Xiujing Han

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.

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.

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...