Sequential symmetric periodic motions in a symmetric discontinuous dynamical system

Abstract

In this paper, grazing-induced sequential symmetric periodic motions in a symmetric discontinuous dynamical system with a symmetric hyperbolic boundary are studied analytically. The analytical switching conditions and grazing conditions at the hyperbola boundary were presented analytically. The sequential symmetric period-(2m−1) motions caused by grazing bifurcations are discussed. Between the two sequential symmetric periodic motions, other symmetric and asymmetric periodic motions exist. The analytical predictions of periodic motions with specific map** structures are carried out, and the corresponding stability of periodic motions are determined through eigenvalue analysis. The complexity of the sequential symmetric periodic motions was illustrated through phase trajectories, and the slow and fast movements in trajectories were observed. Such sequential symmetric period-(2m−1) motions in symmetric discontinuous dynamical systems are similar to sequential periodic motions in the van der Pol oscillator.

Appendix

Consider a linear damped oscillator with dimensionless parameters in the subdomain $\Omega_{i}$

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

(A.1)

With the initial condition ($x_{k} ,y_{k} ,t_{k}$), the solutions for Eq. (A.1) are given as follows:

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

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

  (A.2)

  where

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

  (A.3)

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

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

  (A.4)

  where

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

  (A.5)

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

  $$ \begin{aligned} x^{{(i)}} (t) =\, & e^{{ - \delta ^{{(i)}} (t - t_{k} )/2}} [C_{1} \cos \omega _{d} (t {-} t_{k} ) {+} C_{2} \sin \omega _{d} (t {-} t_{k} )] \\ & + A^{{(i)}} \cos \Omega t + B^{{(i)}} \sin \Omega t + \tfrac{1}{{\alpha ^{{(i)}} }}Q^{{(i)}} , \\ y^{{(i)}} (t) = \,& e^{{ - \delta ^{{(i)}} (t - t_{k} )/2}} [( - \tfrac{1}{2}\delta ^{{(i)}} C_{1} + C_{2} \omega _{d} )\cos \omega _{d} (t - t_{k} ) \\ & - (\tfrac{1}{2}\delta ^{{(i)}} C_{2} + C_{1} \sqrt \Delta )\sin \omega _{d} (t - t_{k} )] \\ & - A^{{(i)}} \Omega \sin \Omega t + B^{{(i)}} \Omega \cos \Omega t \\ \end{aligned} $$

  (A.6)

  where

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

  (A.7)

The Jacobian matrix components are given as follows.

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

  $$ \begin{aligned} \frac{{\partial g_{k1} }}{{\partial x_{k} }} =\, & \frac{{\partial C_{1} }}{{\partial x_{k} }}e^{{\lambda_{1} (t_{k + 1} - t_{k} )}} + \frac{{\partial C_{2} }}{{\partial x_{k} }}e^{{\lambda_{2} (t_{k + 1} - t_{k} )}} , \\ \frac{{\partial g_{k1} }}{{\partial y_{k} }} =\, & \frac{{\partial C_{1} }}{{\partial y_{k} }}e^{{\lambda_{1} (t_{k + 1} - t_{k} )}} + \frac{{\partial C_{2} }}{{\partial y_{k} }}e^{{\lambda_{2} (t_{k + 1} - t_{k} )}} , \\ \frac{{\partial g_{k1} }}{{\partial t_{k} }} =\, & [\frac{{\partial C_{1} }}{{\partial t_{k} }} - C_{1} \lambda_{1} ]e^{{\lambda_{1} (t_{k + 1} - t_{k} )}} \\ &\quad + [\frac{{\partial C_{2} }}{{\partial t_{k} }} - C_{2} \lambda_{2} ]e^{{\lambda_{2} (t_{k + 1} - t_{k} )}} . \\ \end{aligned} $$

