Bursting oscillations in Sprott B system with multi-frequency slow excitations: two novel “Hopf/Hopf”-hysteresis-induced bursting and complex AMB rhythms

In this paper, multi-timescale dynamics and the formation mechanism of a 3D non-autonomous system with two slowly varying periodic excitations are systematically investigated. Interestingly, the system shows novel multibifurcation cascaded bursting oscillations (MBCBOs) when the frequency of the two excitations is much lower than the mean frequency of the original system (MFOS). For instance, periodic, quasi-periodic and chaotic bursting oscillations induced by a variety of cascaded bifurcations are first observed, and the phenomenon of spiking transfer is also revealed. Besides, stability and local bifurcations of the system are comprehensively investigated to analyze the mechanism of the observed MBCBOs, in which bifurcation diagram, Lyapunov exponents, time series, phase portraits, and transformed phase diagrams are used. Finally, through a circuit simulation and hardware experiment, these complex dynamics phenomena are verified physically.

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“…of bifurcations [28]. Therefore, this section will focus on the characteristics of the equilibrium point and the local bifurcation.…”

Section: Local Bifurcation and Stability Analysis Of Equilibrium Pointmentioning

confidence: 99%

Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation

Wang

Zhang

et al. 2021

Nonlinear Dyn

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“…e introduction of the slow-fast dynamics analysis method [19] well explained the bifurcation connection between the spiking state and the quiescent state, so as the mechanism of bursting oscillation. Based on this method, various bursting oscillation modes [20][21][22][23][24][25][26][27][28][29][30][31][32][33] are obtained, such as fold/fold bursting and fold/ Hopf bursting. In nonlinear dynamical system, two time scales can lead to complex dynamical behavior of the system, so it is widely concerned by academic circles domestically and overseas.…”

Section: Introductionmentioning

confidence: 99%

Bursting Oscillation and Its Mechanism of a Generalized Duffing–Van der Pol System with Periodic Excitation

2021

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The complex bursting oscillation and bifurcation mechanisms in coupling systems of different scales have been a hot spot domestically and overseas. In this paper, we analyze the bursting oscillation of a generalized Duffing–Van der Pol system with periodic excitation. Regarding this periodic excitation as a slow-varying parameter, the system can possess two time scales and the equilibrium curves and bifurcation analysis of the fast subsystem with slow-varying parameters are given. Through numerical simulations, we obtain four kinds of typical bursting oscillations, namely, symmetric fold/fold bursting, symmetric fold/supHopf bursting, symmetric subHopf/fold cycle bursting, and symmetric subHopf/subHopf bursting. It is found that these four kinds of bursting oscillations are symmetric. Combining the transformed phase portrait with bifurcation analysis, we can observe bursting oscillations obviously and further reveal bifurcation mechanisms of these four kinds of bursting oscillations.

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“…The fast-slow analysis method [16] first proposed by Rinzelis can be used to reveal the generation mechanism of the bursting response. The bursting oscillation [17][18][19][20][21] is composed of large and small oscillations alternately in a given period. Generally speaking, the large-amplitude oscillation is called the spiking state motion, and its trajectory moves in the large-amplitude limit cycle vector field.…”

Section: Introductionmentioning

confidence: 99%

Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

Zhang

Qian

2021

Mathematics

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In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem.

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Bursting oscillations in Sprott B system with multi-frequency slow excitations: two novel “Hopf/Hopf”-hysteresis-induced bursting and complex AMB rhythms

Cited by 33 publications

References 22 publications

Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation

Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation

Bursting Oscillation and Its Mechanism of a Generalized Duffing–Van der Pol System with Periodic Excitation

Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

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