Abstract
By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.
Highlights
Introduction van der Pol OscillatorsMathematicsSwitched dynamic systems, which are an important class of nonsmooth systems, are described by the interplay between two or more continuous- or discrete-time subsystems, with a switching rule that is defined according to the critical values of the state variable or depends on the fixed time [1,2]
Zhang [14] presented a recursive searching method of the common Lyapunov function (CLF) for the robust stable matrix set, based on which a sliding mode controller was designed, so that the system state remained on the sliding mode surface from the initial time instant
Since different initial conditions may influence the structure of the attractors of the switched system, all the tests presented in this paper have been performed for the initial condition ( x0, y0 ) = (0.5, 0.1)
Summary
The stable limit cycle, LCR , can be obtained, which indicates periodic vibration occurs in the vector field. Another system called the Duffing oscillator, expressed as S2 , can be written in the form. Suppose that the switched system begins to be controlled by subsystem S1 from an initial point X0 , when the trajectory in continuous state space hits the following local surface. After a period of time T + τ1 , the switched system may revert to subsystem S1 until the state variable x hits the surface. Since the whole system alternates between subsystem S1 and S2 , another local surface can be defined, expressed by.
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