  (A.8)
  $$ \begin{aligned} \frac{{\partial g_{{k2}} }}{{\partial x_{k} }} =\, & \lambda _{1} \frac{{\partial C_{1} }}{{\partial x_{k} }}e^{{\lambda _{1} (t_{{k + 1}} - t_{k} )}} + \lambda _{2} \frac{{\partial C_{2} }}{{\partial x_{k} }}e^{{\lambda _{2} (t_{{k + 1}} - t_{k} )}} , \\ \frac{{\partial g_{{k2}} }}{{\partial x_{k} }} =\, & \lambda _{1} \frac{{\partial C_{1} }}{{\partial x_{k} }}e^{{\lambda _{1} (t_{{k + 1}} - t_{k} )}} + \lambda _{2} \frac{{\partial C_{2} }}{{\partial x_{k} }}e^{{\lambda _{2} (t_{{k + 1}} - t_{k} )}} , \\ \frac{{\partial g_{{k2}} }}{{\partial t_{k} }} =\, & [ - \lambda _{1}^{2} C_{1} + \lambda _{1} \frac{{\partial C_{1} }}{{\partial t_{k} }}]e^{{\lambda _{1} (t_{{k + 1}} - t_{k} )}} \\ & + [ - \lambda _{2}^{2} C_{2} + \lambda _{2} \frac{{\partial C_{2} }}{{\partial t_{k} }}]e^{{\lambda _{2} (t_{{k + 1}} - t_{k} )}} \\ \end{aligned} $$

  (A.9)
  $$ \frac{{\partial h_{k} }}{{\partial x_{k} }} = \frac{{\partial h_{k} }}{{\partial y_{k} }} = \frac{{\partial h_{k} }}{{\partial t_{k} }} = 0; $$

  (A.10)
  $$ \begin{aligned} \frac{{\partial g_{{k1}} }}{{\partial x_{{k + 1}} }} =\, & - 1,\frac{{\partial g_{{k1}} }}{{\partial y_{{k + 1}} }} = 0,\frac{{\partial g_{{k2}} }}{{\partial x_{{k + 1}} }} = 0,\frac{{\partial g_{{k2}} }}{{\partial y_{{k + 1}} }} = - 1; \\ \frac{{\partial g_{{k1}} }}{{\partial t_{{k + 1}} }} =\, & \lambda _{1} C_{1} e^{{\lambda _{1} (t_{{k + 1}} - t_{k} )}} + \lambda _{2} C_{2} e^{{\lambda _{2} (t_{{k + 1}} - t_{k} )}} \\ & - A^{{(i)}} \Omega \sin \Omega t_{{k + 1}} + B^{{(i)}} \Omega \cos \Omega t_{{k + 1}} , \\ \frac{{\partial g_{{k2}} }}{{\partial t_{{k + 1}} }} = \,& \lambda _{1}^{2} C_{1} e^{{\lambda _{1} (t_{{k + 1}} - t_{k} )}} + \lambda _{2}^{2} C_{2} e^{{\lambda _{2} (t_{{k + 1}} - t_{k} )}} \\ & - A^{{(i)}} \Omega ^{2} \cos \Omega t_{{k + 1}} - B^{{(i)}} \Omega ^{2} \sin \Omega t_{{k + 1}} \\ \end{aligned} $$

  (A.11)
  $$ \frac{{\partial h_{k} }}{{\partial x_{k + 1} }} = \frac{2}{{a^{2} }}x_{k + 1} ,\;\frac{{\partial h_{k} }}{{\partial y_{k + 1} }} = - \frac{2}{{b^{2} }}y_{k + 1} ,\;\frac{{\partial h_{k} }}{{\partial t_{k + 1} }} = 0 $$

  (A.12)

  where

  $$ \begin{aligned} \frac{{\partial C_{1} }}{{\partial x_{k} }} =\, & \frac{{\lambda _{2} }}{{\lambda _{2} - \lambda _{1} }},\frac{{\partial C_{1} }}{{\partial y_{k} }} = \frac{1}{{\lambda _{1} - \lambda _{2} }}, \\ \frac{{\partial C_{1} }}{{\partial t_{k} }} =\, & \frac{1}{{2w_{d} }} [{(A^{{(i)}} \Omega ^{2} + \lambda _{2} B^{{(i)}} \Omega )\cos \Omega t_{k} } \\ & { + (B^{{(i)}} \Omega ^{2} - \lambda _{2} A^{{(i)}} \Omega )\sin \Omega t_{k} } ]; \\ \end{aligned} $$

