Bursting Oscillations and Mixed-Mode Oscillations in Driven Liénard System

Kingston, S. Leo; Thamilmaran, K.

doi:10.1142/s0218127417300257

Cited by 53 publications

(12 citation statements)

References 38 publications

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“…In this direction Liénard system (1) has been shown to possess multistablility [33,34]. In the following, we investigate the effect of the considered non-feedback methods on multistability.…”

Section: Effect Of Weak Second Periodic Forcing and Constant Bias On ...mentioning

confidence: 99%

“…From this we conclude that the multistability is brought under control when the system is influenced by the constant bias as well as the second periodic forcing. It is important to note that for the considered choice of parameter the sytem possess a double well potential [33]. Now when the system is influenced by a constant bias the system's dynamics is forced to get confined in a single well.…”

Section: Effect Of Weak Second Periodic Forcing and Constant Bias On ...mentioning

confidence: 99%

“…The value of parameters were chosen as in Ref. [33]. In the basin of attraction grey represents the chaotic regions and red represents the periodic orbits.…”

Section: Effect Of Weak Second Periodic Forcing and Constant Bias On ...mentioning

confidence: 99%

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Constant Bias and Weak Second Periodic Forcing : Tools to Mitigate Extreme Events

Sudharsan

Venkatesan²,

Senthilvelan

2021

Preprint

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We propose two potentially viable non-feedback methods, namely (i) constant bias and (ii) weak second periodic forcing as tools to mitigate extreme events. We demonstrate the effectiveness of these two tools in suppressing extreme events in two well-known nonlinear dynamical systems, namely (i) Liénard system and (ii) a non-polynomial mechanical system with velocity dependent potential. As far as the constant bias is concerned, in the Liénard system, the suppression occurs due to the decrease in large amplitude oscillations and in the non-polynomial mechanical system the suppression occurs due to the destruction of chaos into a periodic orbit. As far as the second periodic forcing is concerned, in both the examples, extreme events get suppressed due to the increase in large amplitude oscillations. We also demonstrate that by introducing a phase in the second periodic forcing one can decrease the probability of occurrence of extreme events even further in the non-polynomial system. To provide a support to the complete suppression of extreme events, we present the two parameter probability plot for all the cases. In addition to the above, we examine the feasibility of the aforementioned tools in a parametrically driven version of the non-polynomial mechanical system. Finally, we investigate how these two methods influence the multistability nature in the Liénard system.

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Section: Effect Of Weak Second Periodic Forcing and Constant Bias On ...mentioning

confidence: 99%

Section: Effect Of Weak Second Periodic Forcing and Constant Bias On ...mentioning

confidence: 99%

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Constant Bias and Weak Second Periodic Forcing : Tools to Mitigate Extreme Events

Sudharsan

Venkatesan²,

Senthilvelan

2021

Preprint

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“…Upadhyay et al [2] studied mixed-mode oscillations and the synchronous activity in the noise-induced modified Morris-Lecar neural system. Kingston and amilmaran [3] discussed the bursting oscillations and mixed-mode oscillations in the driven Lienard system. Shimizu et al [4] made a thorough exploration of mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation.…”

Section: Introductionmentioning

confidence: 99%

Mixed-Mode Oscillation in a Class of Delayed Feedback System and Multistability Dynamic Response

Qian

Wang

2020

Complexity

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In this paper, a class of two-parameter mixed-mode oscillation with time delay under the action of amplitude modulation is studied. The investigation is from four aspects. Firstly, a parametric equation is considered as a slow variable. By the time-history diagram and phase diagram, we can find that the system generates a cluster discovery image. Secondly, the Euler method is used to discrete the system and obtain the discrete equation. Thirdly, the dynamic characteristics of the system at different time scales are discussed when the ratio of the natural frequency and the excitation frequency of the system is integer and noninteger. Fourthly, we discuss the influence of time delay on the discovery of clusters of this kind of system. The research shows that the time lag does not interfere with the influence of the cluster image, but the dynamics of the upper and lower parts of the oscillation in each period will be delayed. So, we can improve peak performance by adjusting the time lag and obtain the desired peak. Finally, we explore the multistate dynamic response of a two-dimensional nonautonomous Duffing system with higher order. According to bifurcation diagram and time-history curve, bistable state will appear in the system within the critical range. With the gradual increase of parameters, the chaotic attractor will suddenly disappear which will lead to the destruction of the bistable state.

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“…the external forcing frequency is far smaller than intrinsic frequency of the original system, the oscillator may exhibit typical fast–slow dynamics named as mixed-mode oscillations (MMOs). 15–18 Such oscillatory behaviors often behave in periodic states characterized by a combination of relatively large amplitude (in spiking state) and nearly harmonic small amplitude oscillations (in rest state). 19–22…”

Section: Introductionmentioning

confidence: 99%

Melnikov-threshold-triggered mixed-mode oscillations in a family of amplitude-modulated forced oscillator

Wang

Zhang

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

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This paper deals with the transitions through Melnikov thresholds and the corresponding fast-slow dynamics in a family of bi-parametric mechanical oscillators subjected to an amplitude modulation force both analytically and numerically. Applying Melnikov analytical method, the threshold condition for the occurrence of mixed-mode oscillations is obtained. First, the Melnikov threshold curves are drawn in different parameter space in the unperturbed system. Next, as the control variable crosses above the threshold, two states about mixed-mode oscillations can be identified: the rest state corresponding to local limit cycle from nontrivial equilibrium point and the spiking state corresponding to large amplitude cycle produced by saddle-node bifurcation of limit cycles. Such mixed-mode oscillations are created since the amplitude modulation force slowly and periodically switches between the rest states and spiking states. Furthermore, we elucidate the effect of modulation frequency on phase portraits and dynamical behaviors, and some numerical simulations are also included to illustrate the validity of our study.

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Bursting Oscillations and Mixed-Mode Oscillations in Driven Liénard System

Cited by 53 publications

References 38 publications

Constant Bias and Weak Second Periodic Forcing : Tools to Mitigate Extreme Events

Constant Bias and Weak Second Periodic Forcing : Tools to Mitigate Extreme Events

Mixed-Mode Oscillation in a Class of Delayed Feedback System and Multistability Dynamic Response

Melnikov-threshold-triggered mixed-mode oscillations in a family of amplitude-modulated forced oscillator

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