Stability and chaos for an adjustable excited oscillator with system switch

Abstract A linear spring-damper dynamic oscillator with excitations is studied in the paper, and a set of subsystems are defined by adjusting the constant and magnitude of the periodic external forces. A triangular domain is defined in the phase plane coordinate system, and such a dynamic system will switch to the corresponding subsystem when the flow arrives at the boundary or corner for such a domain. Through employing the theory of discontinuous, the vector fields for the subsystems have been determined which is the necessary conditions of motion ‘bouncing’ within such a triangular domain. To describe the periodic motions of such an oscillator, the generic mappings are constructed. The periodicity and stability for the motion in the steady-state have been discussed. Analytical and numerical predictions have been carried out through phase plane and switching sections to illustrate the effectiveness of the design of the subsystems under the proposed switching scheme. Periodic and chaotic motions have been simulated to institutively illustrate the system switch and stability for such a spring-damper oscillator with adjustable excitations.

[1]  M. Branicky.  Stability of hybrid systems: state of the art , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[2]  V. Utkin.  Variable structure systems with sliding modes , 1977 .

[3]  A. Luo,et al.  Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator , 2012 .

[4]  Huijun Gao,et al.  New results on stabilization of Markovian jump systems with time delay , 2009, Autom..

[5]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[6]  R. Leine,et al.  Bifurcations in Nonlinear Discontinuous Systems , 2000 .

[7]  Albert C. J. Luo,et al.  A theory for flow switchability in discontinuous dynamical systems , 2008 .

[8]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[9]  Ligang Wu,et al.  Asynchronous control for 2-D switched systems with mode-dependent average dwell time , 2017, Autom..

[10]  K. Narendra,et al.  On the stability and existence of common Lyapunov functions for stable linear switching systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[11]  Hai Lin,et al.  Switching Stabilizability for Continuous-Time Uncertain Switched Linear Systems , 2007, IEEE Transactions on Automatic Control.

[12]  Yasuaki Kuroe,et al.  A solution to the common Lyapunov function problem for continuous-time systems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[13]  A. Luo.  Singularity and Dynamics on Discontinuous Vector Fields , 2012 .

[14]  Xilin Fu,et al.  Chatter dynamic analysis for Van der Pol Equation with impulsive effect via the theory of flow switchability , 2014, Commun. Nonlinear Sci. Numer. Simul..

[15]  Zhongyang Fei,et al.  Stabilization of 2-D Switched Systems With All Modes Unstable via Switching Signal Regulation , 2017, IEEE Transactions on Automatic Control.

[16]  Xilin Fu,et al.  Flow switchability of motions in a horizontal impact pair with dry friction , 2017, Commun. Nonlinear Sci. Numer. Simul..

[17]  Xilin Fu,et al.  On periodic motions of an inclined impact pair , 2015, Commun. Nonlinear Sci. Numer. Simul..

[18]  Xilin Fu,et al.  New approach in dynamics of regenerative chatter research of turning , 2014, Commun. Nonlinear Sci. Numer. Simul..

[19]  E. Levitan.  Forced Oscillation of a Spring‐Mass System having Combined Coulomb and Viscous Damping , 1959 .

[20]  A. Luo,et al.  Complex Dynamics of Bouncing Motions on Boundaries and Corners in a Discontinuous Dynamical System , 2017 .

[21]  Albert C. J. Luo,et al.  Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamic systems , 2005 .

[22]  Zhendong Sun.  Reachability analysis of constrained switched linear systems , 2007, Autom..

[23]  K. Narendra,et al.  A common Lyapunov function for stable LTI systems with commuting A-matrices , 1994, IEEE Trans. Autom. Control..

[24]  Erik Mosekilde,et al.  Bifurcations and chaos in piecewise-smooth dynamical systems , 2003 .

[25]  Kumpati S. Narendra,et al.  On the existence of a common quadratic Lyapunov function for two stable second order LTI discrete-time systems , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[26]  Xilin Fu,et al.  Stick motions and grazing flows in an inclined impact oscillator , 2015 .

[27]  R. Decarlo,et al.  Variable structure control of nonlinear multivariable systems: a tutorial , 1988, Proc. IEEE.

[28]  K. Narendra,et al.  A sufficient condition for the existence of a common Lyapunov function for two second order linear systems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.