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

Abstract

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.

Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies

“…However, from the viewpoint of fast-slow dynamics, strong and weak excitations are nothing special. The reason for this is that the slow variables are cos(ω 1 t) and cos(ω 2 t) and both amplitudes β 1 and β 2 belong to the fast subsystem, i.e., like other parameters in the fast subsystem, β 1 and β 2 are general parameters of the fast subsystem, e.g., see the fast subsystems ( 9)- (11). No matter what the magnitudes of the amplitudes are, the amplitudes can change qualitatively properties of the MMOs as long as they vary and pass through bifurcation points of the fast subsystem.…”

“…It has also been shown that a slowly varying periodic parameter can stabilize unstable orbits [9], annihilate one of the coexisting states, and thus results in controlled monostability [10]. In particular, a slowly varying periodic parameter can lead to periodic bifurcation delay behaviors, which have been identified as new routes to repetitive spiking and bursting [11]. However, most of the previous work focused on systems with a single slowly varying periodic parameter and there has been little work done on the analysis of dynamical systems with two slowly varying control parameters.…”

Complex Periodic Mixed-Mode Oscillation Patterns in a Filippov System

“…Chen et al [21] explored multiple fast-slow motions, including "periodic bursts with quasi-periodic spiking", "torus/short transient" mixed mode oscillations, "pitchfork/long transient" periodic mixed mode oscillations, amplitude-modulated and irregular oscillations from numerical method. It can be seen that the behaviors of the mixed-mode oscillation can be produced by many factors, such as different kinds of bifurcation structures [22,23], delay behaviors [24,25], and hysteresis loops [26,27]. However, most of the results about mixed-mode oscillations are made for smooth systems.…”

Melnikov-threshold-triggered mixed-mode oscillations in a family of amplitude-modulated forced oscillator

“…F 1 F 1 > 0 ð Þis the unmodulated forcing amplitude, and F 2 F 2 > 0 ð Þ is the degree of forcing modulation, X X > 0 ð Þ is the forcing frequency, and x x > 0 ð Þis the modulation frequency. Hysteretic cycles can emerge in the reduced form of equation 1analyzed by the singular perturbation methods, [27][28][29][30] given by…”

“…Hysteretic cycles can emerge in the reduced form of equation (1) analyzed by the singular perturbation methods, 27–30 given by where Ω true(0<Ω≪1true). Equation (2) may exhibit the typical fast–slow oscillations as shown in Figure 1, where the fixed parameters are fixed at a=b=1, F1=1, and Ω=0.01 (throughout the paper, we denote truex˙ as y).…”