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For the development of the theoretical framework of slow–fast dynamics, it is essential to study the mechanism of bursting oscillations with high co-dimensional bifurcations in the high-dimensional vector field. This paper is aimed to explore the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at the origin. We establish an eight-dimension system with the slow–fast effect, in which two time scales exist. The fast subsystem is represented as a six-dimensional vector field with triple Hopf bifurcation at the origin, while the slow subsystem is adopted to adjust the variation of parameters in the fast subsystem, which may lead to abundant dynamic behaviors. To better understand the qualitatively different behaviors in the fast subsystem, the trivial solution, single-mode, two-mode, and three-mode movements are introduced based on the projections of trajectory on the three sub-planes. As a result, several types of bursting phenomena, such as Hopf/Hopf bursting oscillations with single-mode, Hopf/Hopf/fold bursting oscillations with single-mode, Hopf/Hopf bursting oscillations with two-mode have been revealed, the mechanism of which is obtained upon the superposition of equilibrium branches together with bifurcations in the fast subsystem as well as transformed phase portrait. In addition, for bursting oscillations on the whole phase space, the amplitude of spiking oscillations may change according to two different limit cycles, which leads to an abrupt change of the oscillating amplitude on the sub-plane, corresponding to different projections of Poincaré. It can be observed the state variables on the different sub-planes alternate between synchronized and non-synchronized states in the fast subsystem, which may be accounted for by the bifurcations. • Hopf bifurcation leads to transitions between different single modes or between single-mode and two-mode. • Synchronization and non-synchronization between state variables on different sub-planes occur. • Complexity mechanisms of the slow–fast dynamics are investigated. • Bursting oscillations behave in quasi-periodic types instead of the high dimensional torus. [ABSTRACT FROM AUTHOR]

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]

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]

The main purpose of the paper is to show that parts of the trajectory for chaotic bursting oscillations may oscillate according to different limit cycles, which implies orderly movement exists in chaos. For a typical jerk oscillator with a cubic nonlinearity, when a slow-varying controlling term is introduced, bursting oscillations may occur, appearing in the combination of large-amplitude and small-amplitude oscillations. Two coexisting stable equilibrium points may evolve to chaos via different cascades of period-doubling bifurcation, respectively, which means different types of stable attractors may coexist with the variation of slow-varying parameter, while several periodic windows exist in the chaotic region. It is found that the spiking oscillations of the full system may change according to different cycles not only for 2-D torus bursting oscillations but also for the chaotic bursting attractor. Furthermore, whether the influence of the short periodic window of the fast subsystem on the bursting oscillations appears or not depends on the size of the slow scale, since the inertia of the movement may cause the delay effect of the bifurcations at both two edges of the window. The findings suggest that coupling of multiple scales may be one of key factors for the diversity of real world. • Coexisted foci via bifurcations respectively lead to coexisted bursting attractors. • Slow-fast dynamics evolves quasi-periodic to chaotic bursting oscillations. • Spiking oscillations of bursting attractor may vary according to different attractors. • Ratio between two scales influences appearance of short periodic window on full system. [ABSTRACT FROM AUTHOR]

This paper aims to report a new route to pulse-shaped explosion (PSE) related to the limit cycle attractor in a mechanical system with multiple-frequency excitations. We show that the compound bursting connected by PSE of equilibria can be observed in this system. Then, with the decrease of the amplitude of parametric excitation, the compound bursting pattern begins to change to other types, accompanied by the generation of PSE of limit cycles. Typically, as a novel sharp transition behavior underlying the occurrence of bursting oscillations, the PSE can be created by the disappearance of critical escape transitions. In present study, based on the two-parameter bifurcation analysis, we find that the PSE can also be created in the branch of limit cycles with the disappearance of fold points of homoclinic bifurcation curves, which is quite different from the ones reported in previous works and can be regarded as a new route to the PSE of limit cycles. Based on this, the compound bursting induced by the PSE of limit cycles is investigated. Besides, it should be pointed out that the limit cycle exhibiting PSE can be regarded as an amplitude-modulated limit cycle attractor, and thus we obtain a newly released bursting rhythm, namely the amplitude-modulated bursting, based on the PSE of limit cycles. These findings have significant implications for the understanding of PSE and the diversity of bursting dynamics induced by it. • A new route to the pulse-shaped explosion of limit cycles. • Compound bursting related to the pulse-shaped explosion of equilibria and limit cycles. • The amplitude-modulated bursting triggered by pulse-shaped explosion of limit cycles. • Transition of bursting patterns from compound bursting to amplitude-modulated 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]