  (A.13)
  $$ \begin{aligned} \frac{{\partial C_{2} }}{{\partial x_{k} }} =\, & \frac{{\lambda _{1} }}{{\lambda _{1} - \lambda _{2} }},{\text{ }}\frac{{\partial C_{2} }}{{\partial y_{k} }} = \frac{1}{{\lambda _{2} - \lambda _{1} }}, \\ \frac{{\partial C_{2} }}{{\partial t_{k} }} =\, & \frac{1}{{2w_{d} }} [ {(A^{{(i)}} \Omega ^{2} + \lambda _{1} B^{{(i)}} \Omega )\cos \Omega t_{k} } \\ & { + (B^{{(i)}} \Omega ^{2} - \lambda _{1} A^{{(i)}} \Omega )\sin \Omega t_{k} } ]. \\ \end{aligned} $$

  (A.14)

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

  $$ \begin{aligned} \frac{{\partial g_{{k1}} }}{{\partial x_{k} }} =\, & [\frac{{\partial C_{1} }}{{\partial x_{k} }} + \frac{{\partial C_{2} }}{{\partial x_{k} }}(t_{{k + 1}} - t_{k} )]e^{{ - \tfrac{1}{2}\delta ^{{(i)}} (t_{{k + 1}} - t_{k} )}} , \\ \frac{{\partial g_{{k1}} }}{{\partial y_{k} }} =\, & [\frac{{\partial C_{2} }}{{\partial y_{k} }}(t_{{k + 1}} - t_{k} )]e^{{ - \tfrac{1}{2}\delta ^{{(i)}} (t_{{k + 1}} - t_{k} )}} , \\ \frac{{\partial g_{{k1}} }}{{\partial t_{k} }} =\, & \{ \tfrac{1}{2}\delta ^{{(i)}} [C_{1} + C_{2} (t_{{k + 1}} - t_{k} )] - C_{2} \\ & + \frac{{\partial C_{1} }}{{\partial t_{k} }} + \frac{{\partial C_{2} }}{{\partial t_{k} }}(t_{{k + 1}} - t_{k} )\} e^{{ - \tfrac{1}{2}\delta ^{{(i)}} (t_{{k + 1}} - t_{k} )}} \\ \end{aligned} $$

  (A.15)
  $$ \begin{aligned} \frac{{\partial g_{k2} }}{{\partial x_{k} }} =\, & [\frac{{\partial \mu_{1} }}{{\partial x_{k} }} + \frac{{\partial \mu_{2} }}{{\partial x_{k} }}(t_{k + 1} - t_{k} )]e^{{ - \tfrac{1}{2}\delta^{(i)} (t_{k + 1} - t_{k} )}} , \\ \frac{{\partial g_{k2} }}{{\partial y_{k} }} =\, & [\frac{{\partial \mu_{1} }}{{\partial y_{k} }} + \frac{{\partial \mu_{2} }}{{\partial y_{k} }}(t_{k + 1} - t_{k} )]e^{{ - \tfrac{1}{2}\delta^{(i)} (t_{k + 1} - t_{k} )}} , \\ \frac{{\partial g_{k2} }}{{\partial t_{k} }} =\, & \{ \tfrac{1}{2}\delta^{(i)} [\mu_{1} + \mu_{2} (t_{k + 1} - t_{k} )]\\ &- \mu_{2} + \frac{{\partial \mu_{1} }}{{\partial t_{k} }} + \frac{{\partial \mu_{2} }}{{\partial t_{k} }}(t_{k + 1} - t_{k} )\} e^{{ - \tfrac{1}{2}\delta^{(i)} (t_{k + 1} - t_{k} )}} \\ \end{aligned} $$

  (A.16)
  $$ \frac{{\partial h_{k} }}{{\partial x_{k} }} = \frac{{\partial h_{k} }}{{\partial y_{k} }} = \frac{{\partial h_{k} }}{{\partial t_{k} }} = 0, $$

  (A.17)
  $$ \frac{{\partial g_{k1} }}{{\partial x_{k + 1} }} = - 1,\frac{{\partial g_{k1} }}{{\partial y_{k + 1} }} = 0,\frac{{\partial g_{k2} }}{{\partial x_{k + 1} }} = 0,\frac{{\partial g_{k2} }}{{\partial y_{k + 1} }} = - 1, $$

