-}=-31/3,\quad y_t^{-}=4, \quad y_{m1}^{-}=11, \quad y_{m2}^{-}=-27/2.
$$
By numerical simulation, we have $y^{-}<y_{m2}^{+}$, and there exists 
a sliding cycle illustrated in Figure \ref{fig5}, which is in agreement 
with the orbit $\mathcal{H}_4$ in Theorem \ref{th3}.

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig5}
\end{center}
\caption{Existence of the sliding cycle of equation \eqref{exm-3} with 
$\mu=-7$}
\label{fig5}
\end{figure}

(3) As $\mu=-6.2$, we have
$$
x_t^{-}=-143/15,\quad y_t^{-}=2.4, \quad y_{m1}^{-}=8.6, \quad
 y_{m2}^{-}=-13.1.
$$
By numerical simulation, we have $y^{-}=y^{+}$, and there exists a 
sliding-zero cycle illustrated in Figures \ref{fig6} and \ref{fig61},
which is in agreement with the orbit $\mathcal{H}_2$ in Theorem \ref{th3}.

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig6}
\end{center}
\caption{Existence of the sliding-zero cycle of equation \eqref{exm-3} 
with $\mu=-6.2$}
\label{fig6}
\end{figure}

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig61}
\end{center}
\caption{Expansion of the sliding-zero cycle in Figure \ref{fig6}}
\label{fig61}
\end{figure}

(4) As $\mu=-6.15$, we have
$$
x_t^{-}=-123/20,\quad y_t^{-}=2.3, \quad y_{m1}^{-}=8.45, \quad
 y_{m2}^{-}=-523/40.
$$
By numerical simulation, we have $y^{-}<y^{+}$, and there exists the 
attractive limit cycle illustrated in Figures \ref{fig7} and 
\ref{fig71}, which is in agreement with the orbit $\mathcal{H}_6$ 
in Theorem \ref{th3}.

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig7}
\end{center}
\caption{Existence of the attractive limit cycle of equation \eqref{exm-3} 
with $\mu=-6.15$}
\label{fig7}
\end{figure}

\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig71}
\end{center}
\caption{Eexpansion of the attractive limit cycle in Figure \ref{fig7}}
\label{fig71}
\end{figure}

It follows that there is a sliding homoclinic bifurcation 
at $\mu=-8.4$ and a sliding cycle bifurcation at $\mu=-6.2$.
\end{example}

\section{Conclusions}\label{conclusion}

In this article, taking the perturbations to the separation boundary into 
consideration, we studied a special class of planar sector-wise linear 
systems, which are separated by two rays starting from the same point. 
More precisely, the two subsystems of this planar sector-wise linear 
systems are the same except for the positions of $\mathbf{x}_e^{\pm}$. 
We mainly discussed the global qualitative dynamics of the system above 
with $\mathbf{x}_e^{\pm}$ being saddles, since the analysis of dynamics with 
$\mathbf{x}_e^{\pm}$ being sink or source points is similar. 
Because the global dynamics of piecewise smooth systems is so complex 
we put $x_e^{+}>0$, $y_e^{+}=-y_e^{-}>0$ and only left $x_e^{-}$ as a 
bifurcation parameter. Based on our results of the existence of sliding 
set and pseudo-equilibrium points, and the properties of the unique 
non-regular point $\mathbf{p}_0=(0,0)$, we obtained that there exist 
sliding cycle bifurcation and sliding homoclinic bifurcation in the above 
system. Moreover, we also got the existence of all important separatrixes 
and their dependence on the bifurcation parameter $x_e^{-}$. 
For example, there is a sliding heteroclinic orbit, which first turns 
a heteroclinic cycle and then a limit cycle, meanwhile, there exists a 
sliding heteroclinic orbit which first turns a limit cycle and then a 
heteroclinic cycle. In addition, we found the coexistence of a sliding
homoclinic cycle and a sliding heteroclinic orbit, while we also found 
the coexistence of a sliding cycle and a limit cycle.

\subsection*{Acknowledgments}
This work is supported by National Natural Science Foundation of China 
(11301196) and by the Fundamental Research Funds for the Central 
Universities, HUST(2015QN128).

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\end{document}

