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

Wang, Qianqian; Yu, Yue; Zhang, Zhengdi; Han, Xiujing

doi:10.1177/1461348419825698

Cited by 10 publications

(6 citation statements)

References 31 publications

(47 reference statements)

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“…However, to date, most of the works have focused on conventional excitations and typical scales, with relatively few reports on low-frequency amplitude modulation, especially in the context of fast-slow dynamics [38,39]. The initial exploration of this topic was presented by Yu et al [40,41], who discovered mixed-mode relaxation oscillations induced by amplitude modulation near the Melnikov curve, and recently, Song et al [42] observed that low-frequency amplitude modulation with an initial phase can extend the duration of the slow process and increase the number of bifurcation points, sparking our interest in further examining amplitude modulation effects. However, these studies primarily focus on the modulation frequency and index in the context of an initial phase difference, neglecting the impact of variations in the initial phase on dynamical behavior.…”

Section: Introductionmentioning

confidence: 99%

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Song,

Jiang,

Han

et al. 2024

Phys. Scr.

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Relaxation oscillations are pervasive in diverse areas of natural sciences and engineering, and exploring the dynamical mechanisms of relaxation oscillations is one of the most significant issues. Typical relaxation oscillations can be observed in the Duffing system. Recently, amplitude modulation has emerged as a novel control mechanism for investigating the behavior of fast-slow dynamics in systemic tension oscillations. It has demonstrated the ability to prolong the quasi-static slow process of the system and increase the number of bifurcation points. However, the exploration of the mechanistic aspects of amplitude modulation is still in its early stages, with many unreported dynamical mechanisms. Among these, investigating the modes of relaxation oscillations induced by amplitude modulation is one of the most important issues. Therefore, this manuscript focuses on studying the effect of amplitude modulation on relaxation oscillations, using the classical forced Duffing system as a representative model. Significantly, we report an intriguing finding for the first time, revealing a new amplitude-modulated mechanism by which the disappearance of relaxation oscillations can be induced. By employing the fast-slow analysis, we have examined the underlying dynamical mechanisms, revealing a strong correlation with the modulation index of amplitude modulation. Notably, when the system operates under low amplitude modulation, an extension of the quasi-static process is observed, manifesting as a prolonged slow process. Conversely, under high amplitude modulation, relaxation oscillations suddenly disappear. Our results serve to enrich the potential mechanisms of amplitude modulation, and our analysis provides a reference for investigating the dynamical behavior induced by amplitude modulation in other dynamical systems.

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Section: Introductionmentioning

confidence: 99%

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Song,

Jiang,

Han

et al. 2024

Phys. Scr.

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“…Furthermore, recent studies by Yu et al [40] applied amplitude modulation dynamics to fast-slow mechanics, revealing complex behaviors like multiple S-shaped manifolds and fast-slow jumping phenomena. In addition, Wang et al [41] used the Melnikov method to study fast-slow dynamics under amplitude modulation, uncovering hybrid relaxation oscillation patterns. Besides, Shi et al [42] proposed a method to predict synchronization bandwidth and frequency stability through phase delay, thereby enabling oscillators to maintain optimal performance.…”

Section: Introductionmentioning

confidence: 99%

Overmodulation causes a variation in the number of jumps in the Duffing system

Song,

Sun,

Han

et al. 2024

Phys. Scr.

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Amplitude modulation, involving normal and overmodulation scenarios, is crucial for information transmission. However, the complex dynamics of how modulation phase shift affects relaxation oscillations, especially under overmodulation conditions, has not been fully elucidated. Thus, this paper aims to explore the dynamical mechanism of relaxation oscillations affected by modulation phase shift under overmodulation conditions. The result shows that minor phase adjustments in low overmodulation phases can change the time series of the signal. Notably, at critical modulation phase thresholds, the number of transitions in each period of relaxation oscillations increases, and this phenomenon can be observed across a range of parameter values. However, further increase in phase will lead to the decrease in the number of transitions in relaxation oscillations, which demonstrates a clear correlation between phase adjustments and fold bifurcations affecting oscillation patterns. Based on the tri-parametric and bi-parametric bifurcation analysis, we explore the effect of overmodulation index on the number of transitions, and find that higher indices induce complex variations in it. These findings highlight the intricate interplay between modulation phase and modulation index in determining relaxation oscillation patterns, which are crucial for understanding amplitude modulation diversity and can serve as a reference for future research on other modulation scenarios.

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“…In recent years, differential equations and their symmetric properties have received much attention, since they have a wide application related to the various phenomena of life. For instance, differential equations appeared in the modeling of population growth, as they were associated with various life sciences such as biology, neural networks and chemical reactions (see [1][2][3] and, for nonlinear dynamic systems [4][5][6]).…”

Section: Introductionmentioning

confidence: 99%

“…The results for Equation (6) were completed in [35][36][37] by using both the Riccati transformation technique and integral averaging technique.…”

Section: Introductionmentioning

confidence: 99%

Third-Order Neutral Differential Equations with Damping and Distributed Delay: New Asymptotic Properties of Solutions

et al. 2022

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In this paper, we are interested in studying the oscillation of differential equations with a damping term and distributed delay. We establish new criteria that guarantee the oscillation of the third-order differential equation in terms of oscillation of the second-order linear differential equation without a damping term. By using the Riccati transformation technique and the principle of comparison, we obtain new results on the oscillation for the studied equation. The results show significant improvement and extend the previous works. Symmetry contributes to determining the correct methods for solving neutral differential equations. Some examples are provided to show the significance of our results.

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

Cited by 10 publications

References 31 publications

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Overmodulation causes a variation in the number of jumps in the Duffing system

Third-Order Neutral Differential Equations with Damping and Distributed Delay: New Asymptotic Properties of Solutions

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