Search

Hopf bifurcation may be one of most common bifurcations, which results in the transitions between quiescence and spiking states in a bursting attractor, since it causes the alternation between an equilibrium point and a limit cycle. Though a lot of bursting attractors related to Hopf bifurcation have been obtained, while how many types of bursting attractors may appear in the neighborhood of a Hopf bifurcation point still remains an open problem. Here we try to answer the question based on the normal form up to the seventh order of a vector field with Hopf bifurcation at the origin. By introducing a low-frequency external excitation to the normal form, we derive the equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying excitation, based on which four typical cases corresponding to qualitatively different evolution of the behaviors are presented. Accordingly, four types of quasi-periodic bursting oscillations, i.e. Hopf/Hopf bursting attractors with (1) and (2) without saddle on the limit cycle bifurcation in the spiking oscillations, respectively, (3) fold/Hopf/fold/Hopf bursting attractors, and (4) bursting attractor with saddle on limit cycle bifurcation at the transition, are obtained with the increase of the exciting amplitude, the mechanism of which is obtained by overlapping equilibrium branches together with bifurcations in generalized autonomous system and transformed phase portrait. It is found that when two coexisted stable attractors are separated by an unstable attractor on the phase plane in generalized autonomous system, the influence of one of the stable attractors on the dynamics of full system may disappear. Furthermore, when the window for a stable attractor between two bifurcation points along slow-varying parameter axis is relatively short, the effect of the attractor as well as the bifurcation can only appear when the external excitation changes slowly enough. The amplitude of the repetitive spiking oscillations may vary according to two different limit cycles separated by saddle on limit cycle bifurcation, at which a sudden change of the oscillating amplitude can be observed. • We derive topological structure of universal unfolding of normal form of Hopf bifurcation. • Four types of 2-D bursting oscillations and their associated mechanism are obtained. • Effect of slow time scale on full dynamics is discussed. • Saddle on limit cycle bifurcation may lead to abrupt change of spiking amplitude. [ABSTRACT FROM AUTHOR]

The mechanism of bursting oscillations in high dimensional system is one of the key topics for the theoretical framework of slow–fast dynamics, especially when higher co-dimensional bifurcation involves the vector field. This paper focuses on a six-dimensional vector field with triple Hopf bifurcation at the origin to explore possible bursting oscillations as well as the mechanism. By introducing a parametric excitation with low-frequency to the normal form, several types of bursting attractors can be obtained, the mechanism of which is presented upon the bifurcation analysis of the generalized autonomous system. With the increase of exciting amplitude, different types of bifurcations may influence the structure of the movement, which may cause a 2-D torus to evolve to 2-D bursting oscillations with single-mode, two-mode and three-mode, respectively, with different projections on three sub-planes. Because of the existence of three independent sub-planes, the bifurcations on different sub-planes may lead to the synchronization and non-synchronization between the associated state variables. Furthermore, the inertia of the movement may lead to the disappearance of the effect of the stable attractor with short window in the generalized autonomous system. • All the possible bursting oscillations associated with triple Hopf bifurcation are presented. • Bursting attractor may evolve from single-mode to two-mode and three-mode oscillations. • Synchronization and non-synchronization between state variables associated with three sub-planes can be observed. • Scale effect may influence the occurrence of attractor with short-window on the dynamics of the bursting oscillations. [ABSTRACT FROM AUTHOR]

Abstract: Bursting oscillations with different waveforms can be observed in the forced Duffing’s equation when the external forcing changes slowly. We present an analysis of bursting oscillations by considering the external forcing as a control parameter and studying its influence on the controlled Duffing’s equation. Furthermore, the effects of forcing amplitude and forcing frequency on bursting oscillations are investigated. The forcing amplitude plays an important role in the generation of bursting oscillations, that is, bursting oscillations may be created for , while if , they do not occur. On the other hand, the time interval between two adjacent spikes of bursting oscillations is dependent on the forcing frequency, which can be computed at . [Copyright &y& Elsevier]

This paper focus on the effects of a codimension-2 non-smooth bifurcation on bursting behaviors in piecewise-smooth systems. A passive circuit with a switched power source is slightly modified by introducing a periodic exciting voltage to establish an example system of Filippov-type. By using Filippov's convex method, the sliding vector field is obtained, and the analytical solution of the sliding motion equation is derived. A codimension-2 non-smooth bifurcation, called "catastrophic boundary focus and catastrophic crossing-sliding bifurcation", is observed, and the unfolding of the bifurcation is discussed. Based on the bifurcation analysis, five bursting oscillations associated with the codimension-2 bifurcation are observed, and the dynamical mechanism is revealed. The study suggests that the bifurcation of boundary equilibrium can be neither a non-smooth fold one nor a persistence one if the sliding vector field is degenerate, and this bifurcation may also lead to jumping behaviors in a bursting. A non-smooth limit cycle may cross the switching manifold transversely, precisely at the boundary of the escaping subregion, causing the limit cycle to disappear catastrophically. This bifurcation of non-smooth limit cycle controls the transition between a quiescent state and a spiking state in a bursting. A grazing-sliding bifurcation in a slow–fast system can form "reentry sliding structures" in a bursting. [ABSTRACT FROM AUTHOR]

