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Highlights: [•] The system which alternates between van der Pol oscillator and Duffing system is investigated. [•] Poincaré map of the whole switched system is defined. [•] The two-parameter bifurcation sets of the switched system are abtained. [•] The evolution of the periodic switching phenomenon to chaos is analyzed in detail. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the problem of impulsive consensus of multi-agent systems, where each agent can be modeled as an identical nonlinear oscillator. Firstly, an impulsive control protocol is designed for directed networks with switching topologies based on the local information of agents. Then sufficient conditions are given to guarantee the consensus of the networked nonlinear oscillators. How to select the discrete instants and impulsive constants is also discussed. Numerical simulations show the effectiveness of our theoretical results. [Copyright &y& Elsevier]

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]

Sharp transitions in relation to the variation of system parameters are frequently encountered in many multiple-timescale systems, and they have been found to be an important factor related to the generation of mixed-mode oscillations (MMOs). The present paper aims to report a novel type of sharp transition, referred to as step-shaped sharp transition, in a nonlinear gyroscope oscillator with multiple-frequency excitations, and investigate the resulting MMOs. We show that step-shaped sharp quantitative changes in relation to the variation of system parameters can be observed in the equilibrium branch, which yields the step-shaped sharp transition. In particular, with the increase of the frequency ratio between the parametric and external excitations, more step-shaped sharp transitions appear in the equilibrium branches, which evolve into the ones displaying different structures. Based on this, the rectangular-pulse-shaped explosion of equilibria is created. Furthermore, these sharp transitions can form active areas for the MMOs, leading to the alternations between large-amplitude and small-amplitude oscillations, and finally the route to MMOs is created. Our findings have significant implications for understanding the fast-slow dynamics of the nonlinear gyroscope oscillator, contributing to the exploration of new routes to MMOs. Thus, the results could provide theoretical support for the potential application of gyroscopes. • The nonlinear gyroscope oscillator with multiple-frequency excitations. • A novel sharp transition type: step-shaped sharp transition of equilibria. • Rectangular-pulse-shaped explosion induced by step-shaped sharp transitions. • Mixed-mode oscillations induced by two novel sharp transitions. [ABSTRACT FROM AUTHOR]

Highlights • An approximation method is proposed for analyzing fast-slow dynamics in parametrically and externally excited systems with two slow incommensurate excitation frequencies. • The validity of the approach is demonstrated by the Duffing and van der Pol systems, respectively. • The proposed approach explains the generation of complex bursting dynamics related to incommensurate excitation frequencies. Abstract This paper aims to report an approximation method, the frequency-truncation fast-slow analysis, for analyzing fast-slow dynamics in parametrically and externally excited systems with two slow incommensurate excitation frequencies (PEESTSIEFs). We obtain truncated, commensurate excitation frequencies, which are approximations of the incommensurate excitation frequencies. Then, we show numerically that bursting behavior in PEESTSIEFs can be approximated in the same systems but with truncated, commensurate excitation frequencies, and therefore bursting dynamics in PEESTSIEFs can be understood by analyzing the same systems with truncated, commensurate excitation frequencies. Based on this, the approximation method for analyzing bursting dynamics in PEESTSIEFs is proposed. The validity of the approach is demonstrated by the Duffing and van der Pol systems, respectively. [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]

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]

Abstract: This paper reports a four-dimension (4D) fast-slow hyperchaotic system with the structure of two time scales by adding a slow state variable w into a three-dimension (3D) chaotic dynamical system, studies the stability and Hopf bifurcation of origin point. Furthermore, based on the fast-slow dynamical bifurcation analysis and the phase planes analysis, different bursting phenomena, symmetric fold/fold bursting, symmetric sub-Hopf/sub-Hopf bursting and chaotic bursting, as well as chaotic and periodic spiking, are observed in the fast-slow hyperchaotic system. Numerical simulations are presented to show these results. [ABSTRACT FROM AUTHOR]

Abstract: This paper is involved with the adaptive modified function projective synchronization (MFPS) problem of hyperchaotic systems with unknown parameters. Based on the Lyapunov stability theorem and adaptive control method, adaptive controllers and parameters update laws can be presented for the MFPS not only between two identical hyperchaotic systems but particularly also between two different hyperchaotic systems with fully unknown or partially unknown parameters. Moreover, the coupling strength can be automatically adapted to a updated law. Numerical simulations are presented to show the effectiveness of the proposed synchronization schemes. [Copyright &y& Elsevier]

In this study, the mechanism for the action of time-invariant delay on a non-autonomous system with slow parametric excitation is investigated. The complex mix-mode oscillations (MMOs) are presented when the parametric excitation item slowly passes through critical bifurcation values of this nonlinear time delay oscillator. We use bifurcation theory to clarify certain generation mechanism related to three complex spiking formations, i.e., ``symmetric sup-pitchfork bifurcation'', ``symmetric sup-pitchfork/sup-Hopf bifurcation'', and ``symmetric sup-pitchfork/sup-Hopf/homoclinic orbit bifurcation''. Such bifurcation behaviors result in various hysteresis loops between the spiking attractor and the quasi-stationary process, which are responsible for the generation of MMOs. We further identify that the occurrence and evolution of such complex MMOs depend on the magnitude of the delay. Specifically, with the increase of time delay, the two limit cycles bifurcated from Hopf bifurcations may merge into an enlarged cycle, which is caused by a saddle homoclinic orbit bifurcation. We can conclude that time delay plays a vital role in the generation of MMOs. Our findings enrich the routes to spiking process and deepen the understanding of MMOs in time delay systems. [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]

Based on Hopf bifurcation delay, we present two novel delay-induced bursting patterns in a modified van der Pol-Duffing circuit system. These delay-induced bursting patterns are classified as compound “delayed supHopf/fold cycle-subHopf/supHopf” bursting and “subHopf/supHopf” bursting via “delayed supHopf/supHopf” hysteresis loop, respectively. Our results show that Hopf bifurcation delay plays a decisive role in the generation of these two bursting patterns, which enriches the routes to bursting and deepens the understanding of underlying mechanisms of bursting. [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]

Abstract: This paper investigates the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Based on LaSalle’s invariance principle, some criteria for the synchronization between two coupled complex networks are achieved via adaptive control. To validate the proposed methods, the unified chaotic system as the nodes of the networks are analyzed in detail, and numerical simulations are given to illustrate the theoretical results. [Copyright &y& Elsevier]