  (A.18)
  $$ \begin{aligned} \frac{{\partial g_{{k1}} }}{{\partial t_{{k + 1}} }} =\, & - e^{{ - \tfrac{1}{2}\delta ^{{(i)}} (t_{{k + 1}} - t_{k} )}} \{ \tfrac{1}{2}\delta ^{{(i)}} [C_{1} + C_{2} (t_{{k + 1}} - t_{k} )] \\ & - C_{2} - A^{{(i)}} \Omega \sin \Omega t_{{k + 1}} + B^{{(i)}} \Omega \cos \Omega t_{{k + 1}} \} , \\ \frac{{\partial g_{{k2}} }}{{\partial t_{{k + 1}} }} =\, & - e^{{ - \tfrac{1}{2}\delta ^{{(i)}} (t_{{k + 1}} - t_{k} )}} \{ \tfrac{1}{2}\delta ^{{(i)}} [\mu _{1} + \mu _{2} (t_{{k + 1}} - t_{k} )] \\ & - \mu _{2} - A^{{(i)}} \Omega ^{2} \cos \Omega t_{{k + 1}} - B^{{(i)}} \Omega ^{2} \sin \Omega t_{{k + 1}} \} ; \\ \end{aligned} $$

  (A.19)
  $$ \frac{{\partial h_{k} }}{{\partial x_{k + 1} }} = \frac{2}{{a^{2} }}x_{k + 1} ,\frac{{\partial h_{k} }}{{\partial y_{k + 1} }} = - \frac{2}{{b^{2} }}y_{k + 1} ,\frac{{\partial h_{k} }}{{\partial t_{k + 1} }} = 0. $$

  (A.20)

  where

  $$ \begin{aligned} \frac{{\partial C_{1} }}{{\partial x_{k} }} = \,& - \lambda _{1} ,{\text{ }}\frac{{\partial C_{1} }}{{\partial y_{k} }} = 1,{\text{ }}\frac{{\partial C_{2} }}{{\partial x_{k} }} = 1, \\ {\text{ }}\frac{{\partial C_{2} }}{{\partial y_{k} }} =\, & 0,\frac{{\partial C_{1} }}{{\partial t_{k} }} = A^{{(i)}} \Omega \sin \Omega t_{k} - B^{{(i)}} \Omega \cos \Omega t_{k} , \\ \frac{{\partial C_{2} }}{{\partial t_{k} }} =\, & ( {A^{{(i)}} \Omega ^{2} - \tfrac{1}{2}\delta ^{{(i)}} B^{{(i)}} \Omega )\cos \Omega t_{k} } \\ & { + (B^{{(i)}} \Omega ^{2} + \tfrac{1}{2}\delta ^{{(i)}} A^{{(i)}} \Omega } )\sin \Omega t_{k} ; \\ \end{aligned} $$

  (A.21)
  $$ \mu_{1} = C_{2} - \tfrac{1}{2}\delta^{(i)} C_{1} ,\mu_{2} = - \tfrac{1}{2}\delta^{(i)} C_{2} ; $$

  (A.22)
  $$ \begin{aligned} \frac{{\partial \mu_{1} }}{{\partial x_{k} }} =\, & 1 + \tfrac{1}{2}\lambda_{1} \delta^{(i)} ,\frac{{\partial \mu_{1} }}{{\partial y_{k} }} = - \tfrac{1}{2}\delta^{(i)} , \, \frac{{\partial \mu_{2} }}{{\partial x_{k} }} = 1, \, \frac{{\partial \mu_{2} }}{{\partial y_{k} }} = 0, \\ \frac{{\partial \mu_{1} }}{{\partial t_{k} }} =\, & \frac{{\partial C_{2} }}{{\partial t_{k} }} - \tfrac{1}{2}\delta^{(i)} \frac{{\partial C_{1} }}{{\partial t_{k} }},\frac{{\partial \mu_{2} }}{{\partial t_{k} }} = - \tfrac{1}{2}\delta^{(i)} \frac{{\partial C_{2} }}{{\partial t_{k} }}. \\ \end{aligned} $$

  (A.23)

