Periodic orbits and bifurcations in discontinuous systems with a hyperbolic boundary

Abstract

The periodic orbits and bifurcations in a class of second-order discontinuous systems with a hyperbolic boundary are studied in this paper. Specifically, a periodically forced discontinuous system described by three different linear subsystems is considered mainly to demonstrate the methodology. Analytical conditions for the reachability of local flows in each sub-domain and the passability of flows on the hyperbolic boundary are developed first. Then, through the return mapping structure of motions, the periodic orbits with or without sliding motions are analytically predicted, and the corresponding bifurcation analyses are carried out. Finally, numerical illustrations of periodic orbits are presented and the G-function is proposed to show the analytical criteria. The results obtained in this paper can be applied to the sliding mode control in discontinuous dynamical systems.

Appendix

Consider second order differential equation in Eq. (2) with initial condition $(x_0, \dot{x}_0, t_0)\in R^3$, whose solution $(x^{(i)}(t; x_0, \dot{x}_0, t_0), \dot{x}^{(i)}(t; x_0, \dot{x}_0, t_0))$ (for simplicity, denote by $(x^{(i)}(t), \dot{x}^{(i)}(t))$) in corresponding domains $\Omega _i$ ($i\in \{1, 2, 3\}$) is given as follows.

Case 1      $d_i^2>c_i$.

$$\begin{aligned} x^{(i)}(t)= & {} C_1^{(i)} \mathrm{e}^{\lambda _1^{(i)}(t-t_0)}+ C_2^{(i)} \mathrm{e}^{\lambda _2^{(i)}(t-t_0)}\\&+\,A^{(i)}\cos (\Omega t+\phi )+B^{(i)}\sin (\Omega t+\phi )+C^{(i)},\\ \dot{x}^{(i)}(t)= & {} \lambda _1^{(i)} C_1^{(i)} \mathrm{e}^{\lambda _1^{(i)}(t-t_0)}+\lambda _2^{(i)} C_2^{(i)} \mathrm{e}^{\lambda _2^{(i)}(t-t_0)}\\&-\,A^{(i)}\Omega \sin (\Omega t+\phi )+B^{(i)}\Omega \cos (\Omega t+\phi ), \end{aligned}$$

where $\lambda _{1,2}^{(i)}=-d_i\pm \sqrt{d_i^2-c_i},\,\,\omega _d^{(i)}=\sqrt{d_i^2-c_i}$, and

$$\begin{aligned} A^{(i)}= & {} \frac{A_0(c_i-\Omega ^2)}{(c_i-\Omega ^2)^2+(2d_i\Omega )^2},\\ B^{(i)}= & {} \frac{2d_i\Omega A_0}{(c_i-\Omega ^2)^2+(2d_i\Omega )^2},\,\,\,\,C^{(i)}=-\frac{b_i}{c_i},\end{aligned}$$

$$\begin{aligned} C_1^{(i)}= & {} \frac{1}{2\omega _d^{(i)}}\left\{ -\left[ B^{(i)}\Omega +A^{(i)}(d_i+\omega _d^{(i)})\right] \cos (\Omega t_0+\phi )\right. \\&+\,\left[ A^{(i)}\Omega -B^{(i)}(d_i+\omega _d^{(i)})\right] \sin (\Omega t_0+\phi )\\&\left. -\,(d_i+\omega _d^{(i)})(C^{(i)}-x_0)+\dot{x}_0 \right\} ,\\ C_2^{(i)}= & {} \frac{1}{2\omega _d^{(i)}}\left\{ \left[ B^{(i)}\Omega +A^{(i)}(d_i-\omega _d^{(i)})\right] \cos (\Omega t_0+\phi )\right. \\&-\,\left[ A^{(i)}\Omega +B^{(i)}(-d_i+\omega _d^{(i)})\right] \sin (\Omega t_0+\phi )\\&\left. +\,(d_i-\omega _d^{(i)})(C^{(i)}-x_0)-\dot{x}_0 \right\} . \end{aligned}$$

Case 2     $d_i^2<c_i$.

$$\begin{aligned} x^{(i)}(t)= & {} \left[ C_1^{(i)}\cos \omega _d^{(i)}(t-t_0)+ C_2^{(i)}\sin \omega _d^{(i)}(t-t_0) \right] \\&\times \,\mathrm{e}^{-d_i(t-t_0)}+\,A^{(i)}\cos (\Omega t+\phi )\\&+B^{(i)}\sin (\Omega t+\phi )+C^{(i)},\\ \dot{x}^{(i)}(t)= & {} \left[ (-d_iC_1^{(i)}+\omega _d^{(i)}C_2^{(i)})\cos \omega _d^{(i)}(t-t_0)\right. \\&\left. -\, (\omega _d^{(i)}C_1^{(i)}+d_iC_2^{(i)})\sin \omega _d^{(i)}(t-t_0)\right] \mathrm{e}^{-d_i(t-t_0)}\\&-\,A^{(i)}\Omega \sin (\Omega t+\phi )+B^{(i)}\Omega \cos (\Omega t+\phi ), \end{aligned}$$

where $\omega _d^{(i)}=\sqrt{c_i-d_i^2}$, constants $A^{(i)},\,\,B^{(i)}$ and $C^{(i)}$ are the same as case 1), and

$$\begin{aligned} C_1^{(i)}= & {} x_0-\left[ A^{(i)}\cos (\Omega t_0\!+\!\phi )+B^{(i)}\sin (\Omega t_0+\phi )+C^{(i)}\!\right] \!, \\ C_2^{(i)}= & {} \frac{1}{\omega _d^{(i)}}\left[ \dot{x}_0-(A^{(i)}d_i+B^{(i)}\Omega )\cos (\Omega t_0+\phi )\right. \\&\left. +\,(A^{(i)}\Omega -B^{(i)}d_i)\sin (\Omega t_0+\phi )+d_i(x_0-C^{(i)})\right] . \end{aligned}$$

Case 3      $d_i^2=c_i$.

$$\begin{aligned} x^{(i)}(t)= & {} [C_1^{(i)}+ C_2^{(i)}(t-t_0)]\mathrm{e}^{\lambda _1^{(i)}(t-t_0)}\\&+\,A^{(i)}\cos (\Omega t+\phi )+B^{(i)}\sin (\Omega t+\phi )+C^{(i)},\\ \dot{x}^{(i)}(t)= & {} [\lambda _1^{(i)}C_1^{(i)}+ C_2^{(i)}+\lambda _1^{(i)}C_2^{(i)}(t-t_0)]\mathrm{e}^{\lambda _1^{(i)}(t-t_0)}\\&-\,A^{(i)}\Omega \sin (\Omega t+\phi )+B^{(i)}\Omega \cos (\Omega t+\phi ), \end{aligned}$$

where $\lambda _1^{(i)}=-2d_i$, constants $A^{(i)},\,\,B^{(i)}$ and $C^{(i)}$ are the same as case 1), and

$$\begin{aligned} C_1^{(i)}= & {} x_0-\left[ A^{(i)}\cos (\Omega t_0+\phi )\!+\!B^{(i)}\sin (\Omega t_0+\phi )+C^{(i)}\!\right] \!, \\ C_2^{(i)}= & {} \dot{x}_0-(A^{(i)}d_i+B^{(i)}\Omega )\cos (\Omega t_0+\phi )+(A^{(i)}\Omega \\&-\,B^{(i)}d_i)\sin (\Omega t_0+\phi )+d_i(x_0-C^{(i)}). \end{aligned}$$