COMPLICATED BEHAVIORS AS WELL AS THE MECHANISM IN BVP OSCILLATOR WITH SWITCHES RELATED TO TWO STATES

The dynamics of nonlinear switched systems which possess wide engineering background and cannot be explored directly by traditional nonlinear theory, become one of hot and frontier tasks at home and abroad for the time being. The complicated behaviors as well as the mechanism of the vector field alternated between two subsystems by two different critical states are investigated in this paper. Upon employing the typical generalized BVP oscillator as an example, by introducing bilateral switch, the nonlinear dynamical model alternated between two subsystems related two states is established, the different movement forms as well as the dynamical evolution of which caused by switches are explored in details. Based on the Poincaré theory of nonlinear system, the computational equation of Lyapunov exponents of switched system is derived. Combined with the bifurcation analysis of subsystems, different oscillations of the system are discussed, upon which the nonlinear behaviors such as sudden changes of period in periodic oscillations and the route to chaos with period-doubling bifurcations as well as the related essence are presented. Different from the systems with fixed time or single state switch, much more nonlinear phenomena may be observed in the dynamic systems with two state switches in which there may exist more switch points with changeable positions. Furthermore, different from the cascading of period-doubling bifurcations in smooth systems, the period-doubling bifurcations in switched systems correspond to the doubling of the number of switch points, which usually does not correspond to the doubling of the real periodic length of the movements.