Relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork–Hopf bifurcation

Xia, Yibo; Zhang, Zhengdi; Bi, Qinsheng

doi:10.1007/s11071-020-05795-0

Cited by 21 publications

(3 citation statements)

References 21 publications

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“…two scales in frequency domain, bursting oscillations can also be observed, while the bursting mechanism can't be obtained directly by the traditional slow-fast analysis method. In recent years, Bi et al [7,[26][27][28] presented a modified slow-fast method with the conceptions of generalized autonomous system and transformed phase portrait, which have been demonstrated to be an effective tool to analyze the generation mechanism of bursting oscillations in dynamical systems with a single slow excitation. The one periodic excitation dynamical systems can be expressed in the form ẋ = f[x, µ µ µ, Acos(ωt)],…”

Section: Introductionmentioning

confidence: 99%

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Zhang

Peng

2022

Nonlinear Dyn

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The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise resistor and a harmonically changed electric source into a typical Chua's circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation.

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Section: Introductionmentioning

confidence: 99%

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Zhang

Peng

2022

Nonlinear Dyn

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Section: Introductionmentioning

confidence: 99%

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Xu¹,

Zhang²,

Peng³

2021

Preprint

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The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation

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“…Ambrosio et al addressed the canard phenomenon in a slow-fast modified Leslie-Gower model [20]. Xia et al discussed relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork-Hopf bifurcation [21]. Atabaigi and Barati studied relaxation oscillations and canard explosion in a predatorprey system of Holling and Leslie types [22].…”

Section: Introductionmentioning

confidence: 99%

Relaxation Oscillations and Dynamical Properties in Two Time-Delay Slow-Fast Modified Leslie-Gower Models

2020

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In this paper, we consider two kinds of time-delay slow-fast modified Leslie-Gower models. For the first system, we prove the existence and uniqueness of relaxation oscillation cycle through the geometric singular perturbation theory and entry-exit function. For the second system, we put forward a conjecture that the relaxation oscillation of the system is unique. Numerical simulation also verifies our results for the systems.

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Relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork–Hopf bifurcation

Cited by 21 publications

References 21 publications

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

Relaxation Oscillations and Dynamical Properties in Two Time-Delay Slow-Fast Modified Leslie-Gower Models

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