Qinsheng Bi

Dynamical analysis of two coupled parametrically excited van der Pol oscillators

The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to different types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor.

A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme.

Delayed Bifurcations to Repetitive Spiking and Classification of Delay-Induced Bursting

Three new routes to repetitive spiking, i.e. the delayed transcritical bifurcation, the delayed supercritical pitchfork bifurcation and the delayed subcritical pitchfork bifurcation, are revealed in this paper. We use bifurcation theory to classify bursting patterns related to three such delayed bifurcations. Then many new bursting patterns are obtained, including 24 new bursting patterns of point-point type, 27 new bursting patterns of point-cycle type and three new bursting patterns of point-torus type. Our study suggests that the classification of bursting remains to be further explored, since many new bursting patterns may be obtained based on new routes to repetitive spiking, even though we just consider codimension-1 bifurcations.

Bifurcation mechanism of bursting oscillations in parametrically excited dynamical system

The evolution of bursting oscillations in a parametrically excited dynamical system with order gap between the excited frequency and the natural frequency is investigated in this paper. By regarding the periodic excited term as a slow-varying parameter, different forms of bifurcations of the system are obtained. Base on the overlap between the bifurcation diagram and the phase portrait, the mechanism of different types of bursting oscillations are obtained. Furthermore, some phenomena in bursting oscillations such as symmetry breaking behavior are explained through the bifurcations occurring at the transitions between the quiescent state (QS) and spiking state (SP).

Bifurcations and some new traveling wave solutions for the CH-γ equation

In this paper, the CH-@c equation is investigated by employing the bifurcation theory and the method of phase portraits analysis. The dynamical behavior of equilibrium points and the bifurcations of phase portraits of the traveling wave system corresponding to this equation are discussed. Under some parameter conditions, some bounded traveling wave solutions such as solitary waves, peakons and periodic cusp waves are presented. Furthermore, based on the auxiliary equation, various new traveling wave solutions of parametric form are given. The previous results for this equation are extended.

Obtaining amplitude-modulated bursting by multiple-frequency slow parametric modulation

Amplitude-modulated bursting (AMB), characterized by oscillations appearing in the envelope of the active phase of bursting, is a novel class of bursting rhythms reported recently. The present paper aims to report a simple and effective method, i.e., the multiple-frequency slow parametric modulation (MFSPM) method, for obtaining such a bursting pattern. We show that the MFSPM can be well controlled so that it may exhibit multiple continuous ups and downs in the active area. Then, the amplitude of the traced active state alternates between increases and decreases accordingly, which leads to oscillations in the envelope of the active phase, and AMB is thus created. Based on this, the route to AMB by the MFSPM is presented. The validity of the approach is demonstrated by several examples. The proposed approach does not depend on specific systems or bifurcations and thus is a general method.

Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations

This paper aims to report two novel bursting patterns, the turnover-of-pitchfork-hysteresis-induced bursting and the compound pitchfork-hysteresis bursting, demonstrated for the Duffing system with multiple-frequency parametric excitations. Typically, a hysteresis behavior between the origin and non-zero equilibria of the fast subsystem can be observed due to delayed pitchfork bifurcation. Based on numerical analysis, we show that the stable equilibrium branches, related to the non-zero equilibria resulted from the pitchfork bifurcation, may become the ones with twists and turns. Then, the novel bursting pattern turnover-of-pitchfork-hysteresis-induced bursting is revealed accordingly. In particular, we show that additional pitchfork bifurcation points may appear in the fast subsystem under certain parameter conditions. This creates multiple delay-induced hysteresis behavior and helps us to reveal the other novel bursting pattern, the compound pitchfork-hysteresis bursting. Besides, effects of parameters on the bursting patterns are studied to explore the relation of these two novel bursting patterns.

Boundary-Crisis-Induced Complex Bursting Patterns in a Forced Cubic Map

This paper reports novel routes to complex bursting patterns based on a forced cubic map, in which boundary-crisis-induced novel bursting patterns are investigated. Typically, the cubic map exhibits stable upper and lower branches of fixed points, which may evolve into chaos in opposite parameter directions by a cascade of period-doubling bifurcations. We show that the chaotic attractors on the stable branches may suddenly disappear by boundary crisis, thus leading to fast transitions from chaos to other attractors and giving rise to switchings between the stable branches of solutions of the cubic map. In particular, the attractors that the trajectory switches to by boundary crisis can be fixed points, periodic orbits and chaos, dependent on parameter values of the cubic map, and this helps us to reveal three general types of boundary-crisis-induced bursting, i.e. bursting of chaos-point type, bursting of chaos-cycle type and bursting of chaos-chaos type. Moreover, each bursting type may contain various b...