 orbits is one of important and challenging topics in
both smooth systems and Filippov systems. However, in Filippov systems there are
three classes of new periodic orbits which do not appear in smooth systems because of switching surfaces.
They are so-called crossing periodic orbits, grazing periodic orbits and sliding periodic orbits(see, e.g., \cite{ChL-DB, ChL-YA}).

Many models in applications are linear Filippov systems with a line of discontinuity, such as
the direct voltage control system of the buck converter(see \cite{CO, ChL-EE})
$$
\begin{aligned}
\left(
\begin{array}{c}
\dot x\\
\dot y
\end{array}
\right)=
\left(
\begin{array}{cc}
-a &1\\
-1 &-b
\end{array}
\right)
\left(
\begin{array}{c}
x\\
y
\end{array}
\right)+
\left\{
\begin{aligned}
&\left(
\begin{array}{c}
0\\
0
\end{array}
\right) &&{\rm if}~x>x_{ref},\\
& \left(
\begin{array}{c}
0\\
1
\end{array}
\right) &&{\rm if}~x<x_{ref},
\end{aligned}
\right.
\end{aligned}
$$
where $a>0, b>0$ are normalized parameters of the circuit, $x, y$ are the normalized load
voltage and impedance current, $x_{ref}$ is the desired normalized load voltage.
Another system
$$\ddot x+\alpha\dot x+\mu x{\rm sgn}\dot x+\beta x=0$$
models an unforced mechanical oscillator with dry friction $\mu x{\rm sgn}\dot x$ (see \cite{LT}),
where $\alpha\dot x$ and $\beta x$ denote respectively the viscous damping and restoring force.
This system is also written as a linear  Filippov system
$$
\begin{aligned}
\left(
\begin{array}{c}
\dot x\\
\dot y
\end{array}
\right)=
\left\{
\begin{aligned}
&\left(
\begin{array}{cc}
0 &1\\
-\beta-\mu &-\alpha
\end{array}
\right)\left(
\begin{array}{c}
x\\
y
\end{array}
\right) &&{\rm if}~y>0,\\
& \left(
\begin{array}{cc}
0 &1\\
-\beta+\mu &-\alpha
\end{array}
\right)\left(
\begin{array}{c}
x\\
y
\end{array}
\right) &&{\rm if}~y<0.
\end{aligned}
\right.
\end{aligned}
$$
Thus, the investigation of linear Filippov systems is one of important topics in the field of nonsmooth dynamical systems.

Generally, a linear Filippov system of dimension $2$ with a line of discontinuity is of form
\begin{eqnarray}
\left\{
\begin{aligned}
  \dot z&=A^+z+b^+~~~~&&{\rm if~}z\in\Sigma^+,\\
  \dot z&=A^-z+b^-~ ~&&{\rm if~}z\in\Sigma^-,
\end{aligned}
\right.
\label{PFS}
\end{eqnarray}
where $z:=(x, y)^\top\in\mathbb{R}^2$, $\Sigma^+:=\{z\in\mathbb{R}^2: H(z)>0\}, \Sigma^-:=\{z\in\mathbb{R}^2: H(z)<0\}$,
\begin{eqnarray*}
\begin{aligned}
&A^\pm:=\left(
\begin{array}{cc}
  a^\pm_{11}&a^\pm_{12}\\
  a^\pm_{21}&a^\pm_{22}\\
\end{array}
\right)\in \mathbb{R}^{2\times 2},~~~~b^\pm:=\left(
\begin{array}{c}
  b^\pm_1\\
  b^\pm_2\\
\end{array}
\right)\in \mathbb{R}^2.
\end{aligned}
\end{eqnarray*}
Here $H(z):=c\cdot z+d$, $c=(c_1,c_2)\in\mathbb{R}^2\backslash\{(0,0)\}$, $d\in\mathbb{R}$.
Let
$$\Sigma:=\{z\in\mathbb{R}^2: H(z)=0\}.$$
Sometimes $\Sigma$ is called the {\it switching line} or {\it discontinuity line} of system (\ref{PFS}).
For system (\ref{PFS}), solutions without points in $\Sigma$ are naturally determined by the vector filed $A^+z+b^+$ or $A^-z+b^-$. However, if a solution reaches $\Sigma$ at a time, then a new rule must be adopted to define its evolution. A widely use method is the so-called Filippov convention (see \cite{ChL-AF, ChL-YA}). In particular, as seen in \cite{ChL-YA}, $\Sigma$ is divided into the {\it crossing region}
$$\Sigma^c:=\{z\in \Sigma: ~(c\cdot(A^+z+b^+))(c\cdot(A^-z+b^-))>0\}$$
and the {\it sliding region}
$$\Sigma^s:=\{z\in\Sigma: ~(c\cdot(A^+z+b^+))(c\cdot(A^-z+b^-))\le0\}.$$
In addition,
$$
\begin{aligned}
\Sigma^s_a:&=\{z\in \Sigma^s:~c\cdot(A^+z+b^+)<0,~ c\cdot(A^-z+b^-)>0\},\\
\Sigma^s_r:&=\{z\in \Sigma^s:~c\cdot(A^+z+b^+)>0,~ c\cdot(A^-z+b^-)<0\}
\end{aligned}
$$
are called {\it attractive siding region} and {\it repulsive sliding region}, respectively.
In $\Sigma^c$, both two vector fields are transversal to $\Sigma$ and their normal components have same sign.
Thus the solution passing through a point in $\Sigma^c$ crosses $\Sigma$ at the point. In $\Sigma^s$, either their normal components have opposite sign or at least one of them vanishes. In this case, there exists the so-called {\it sliding solution}, which is the flow of a differential equation
\begin{eqnarray}
\dot z=F^s(z)~~~~~~~{\rm for}~~ z\in\Sigma^s.
\label{sliding}
\end{eqnarray}
For $z\in\Sigma^s$ satisfying $c\cdot(A^-z+b^-)\ne c\cdot(A^+z+b^+)$, $F^s(z)$ is given by
\begin{eqnarray}
F^s(z)=
\frac{c\cdot(A^-z+b^-)(A^+z+b^+)-c\cdot(A^+z+b^+)(A^-z+b^-)}
{c\cdot(A^-z+b^-)-c\cdot(A^+z+b^+)}
\label{sliding0}
\end{eqnarray}
However, for $z\in\Sigma^s$ satisfying $c\cdot(A^-z+b^-)=c\cdot(A^+z+b^+)$, which termed as a {\it singular sliding point},
$F^s(z)$ either is defined by extending (\ref{sliding0}) if an extension is possible or is defined by $0$ (see \cite{ChL-YA}).
Usually, $F^s(z)$ is called the {\it sliding vector field} and its equilibria are called {\it pseudo-equilibria} of system (\ref{PFS}).
In conclusion, the solution of system (\ref{PFS}) can be defined by concatenating the flows of $A^+z+b^+, A^-z+b^-$ and $F^s(z