Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation

The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

In this paper, we study the global dynamics of a nonsmooth Rayleigh-Duffing equation x¨ + ax˙ + bx˙|x˙| + cx + dx3 = 0 for the case d > 0, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh-Duffing oscillator for the case d < 0, i.e., the saddle case, has been studied completely in the companion volume [Int. J. Non-Linear Mech., 129 (2021) 103657]. The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent occurs. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

Limit Cycle Bifurcations Near a Cuspidal Loop

In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the first order Melnikov functions. We give a method to compute more coefficients of the expansions to find more limit cycles near the cuspidal loop. As an application example, we considered a polynomial near-Hamiltonian system and found 12 limit cycles near the cuspidal loop and the center.

Механизмы перехода от ритмической активности к пачечной в модели ноцицептивного нейрона

The transitions from tonic spiking to bursting for the nociceptive neuron model have been studied with changing the external stimulus value. The presence of the fold limit cycle bifurcation in the structure of the bifurcation diagram of the fast subsystem and the torus bifurcation in the structure of the bifurcation diagram of the full system lead to the emergence of special solutions of the type torus canards in these transitions. This confirms the assumption that torus canards are an obligatory feature for transitions between rhythmic and burst discharges

Global Dynamics of Memristor Oscillator

In this paper, we investigate the global dynamics of a memristor oscillator [Formula: see text] which comes from [Corinto et al., 2011], where [Formula: see text], and [Formula: see text]. Clearly, the case [Formula: see text] is trivial. So far, all results of this oscillator were given only for the case [Formula: see text], where the set of equilibria may change among a singleton, three points and a singular continuum and at most one limit cycle can arise and no limit cycles arise from the continuum. Compared with the case [Formula: see text], this oscillator displays more complicated dynamics for the case when [Formula: see text]. More clearly, one limit cycle may arise from the continuum and at most three limit cycles appear in the case of three equilibria, where generalized pitchfork bifurcation, saddle-node bifurcation, generalized Hopf bifurcation, double limit cycle bifurcation and homoclinic bifurcation may occur. Finally all global phase portraits are given for [Formula: see text] cases on the Poincaré disc, where a generalized normal sector method is applied. Moreover, our partial analytical results are demonstrated by numerical examples.