This paper aims at the bursting oscillations as well as the dynamical mechanism in non-smooth systems. A fourth-order Chua's circuit is modified to establish a non-smooth system by introducing a periodic exciting current and a piecewise nonlinear resistor. Under the condition that the dimensionless excitation frequency is far less than 1, the system is a slow–fast system, and five representative bursting patterns are observed via changing the excitation amplitude. By regarding the whole term of periodic exciting current as a bifurcation parameter, several kinds of smooth or non-smooth bifurcations are obtained in the fast subsystem via numerical or theoretic method. Particularly, a so-called saddle-focus-induced catastrophic bifurcation(SFIC), which is generated by collision between a non-smooth chaotic attractor and a saddle-focus, is observed. In addition, bifurcation diagrams and the associated maximal Lyapunov exponent spectrum are also obtained numerically, which can illustrate the scenario from periodic solutions to chaotic solutions via period-doubling bifurcations. Based on the bifurcation analysis, the generation mechanism of the bursting patterns is revealed. Research shows that the SFIC bifurcation in the fast subsystem may result in a sudden jumping behavior in the slow–fast system from a chaotic attractor to another. A spiking attractor may be constructed by chaotic attractors and limit cycle attractors together, which can lead to chaotic–periodic bursting oscillations. Slow passage effect induced by Hopf bifurcation can lead to delay phenomenon of other bifurcation in the slow–fast system. • A saddle-focus-induced catastrophic bifurcation of the chaotic attractor is obtained. • The maximal Lyapunov exponent spectrum of bifurcation diagram is calculated. • Five typical bursting patterns are observed, and the mechanism of them is revealed. [ABSTRACT FROM AUTHOR]

In this paper, a modified Chua's circuit, possessing an external excitation current and a piecewise nonlinear resistor, is considered to investigate bursting oscillations and the dynamical mechanism in the Filippov system, focusing on the effects of sliding bifurcations on the bursting dynamics. Five typical representative bursting oscillations are observed when there is a gap between the external excitation frequency and the natural one. Codimension-1 bifurcations of the fast subsystem are discussed by regarding the whole excitation term as a bifurcation parameter. The necessary conditions of conventional bifurcations of the equilibria and the bifurcations of boundary equilibria are obtained via theoretical analysis. Both the local adding-sliding bifurcation, crossing-sliding bifurcation, non-smooth fold bifurcation, the fold bifurcation of non-smooth limit cycle in the fast subsystem and the adding-sliding bifurcation in the slow–fast coupling system are observed via numerical method. Based on slow–fast analysis method and the bifurcation analysis, the dynamical mechanism is discovered. Research found that the non-smooth fold bifurcation of the boundary equilibrium can lead to jumping behaviors of the trajectory from the switching manifold to the attractors of the subsystem. The crossing-sliding bifurcation may not cause the transition between spiking state and quiescent state. The local adding-sliding bifurcation in the fast subsystem can lead to adding-sliding bifurcation of non-smooth limit cycle in the coupling system. There may exist adding-sliding structure in bursting oscillations before or after the adding-sliding bifurcation. • Five typical bursting oscillations and the mechanism in a Filippov-type Chua's circuit are investigated. • Local adding-sliding bifurcation in the fast subsystem and adding-sliding structure in the coupling system are observed. • Jumping behaviors from the switching manifold to the attractors of fast subsystem, which are induced by the non-smooth fold bifurcation, are observed. • It is found that the crossing-sliding bifurcation will not cause the transition between spiking state and quiescent state. [ABSTRACT FROM AUTHOR]

The aim of this paper is to reveal the dynamical mechanism of bursting oscillations in non-smooth dynamical systems, and the interest is focused on the effects of period-doubling bifurcation and chaotic attractor on the bursting oscillations. By modifying a fourth-order Chua's circuit, a dynamical system possessing non-smooth switching manifold and multiple scale variables, is established. By taking the slow-varying variable as a bifurcation parameter, Subcritical non-smooth Hopf bifurcation, C-bifurcation, and period-doubling bifurcation are obtained in the fast subsystem. Further more, chaotic attractors, which are generated from a series of period-doubling bifurcations, are also observed. Based on the numerical simulations and bifurcation analysis of the fast subsystem, eight typical bursting patterns are obtained, and their dynamical mechanism is revealed. Study shows that period-2 limit cycle in the fast subsystem may cause the slow–fast system to oscillates in alternating small and large amplitudes; A complex spiking attractors made up of limit cycles and chaotic attractors may lead to spiking states changing from chaotic oscillations to periodic ones, and the trajectory may jump directly from a chaotic attractor to other stable one without undergoing any bifurcation of the fast subsystem. [ABSTRACT FROM AUTHOR]

In this paper, a novel three-dimensional modified van der Pol-Duffing circuit with a quintic nonlinear resistor is proposed, and the multiple stable attractors are observed. The complex bursting patterns are investigated, various patterns of bursting oscillations are obtained under slowly changing frequency, and the transition mechanism of which can be revealed via the modified slow–fast analysis as well as the transformed phase portrait. It can be found that for the system with mono-stability, only one bursting pattern exists, while for the case with multi-stability, the trajectory can enter into different attraction areas in the transition region, leading to complex forms of bursting oscillations. Furthermore, the multi-valued characteristic of the system are observed by the domain of attraction. • The multiple stable attractors of the modified van der Pol-Duffing circuit are observed. • Various patterns of bursting oscillations are obtained under slowly changing frequency. • The transition mechanism is revealed via the modified slow–fast analysis. • The multi-valued characteristic of the system are observed by the domain of attraction. [ABSTRACT FROM AUTHOR]

Highlights: [•] The time delay has been taken as a variable parameter to investigate its effect on the dynamics of bursting. [•] Two bursters, i.e. the symmetric fold/fold burster and symmetric Hopf/Hopf burster are discussed. [•] The results show we can control different bursting motions in different ways with delayed feedback. [ABSTRACT FROM AUTHOR]