• We systematically and analytically show the distinctive roles of hysteresis, amplitude death and oscillation death mechanisms in generating fast-slow behavior. • In the Duffing system, we show that hysteresis is the main mechanism in generating fast-slow phenomena. • In Van der Pol system with slow forcing, amplitude death and oscillation death are the fundamental mechanisms because of Hopf bifurcation. • We further provide a rather simple model to implementing fast-slow switching relaying scenarios among subsystems. Fast-slow phenomena are often referred to as time series showing clearly separated time scales of alternations between consecutive high-frequency large-amplitude oscillations and much smaller amplitudes or much more infrequent or quiescent intervals, which can be observed in many nonlinear systems. In this work, we show distinctive roles of hysteresis, amplitude death and oscillation death mechanisms in generating fast-slow switching behavior by using the paradigmatic models of Duffing and Van der Pol systems which are driven by both parametric and external excitations for the case when the frequencies of excitations are 1 : 1 and 1 : 2 rationally correlated. In the Duffing system, we show that hysteresis is the main mechanism to generate fast-slow behavior and the fast oscillations are resulted from the transient processes between different steady states. On the other hand, in the Van der Pol system, amplitude death and oscillation death are the fundamental mechanisms because of Hopf bifurcation. In this case, fast oscillations mainly correspond to the limit cycles, while slow quiescent intervals are due to the stable fixed points. In addition, we provide a rather simple model to implement responding lags of Van der Pol oscillators to the same external forcing, resulting in fast-slow switching relaying scenarios among subsystems. All theoretical results have been confirmed precisely by numerical simulations. [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 multi-time-scale-coupled nonlinear dynamic system is constructed by introducing an external excitation into a charge-controlled memcapacitive system. We report the complex bursting oscillation mechanism of this system under different parameter conditions by taking advantage of the fast-slow analysis method. Firstly, we discuss several different types of periodic bursting behaviors consisting of "asymmetric and symmetric 'delayed pitchfork/Hopf/delayed Homoclinic/Homoclinic/Hopf' bursting type", "asymmetric and symmetric 'delayed Hopf/delayed Homoclinic/Homoclinic/Hopf' mixed-mode bursting oscillation", "symmetric 'delayed Hopf/delayed Homoclinic/delayed pitchfork' mixed-mode bursting oscillation" and "asymmetric 'cycle/point' mixed-mode bursting type via 'delayed Hopf/delayed pitchfork' hysteresis loop". The numerical simulation results express that different delay-bifurcation intervals under the slow passage effect will induce completely various bursting types. Furthermore, we explore a novel route with complex structure from periodic bursting to chaotic bursting. By gradually increasing the value of control parameter, we can observe a variety of dynamic behaviors including complex structure such as symmetric breaking bifurcations, intermittent period-doubling bursting type, period-doubling bifurcations and inverse period-doubling bifurcations. With the further increase of the parameter, the system generates crisis effect, and we can obtain the chaotic attractors in the periodic window. We discover that different parameter conditions can produce chaotic attractors which can induce "'point/chaos/period/chaos/point' intermittent chaos type", "'point/chaos/point' intermittent chaos type" and "'cycle/chaos/cycle' intermittent chaos type" until complete chaos. Chaotic bursting types are corroborated by the maximum Lyapunov exponents. Our research results enrich the route to bursting oscillation and deepen the understanding of the mechanism of complex bursting oscillation types. [ABSTRACT FROM AUTHOR]