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

  $$ \begin{aligned} \frac{{\partial g_{{k1}} }}{{\partial x_{k} }} =\, & e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} [ {\frac{{\partial C_{1} }}{{\partial x_{k} }}\cos \omega _{d} (t_{{k + 1}} - t_{k} )} \\ & { + \frac{{\partial C_{2} }}{{\partial x_{k} }}\sin \omega _{d} (t_{{k + 1}} - t_{k} )}], \\ \frac{{\partial g_{{k1}} }}{{\partial y_{k} }} =\, & e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} \frac{{\partial C_{2} }}{{\partial y_{k} }}\sin \omega _{d} (t_{{k + 1}} - t_{k} ), \\ \frac{{\partial g_{{k1}} }}{{\partial t_{k} }} =\, & e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} [ {(\eta _{1} + \frac{{\partial C_{1} }}{{\partial t_{k} }})\cos \omega _{d} (t_{{k + 1}} - t_{k} )} \\ & { + (\eta _{2} + \frac{{\partial C_{2} }}{{\partial t_{k} }})\sin \omega _{d} (t_{{k + 1}} - t_{k} )}]; \\ \end{aligned} $$

  (A.24)
  $$ \begin{aligned} \frac{{\partial g_{{k2}} }}{{\partial x_{k} }} =\, & - e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} [ {\frac{{\partial \eta _{1} }}{{\partial x_{k} }}\cos \omega _{d} (t_{{k + 1}} - t_{k} )} \\ & { + \frac{{\partial \eta _{2} }}{{\partial x_{k} }}\sin \omega _{d} (t_{{k + 1}} - t_{k} )]} ], \\ \frac{{\partial g_{{k2}} }}{{\partial y_{k} }} =\, & - e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} [ {\frac{{\partial \eta _{1} }}{{\partial y_{k} }}\cos \omega _{d} (t_{{k + 1}} - t_{k} )} \\ & { + \frac{{\partial \eta _{2} }}{{\partial y_{k} }}\sin \omega _{d} (t_{{k + 1}} - t_{k} )} ], \\ \frac{{\partial g_{{k2}} }}{{\partial t_{k} }} =\, & - e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} [ {(\gamma _{1} + \frac{{\partial \eta _{1} }}{{\partial t_{k} }})\cos \omega _{d} (t_{{k + 1}} - t_{k} )} \\ & { + (\gamma _{2} + \frac{{\partial \eta _{2} }}{{\partial t_{k} }})\sin \omega _{d} (t_{{k + 1}} - t_{k} )} ]; \\ \end{aligned} $$

  (A.25)
  $$ \frac{{\partial h_{k} }}{{\partial x_{k} }} = \frac{{\partial h_{k} }}{{\partial y_{k} }} = \frac{{\partial h_{k} }}{{\partial t_{k} }} = 0, $$

  (A.26)
  $$ \begin{aligned} \frac{{\partial g_{{k1}} }}{{\partial x_{{k + 1}} }} =\, & - 1,\;\frac{{\partial g_{{k1}} }}{{\partial y_{{k + 1}} }} = 0,\;\frac{{\partial g_{{k2}} }}{{\partial x_{{k + 1}} }} = 0,\;\frac{{\partial g_{{k2}} }}{{\partial y_{{k + 1}} }} = - 1; \\ \frac{{\partial g_{{k1}} }}{{\partial t_{{k + 1}} }} =\, & - e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} \left[ {\eta _{1} \cos \omega _{d} (t_{{k + 1}} - t_{k} )} \right. \\ & \left. { + \eta _{2} \sin \omega _{d} (t_{{k + 1}} - t_{k} )} \right] - A^{{(i)}} \Omega \sin \Omega t_{{k + 1}} \\ &\quad + B^{{(i)}} \Omega \cos \Omega t_{{k + 1}} , \\ \frac{{\partial g_{{k2}} }}{{\partial t_{{k + 1}} }} =\, & e^{{ - \delta ^{{(i)}} (t_{{k + 1}} - t_{k} )/2}} \left[ {\gamma _{1} \cos \omega _{d} (t_{{k + 1}} - t_{k} )} \right. \\ & \left. { + \gamma _{2} \sin \omega _{d} (t_{{k + 1}} - t_{k} )} \right] - A^{{(i)}} \Omega ^{2} \cos \Omega t_{{k + 1}} \\ &\quad - B^{{(i)}} \Omega ^{2} \sin \Omega t_{{k + 1}} . \\ \end{aligned} $$

  (A.27)
  $$ \frac{{\partial h_{k} }}{{\partial x_{k + 1} }} = \frac{2}{{a^{2} }}x_{k + 1} ,\;\frac{{\partial h_{k} }}{{\partial y_{k + 1} }} = - \frac{2}{{b^{2} }}y_{k + 1} ,\;\frac{{\partial h_{k} }}{{\partial t_{k + 1} }} = 0 $$

