This paper investigates the generation of complex bursting patterns in Van der Pol system with two slowly changing external forcings.Complex bursting patterns,including complex periodic bursting and chaotic bursting,are presented for the cases when the two frequencies are commensurate and incommensurate.These complex bursting patterns are novel and have not been reported in previous work.Based on the fast-slow dynamics,the evolution processes of the slow forcing are presented to reveal the dynamical mechanisms underlying the appearance of these complex bursting patterns.With the change of amplitudes and frequencies,the slow forcing may visit the spiking and rest areas in different ways,which leads to the generation of different complex bursting patterns.
The fast-slow effect can be observed in a typical non-smooth electric circuit with order gap between the natural frequency and the excitation frequency.Numerical simulations are employed to show complicated behaviours,especially different types of busting phenomena.The bifurcation mechanism for the bursting solutions is analysed by assuming the forms of the solutions and introducing the generalized Jacobian matrix at the non-smooth boundaries,which can also be used to account for the evolution of the complicated structures of the phase portraits with the variation of the parameter.Period-adding bifurcation has been explored through the computation of the eigenvalues related to the solutions.At the non-smooth boundaries the so-called 'single crossing bifurcation' can occur,corresponding to the case where the eigenvalues jump only once across the imaginary axis,which leads the periodic burster to have a quasi-periodic oscillation.
The problem of reliable impulsive synchronization for a class of nonlinear chaotic systems has been investigated in this paper. Firstly a reliable impulsive controller is designed by using the impulsive control theory. Then by the uniform asymptotic stability criteria of systems with impulsive effects, some sufficient conditions for reliable impulsive synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.
