Approximate analytical solution in slow-fast system based on modified multi-scale method

Li, Xianghong; Tang, Jianhua; Wang, Yanli; Shen, Yongjun

doi:10.1007/s10483-020-2598-9

Cited by 6 publications

(1 citation statement)

References 40 publications

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“…Most of the research for the bursting oscillations in the dynamical systems with periodic excitation focused on the case with no more than three equilibrium branches [15,16]. Furthermore, only two relatively simple types of bifurcations, i.e., the fold and Hopf bifurcations, are considered for the transitions between the quiescent states and spiking states [17].…”

Section: Introductionmentioning

confidence: 99%

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Zhang

Han

et al. 2022

Nonlinear Dyn

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The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol-Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slowvarying parameter, on which different types of bifurcations, such as Hopf and pitch fork bifurcations, can be observed. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously dif-

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Section: Introductionmentioning

confidence: 99%

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Zhang

Han

et al. 2022

Nonlinear Dyn

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Cluster oscillation and bifurcation of fractional-order Duffing system with two time scales

Wang

Shen

2020

Acta Mech. Sin.

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Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

Xie,

Wan,

Zhao

et al. 2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The paper investigates the influence of parameters on the stability of fractional order differential quasiperiodic Mathieu equations. First, we use the perturbation method to obtain approximate expressions (i.e., transition curves) for the stability and unstable region boundaries of the equation. After obtaining the approximate expression of the transition curve, we use Lyapunov's first method to analyze the stability of the fractional order differential quasiperiodic Mathieu system, thereby obtaining the conditions for the stability of the fractional order differential quasiperiodic Mathieu equation system. Second, by comparing the approximate expressions of the transition curve of the steady-state periodic solution of the quasiperiodic Mathieu oscillator under different parameter conditions, we obtained the conclusion that the fractional order differential term exists in the form of equivalent stiffness and equivalent damping in the fractional order differential quasiperiodic Mathieu system. By comparison, we have summarized the general forms of equivalent linear damping and equivalent stiffness of the system. Through this general form, we can define an approximate expression for the thickness of unstable regions to better study the characteristics of fractional order differential quasiperiodic Mathieu systems. Finally, the influence of the parameters of the fractional order differential quasiperiodic Mathieu equation on the transition curve of the equation was intuitively analyzed through numerical simulation, to analyze the stability changes in the equation.

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Approximate analytical solution in slow-fast system based on modified multi-scale method

Cited by 6 publications

References 40 publications

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Cluster oscillation and bifurcation of fractional-order Duffing system with two time scales

Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

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