  (A.28)

  where

  $$ \begin{aligned} \frac{{\partial C_{1} }}{{\partial x_{k} }} =\, & 1,{\text{ }}\frac{{\partial C_{1} }}{{\partial t_{k} }} = A^{{(i)}} \Omega \sin \Omega t_{k} - B^{{(i)}} \Omega \cos \Omega t_{k} , \\ \frac{{\partial C_{2} }}{{\partial x_{k} }} =\, & \frac{{\delta ^{{(i)}} }}{{2\omega _{d} }},{\text{ }}\frac{{\partial C_{2} }}{{\partial y_{k} }} = \frac{1}{{\omega _{d} }}; \\ \frac{{\partial C_{2} }}{{\partial t_{k} }} = & \frac{1}{{\omega _{d} }}\left[ {(A^{{(i)}} \Omega ^{2} - \tfrac{1}{2}B^{{(i)}} \delta ^{{(i)}} )\cos \Omega t_{k} } \right. \\ & \left. { + (B^{{(i)}} \Omega ^{2} + \tfrac{1}{2}A^{{(i)}} \delta ^{{(i)}} )\sin \Omega t_{k} } \right]; \\ \end{aligned} $$

  (A.29)
  $$ \eta_{1} = \tfrac{1}{2}\delta^{(i)} C_{1} - \omega_{d} C_{2} ,\eta_{2} = \tfrac{1}{2}\delta^{(i)} C_{2} + \omega_{d} C_{1} ; $$

  (A.30)
  $$ \begin{aligned} \frac{{\partial \eta _{1} }}{{\partial x_{k} }} =\, & 0,{\text{ }} \\ \frac{{\partial \eta _{2} }}{{\partial x_{k} }} =\, & \tfrac{1}{{4\omega _{d} }}(\delta ^{{(i)}} )^{2} + \omega _{d} ; \\ \frac{{\partial \eta _{1} }}{{\partial y_{k} }} =\, & - 1,{\text{ }}\frac{{\partial \eta _{2} }}{{\partial y_{k} }} = \tfrac{1}{{2\omega _{d} }}\delta ^{{(i)}} ; \\ \end{aligned} $$

  (A.31)
  $$ \begin{aligned} \frac{{\partial \eta _{1} }}{{\partial t_{k} }} =\, & [\tfrac{1}{2}\delta ^{{(i)}} B^{{(i)}} (1 - \Omega ) - A^{{(i)}} \Omega ^{2} ]\cos \Omega t_{k} \\ & - [\tfrac{1}{2}\delta ^{{(i)}} A^{{(i)}} (1 - \Omega ) + B^{{(i)}} \Omega ^{2} ]\sin \Omega t_{k} , \\ \frac{{\partial \eta _{2} }}{{\partial t_{k} }} = & [\tfrac{1}{{2\omega _{d} }}\delta ^{{(i)}} (A^{{(i)}} \Omega ^{2} - \tfrac{1}{2}\delta ^{{(i)}} B^{{(i)}} ) - \omega _{d} \Omega B^{{(i)}} ]\cos \Omega t_{k} \\ & + [\tfrac{1}{{2\omega _{d} }}\delta ^{{(i)}} (\Omega ^{2} B^{{(i)}} + \tfrac{1}{2}\delta ^{{(i)}} A^{{(i)}} ) \\ & + \omega _{d} \Omega A^{{(i)}} ]\sin \Omega t_{k} ; \\ \end{aligned} $$

  (A.32)
  $$ \gamma_{1} = \tfrac{1}{2}\delta^{(i)} \eta_{1} - w_{d} \eta_{2} ,\gamma_{2} = \tfrac{1}{2}\delta^{(i)} \eta_{2} + \omega_{d} \eta_{1} . $$

  (A.33)

• Case IV: Sliding motion

  $$ \lambda_{0} = \frac{a}{b},C_{1} = \tfrac{1}{2}(x_{k} + \tfrac{1}{{\lambda_{0} }}y_{k} ),C_{2} = \tfrac{1}{2}(x_{k} - \tfrac{1}{{\lambda_{0} }}y_{k} ) $$

