Analytical periodic motions in a discontinuous system with a switching hyperbola

Abstract

In this paper, studied are periodic motions in a discontinuous system with three vector fields switching through two branches of a hyperbola. The switchability conditions of flows at the discontinuous boundaries are presented through G-functions. Such conditions are presented for passable motions and grazing motions at the boundary with sliding motions on the boundary. With such analytical conditions, periodic motions with specific map** structures are analytically predicted, and the corresponding stability and bifurcations are presented through eigenvalue analysis. Numerical illustrations of periodic motions are carried out, and the corresponding G-functions are presented for illustration of motion switchability at the hyperbolic boundaries. The discontinuity of periodic motions is shown in phase trajectories. This study tells the complex motions can be obtained through a few individual systems with arbitrary switching boundaries, and the corresponding discontinuous dynamics can be discussed for dynamical system design and control.

Appendix

Consider a linear oscillator in domain $\Omega_{\alpha }$

$$ \ddot{x}^{(i)} + \delta^{(i)} \dot{x}^{(i)} + \alpha^{(i)} x^{(i)} = Q^{(i)} + Q_{0}^{(i)} \cos \Omega t. $$

(51)

With the initial condition $(x_{k} ,\dot{x}_{k} ,t_{k} )$, the solutions to Eq. (51) are presented as follows. Let ${\mathbf{x}}^{(i)} = (x^{(i)} ,\dot{x}^{(i)} )^{\text{T}} = (x^{(i)} ,y^{(i)} )^{\text{T}}$.

Case I: $\Delta = \delta^{(i)2} - 4\alpha^{(i)} > 0$

$$ \begin{aligned} x^{(i)} (t) & = C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )e^{{\lambda_{1}^{(i)} (t - t_{k} )}} + C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )e^{{\lambda_{2}^{(i)} (t - t_{k} )}} \\ & \;\; + A^{(i)} \cos \Omega t + B^{(i)} \sin \Omega t + \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} , \\ \end{aligned} $$

(52)

$$ \begin{aligned} y^{(i)} (t) & = \lambda_{1}^{(i)} C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )e^{{\lambda_{1}^{(i)} (t^{(i)} - t_{k} )}} \\ & \;\; + \lambda_{2}^{(i)} C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )e^{{\lambda_{2}^{(i)} (t^{(i)} - t_{k} )}} \\ & \;\; - A^{(i)} \Omega \sin \Omega t + B^{(i)} \Omega \cos \Omega t \\ \end{aligned} $$

(53)

where

$$ \begin{aligned} \lambda_{1}^{(i)} & = \tfrac{1}{2}( - \delta^{(i)} + \sqrt \Delta ),\lambda_{2}^{(i)} = \tfrac{1}{2}( - \delta^{(i)} - \sqrt \Delta ); \\ A^{(i)} & = \frac{{Q_{0}^{(i)} (\alpha^{(i)} - \Omega^{2} )}}{{(\delta^{(i)} \Omega )^{2} + (\alpha^{(i)} - \Omega^{2} )^{2} }}, \\ B^{(i)} & = \frac{{Q_{0}^{(i)} \delta^{(i)} \Omega }}{{(\delta^{(i)} \Omega )^{2} + (\alpha^{(i)} - \Omega^{2} )^{2} }}; \\ C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) & = \frac{1}{\sqrt \Delta }[y_{k}^{(i)} - \lambda_{2}^{(i)} x_{k}^{(i)} + \lambda_{2}^{(i)} \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} \\ & \;\; + (\lambda_{2}^{(i)} A^{(i)} - B^{(i)} \Omega )\cos \Omega t_{k} \\ & \;\; + (\lambda_{2}^{(i)} B^{(i)} + A^{(i)} \Omega )\sin \Omega t_{k} ], \\ C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) & = \frac{1}{\sqrt \Delta }[\lambda_{1}^{(i)} x_{k}^{(i)} - y_{k}^{(i)} - \lambda_{1}^{(i)} \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} \\ & \;\; - (\lambda_{1}^{(i)} A^{(i)} - B^{(i)} \Omega )\cos \Omega t_{k} \\ & \;\; - (\lambda_{1}^{(i)} B^{(i)} + A^{(i)} \Omega )\sin \Omega t_{k} ]. \\ \end{aligned} $$

(54)

Case II: $\Delta = \delta^{(i)2} - 4\alpha^{(i)} = 0$

$$ \begin{aligned} x^{(i)} (t) & = [C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )(t^{(i)} - t_{k} ) + C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )]e^{{\lambda_{1}^{(i)} (t^{(i)} - t_{k} )}} \\ & \;\; + A^{(i)} \cos \Omega t + B^{(i)} \sin \Omega t + \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} , \\ \end{aligned} $$

(55)

$$ \begin{aligned} \dot{x}^{(i)} (t) & = \left[ {\lambda_{1}^{(i)} C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )(t^{(i)} - t_{k} )} \right. \\ & \;\;\left. { + \lambda_{1}^{(i)} C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) + C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )} \right]e^{{\lambda_{1}^{(i)} (t^{(i)} - t_{k} )}} \\ & \;\; - A^{(i)} \Omega \sin \Omega t + B^{(i)} \Omega \cos \Omega t \\ \end{aligned} $$

