Sliding homoclinic orbits and bifurcations of three-dimensional piecewise affine systems

Abstract

Sliding dynamics is a peculiar phenomenon to discontinuous dynamical systems, while homoclinic orbits play a role in studying the global dynamics of dynamical systems. This paper provides a method to ensure the existence of sliding homoclinic orbits of three-dimensional piecewise affine systems. In addition, sliding cycles are obtained by bifurcations of the systems with sliding homoclinic orbits to saddles. Two examples with simulations of sliding homoclinic orbits and sliding cycles are provided to illustrate the effectiveness of the results.

Appendices

Appendix A: The proof of Lemma 1

The sliding system (12) can be written as

$$\begin{aligned}{} & {} \left( \begin{array}{cc}\dot{y} \\ \dot{z}\\ \end{array}\right) = \left( \begin{array}{cc} a_{22}y+a_{23}z+a_2\\ a_{32}y+a_{33}z+a_3\\ \end{array}\right) \\ {}{} & {} -\frac{a_{12}y+a_{13}z+a_1}{(a_{12}-b_{12})y+(a_{13}-b_{13})z+a_1-b_1}\times \\{} & {} \left( \begin{array}{cc} (a_{22}-b_{22})y+(a_{23}-b_{23})z+a_2-b_2\\ (a_{32}-b_{32})y+(a_{33}-b_{33})z+a_3-b_3\\ \end{array}\right) , \end{aligned}$$

for $\textbf{x}_s=(0\quad y\quad z)^\intercal \in \Sigma _{s}$, where $\textbf{A}=(a_{ij})_{3\times 3}$, $\mathbf {B(\mu )}=(b_{ij})_{3\times 3}$, $\textbf{a}=(a_i)_{3\times 1}$ and $\textbf{b}(\mu )=(b_i)_{3\times 1}$, $i, j=1, 2, 3$.

Hence, system (12) is an affine system if and only if there exist constants $k_1$, $k_2$ and $k_3$ such that

$$\begin{aligned} \frac{a_{12}y+a_{13}z+a_1}{(a_{12}-b_{12})y+(a_{13}-b_{13})z+a_1-b_1}=k_1, \end{aligned}$$

(1)

or

$$\begin{aligned}{} & {} \frac{(a_{22}-b_{22})y+(a_{23}-b_{23})z+a_2-b_2}{(a_{12}-b_{12})y+(a_{13}-b_{13})z+a_1-b_1}=k_2, \end{aligned}$$

(2)

$$\begin{aligned}{} & {} \frac{(a_{32}-b_{32})y+(a_{33}-b_{33})z+a_3-b_3}{(a_{12}-b_{12})y+(a_{13}-b_{13})z+a_1-b_1}=k_3. \end{aligned}$$

(3)

Since $\textbf{x}_s=(0\quad y\quad z)^\intercal \in \Sigma _{s}$, then

$$\begin{aligned} (a_{12}y+a_{13}z+a_1)(b_{12}y+b_{13}z+a_1)<0, \end{aligned}$$

and $0<k_1<1$. If Eq. (1) holds, then

$$\begin{aligned} \frac{a_{12}}{b_{12}}=\frac{a_{13}}{b_{13}}=\frac{a_1}{b_1}=\frac{k_1}{k_1-1}<0. \end{aligned}$$

(4)

If Eqs. (2) and (3) hold, then

$$\begin{aligned}{} & {} \frac{a_{22}-b_{22}}{a_{12}-b_{12}}=\frac{a_{23}-b_{23}}{a_{13}-b_{13}}=\frac{a_2-b_2}{a_1-b_1}=k_2, \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \frac{a_{32}-b_{32}}{a_{12}-b_{12}}=\frac{a_{33}-b_{33}}{a_{13}-b_{13}}=\frac{a_2-b_2}{a_1-b_1}=k_3. \end{aligned}$$

(6)

And then the sliding system (12) is

$$\begin{aligned} \left( \begin{array}{cc}\dot{y} \\ \dot{z}\\ \end{array}\right) =\left( \begin{array}{cc} a_{22}y+a_{23}z+a_2\\ a_{32}y+a_{33}z+a_3\\ \end{array}\right) \nonumber \\ -(a_{12}y+a_{13}z+a_1) \left( \begin{array}{cc} \frac{a_2-b_2}{a_1-b_1}\\ \frac{a_2-b_2}{a_1-b_1}\\ \end{array}\right) . \end{aligned}$$

(7)

Lemma 1 is proved.