  (A.34)
  $$ \begin{aligned} \frac{{\partial g_{k1} }}{{\partial x_{k} }} =\, & \tfrac{1}{2}(C_{1} e^{{\lambda_{0} (t_{k + 1} - t_{k} )}} + C_{2} e^{{ - \lambda_{0} (t_{k + 1} - t_{k} )}} ), \\ \frac{{\partial g_{k1} }}{{\partial y_{k} }} =\, & \tfrac{1}{{2\lambda_{0} }}(e^{{\lambda_{0} (t_{k + 1} - t_{k} )}} - e^{{ - \lambda_{0} (t_{k + 1} - t_{k} )}} ), \\ \frac{{\partial g_{k1} }}{{\partial t_{k} }} =\, & \tfrac{1}{2}( - C_{1} e^{{\lambda_{0} (t_{k + 1} - t_{k} )}} + C_{2} e^{{ - \lambda_{0} (t_{k + 1} - t_{k} )}} ); \\ \end{aligned} $$

  (A.35)
  $$ \begin{aligned} \frac{{\partial g_{k2} }}{{\partial x_{k} }} =\, & \tfrac{1}{2}\lambda_{0} (e^{{\lambda_{0} (t_{k + 1} - t_{k} )}} - e^{{ - \lambda_{0} (t_{k + 1} - t_{k} )}} ), \\ \frac{{\partial g_{k2} }}{{\partial y_{k} }} =\, & \tfrac{1}{2}(e^{{\lambda_{0} (t_{k + 1} - t_{k} )}} + e^{{ - \lambda_{0} (t_{k + 1} - t_{k} )}} ), \\ \frac{{\partial g_{k2} }}{{\partial t_{k} }} =\, & \tfrac{1}{2}\lambda_{0} ( - C_{1} e^{{\lambda_{0} (t_{k + 1} - t_{k} )}} - C_{2} e^{{ - \lambda_{0} (t_{k + 1} - t_{k} )}} ); \\ \end{aligned} $$

  (A.36)
  $$ \begin{gathered} \frac{{\partial g_{k1} }}{{\partial x_{k + 1} }} = - 1,\;\frac{{\partial g_{k1} }}{{\partial y_{k + 1} }} = 0,\;\frac{{\partial g_{k1} }}{{\partial t_{k + 1} }} = - \frac{{\partial g_{k1} }}{{\partial t_{k} }}; \hfill \\ \frac{{\partial g_{k2} }}{{\partial x_{k + 1} }} = 0,\;\;\frac{{\partial g_{k2} }}{{\partial y_{k + 1} }} = - 1,\;\frac{{\partial g_{k2} }}{{\partial t_{k + 1} }} = - \frac{{\partial g_{k2} }}{{\partial t_{k} }}. \hfill \\ \end{gathered} $$

  (A.37)
  $$ \begin{aligned} \frac{{\partial h_{k} }}{{\partial x_{k} }} =\, & \frac{{\partial h_{k} }}{{\partial y_{k} }} = \frac{{\partial h_{k} }}{{\partial t_{k} }} = 0, \\ {\text{ }}\frac{{\partial h_{k} }}{{\partial x_{{k + 1}} }} =\, & ( {\frac{{\alpha ^{{(i)}} a}}{b} + \frac{b}{a}} )\frac{{y_{{k + 1}}^{2} }}{{\sqrt {y_{{k + 1}}^{2} + b^{2} } }}, \\ \frac{{\partial h_{k} }}{{\partial t_{{k + 1}} }} =\, & \frac{{\delta ^{{(i)}} ay_{{k + 1}} }}{{b\sqrt {y_{{k + 1}}^{2} + b^{2} } }}, \\ \frac{{\partial h_{k} }}{{\partial y_{{k + 1}} }} =\, & \frac{a}{b}\frac{{y_{{k + 1}}^{2} + b^{2} - 2y_{{k + 1}} }}{{(y_{{k + 1}}^{2} + b^{2} )^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} }} [(\alpha ^{{(i)}} + \frac{{b^{2} }}{{a^{2} }})x_{{k + 1}} \\ &\quad + \delta ^{{(i)}} y_{{k + 1}} - Q^{{(i)}} \cos \Omega t_{{k + 1}} - Q_{0}^{{(i)}} ] \\ &\quad + \frac{{ay_{{k + 1}} }}{{b\sqrt {y_{{k + 1}}^{2} + b^{2} } }}\delta ^{{(i)}} . \\ \end{aligned} $$

  (A.38)