(56)

where

$$ \begin{aligned} \lambda_{1}^{(i)} & = \lambda_{2}^{(i)} = - \tfrac{1}{2}\delta^{(i)} ; \\ C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) & = y_{k}^{(i)} - \lambda_{1}^{(i)} x_{k}^{(i)} + \lambda_{1}^{(i)} \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} \\ & \;\; + (\lambda_{1}^{(i)} B^{(i)} + A^{(i)} \Omega )\sin \Omega t_{k} \\ & \;\; + (\lambda_{1}^{(i)} A^{(i)} - B^{(i)} \Omega )\cos \Omega t_{k} ], \\ C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) & = x_{k} - A^{(i)} \cos \Omega t_{k} - B^{(i)} \sin \Omega t_{k} - \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} . \\ \end{aligned} $$

(57)

Case III: $\Delta = \delta^{(i)2} - 4\alpha^{(i)} < 0$

$$ \begin{aligned} x^{(i)} (t) & = e^{{ - \lambda^{(i)} (t^{(i)} - t_{k} )}} \{ C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )\cos [\omega_{d}^{(i)} (t - t_{k} )] \\ & \;\; + C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )\sin [\omega_{d}^{(i)} (t - t_{k} )]\} \\ & \;\; + A^{(i)} \cos \Omega t + B^{(i)} \sin \Omega t + \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} , \\ \end{aligned} $$

(58)

$$ \begin{aligned} y^{(i)} (t) & = e^{{ - \lambda^{(i)} (t^{(i)} - t_{k} )}} \{ [ - \lambda^{(i)} C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) \\ & \;\; + \omega_{d}^{(i)} C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )]\cos [\omega_{d}^{(i)} (t^{(i)} - t_{k} )] \\ & \;\; - [\lambda^{(i)} C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) + \omega_{d}^{(i)} C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} )] \\ & \;\; \times \sin [\omega_{d}^{(i)} (t^{(i)} - t_{k} )]\} \\ & \;\; - A^{(i)} \Omega \sin \Omega t + B^{(i)} \Omega \cos \Omega t \\ \end{aligned} $$

(59)

where

$$ \begin{aligned} \lambda^{(i)} & = \tfrac{1}{2}\delta^{(i)} ,\omega_{d}^{(i)} = \tfrac{1}{2}\sqrt { - \Delta } ; \\ C_{1}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) & = x_{k} - A^{(i)} \cos \Omega t_{k} - B^{(i)} \sin \Omega t_{k} - \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} ; \\ C_{2}^{(i)} ({\mathbf{x}}_{k}^{(i)} ,t_{k} ) & = \frac{1}{{\omega_{d}^{(i)} }}[y_{k} + \lambda^{(i)} x_{k} - \lambda^{(i)} \tfrac{1}{{\alpha^{(i)} }}Q^{(i)} \\ & \;\; + (A^{(i)} \Omega - \lambda^{(i)} B^{(i)} )\sin \Omega t_{k} \\ & \;\; - (B^{(i)} \Omega + \lambda^{(i)} A^{(i)} )\cos \Omega t_{k} ]. \\ \end{aligned} $$

(60)

The dynamic system on the hyperbolic boundary is

$$ {\dot{\mathbf{x}}}^{(0)} = \left( {\begin{array}{*{20}c} {y^{(0)} } \\ {\lambda_{0}^{2} x^{(0)} } \\ \end{array} } \right) $$

(61)

where

$$ \lambda_{0} = \tfrac{b}{a},\;{\mathbf{x}}^{(0)} = (x,y)^{\text{T}} \in \partial \Omega_{12} \cup \partial \Omega_{23} . $$

(62)

The corresponding solution is

$$ x^{(0)} (t) = C_{1} ({\mathbf{x}}_{k}^{(0)} ,t_{k} )e^{{\lambda_{0} (t - t_{k} )}} + C_{2} ({\mathbf{x}}_{k}^{(0)} ,t_{k} )e^{{ - \lambda_{0} (t - t_{k} )}} , $$

(63)

$$ \begin{aligned} y^{(0)} (t) & = \lambda_{0} \left[ {C_{1}^{(0)} ({\mathbf{x}}_{k}^{(0)} ,t_{k} )e^{{\lambda_{0} (t - t_{k} )}} } \right. \\ & \;\;\left. { - C_{2}^{(0)} ({\mathbf{x}}_{k}^{(0)} ,t_{k} )e^{{ - \lambda_{0} (t - t_{k} )}} } \right] \\ \end{aligned} $$

(64)

where

$$ \begin{aligned} & C_{1}^{(0)} ({\mathbf{x}}_{k}^{(0)} ,t_{k} ) = \frac{1}{{2\lambda_{0} }}(\lambda_{0} x_{k}^{(0)} + y_{k}^{(0)} ), \\ & C_{2}^{(0)} ({\mathbf{x}}_{k}^{(0)} ,t_{k} ) = \frac{1}{{2\lambda_{0} }}(\lambda_{0} x_{k}^{(0)} - y_{k}^{(0)} ). \\ \end{aligned} $$

(65)