Appendix B: The proof of Lemma 2

The sliding system (13) can be written as

$$\begin{aligned} \left( \begin{array}{ccc}\dot{y} \\ \dot{z}\\ \end{array}\right)= & {} (b_{12}y+b_{13}z+b_1)\left( \begin{array}{cc} a_{22}y+a_{23}z+a_2\\ a_{32}y+a_{33}z+a_3\\ \end{array}\right) \\+ & {} (a_{12}y+a_{13}z+a_1)\left( \begin{array}{cc} b_{22}y+b_{23}z+b_2\\ b_{32}y+b_{33}z+b_3\\ \end{array}\right) ,\\= & {} \left( \begin{array}{ccc} (b_{12}a_{22}-a_{12}b_{22})y^2+(b_{13}a_{23}-a_{13}b_{23})z^2\\ (b_{12}a_{32}-a_{12}b_{32})y^2+(b_{13}a_{33}-a_{13}b_{33})z^2\\ \end{array}\right) \\+ & {} \left( \begin{array}{ccc} (b_{13}a_{22}+b_{12}a_{23}-a_{12}b_{23}-a_{13}b_{22})yz\\ (b_{13}a_{32}+b_{12}a_{33}-a_{12}b_{33}-a_{13}b_{32})yz\\ \end{array}\right) \\+ & {} \left( \begin{array}{ccc} (b_1a_{22}+b_{12}a_2-a_1b_{22}-a_{12}b_2)y\\ (b_1a_{32}+b_{12}a_3-a_1b_{32}-a_{12}b_3)y\\ \end{array}\right) \\+ & {} \left( \begin{array}{ccc} (b_1a_{23}+b_{13}a_2-a_1b_{23}-a_{13}b_2)z+b_1a_2-a_1b_2\\ (b_1a_{33}+b_{13}a_3-a_1b_{33}-a_{13}b_3)z+b_1a_3-a_1b_3\\ \end{array}\right) . \end{aligned}$$

for $\textbf{x}_s=(0\quad y\quad z)^\intercal \in \Sigma _{s}$, where $\textbf{A}=(a_{ij})_{3\times 3}$, $\mathbf {B(\mu )}=(b_{ij})_{3\times 3}$, $\textbf{a}=(a_i)_{3\times 1}$ and $\textbf{b}(\mu )=(b_i)_{3\times 1}$, $i, j=1, 2, 3$.

Hence, system (13) is an affine system if and only if

$$\begin{aligned}{} & {} b_{12}a_{22}-a_{12}b_{22}=0,b_{12}a_{32}-a_{12}b_{32}=0, \\{} & {} b_{13}a_{23}-a_{13}b_{23}=0,b_{13}a_{33}-a_{13}b_{33}=0, \\{} & {} b_{13}a_{22}+b_{12}a_{23}-a_{12}b_{23}-a_{13}b_{22}=0, \\{} & {} b_{13}a_{32}+b_{12}a_{33}-a_{12}b_{33}-a_{13}b_{32}=0. \end{aligned}$$

And then the sliding system (13) is

$$\begin{aligned} \left( \begin{array}{ccc}\dot{y} \\ \dot{z}\\ \end{array}\right)= & {} \left( \begin{array}{ccc} (b_1a_{22}+b_{12}a_2-a_1b_{22}-a_{12}b_2)y\\ (b_1a_{32}+b_{12}a_3-a_1b_{32}-a_{12}b_3)y\\ \end{array}\right) \nonumber \\+ & {} \left( \begin{array}{ccc} (b_1a_{23}+b_{13}a_2-a_1b_{23}-a_{13}b_2)z+b_1a_2-a_1b_2\\ (b_1a_{33}+b_{13}a_3-a_1b_{33}-a_{13}b_3)z+b_1a_3-a_1b_3\\ \end{array}\right) . \nonumber \\ \end{aligned}$$

(1)

Lemma 2 is proved.