Bogdanov–Takens resonance in time-delayed systems

Coccolo, Mattia; Zhu, Beibei; Sanjuán, Miguel A. F.; Sanz‐Serna, J. M.

doi:10.1007/s11071-017-3992-1

Cited by 10 publications

(4 citation statements)

References 27 publications

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“…Jeevarathinam et al (2011) and Daza et al (2013b) investigate in a similar way the forced, delayed Duffing oscillator. A new kind of nonlinear resonance of periodically forced delay systems has recently been described by Coccolo et al (2018).…”

Section: Introductionmentioning

confidence: 99%

A stroboscopic averaging algorithm for highly oscillatory delay problems

Sanz‐Serna

Zhu

2018

IMA Journal of Numerical Analysis

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Section: Introductionmentioning

confidence: 99%

A stroboscopic averaging algorithm for highly oscillatory delay problems

Sanz‐Serna

Zhu

2018

IMA Journal of Numerical Analysis

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“…Interestingly, other related phenomena commonly attributed to the microscopic realm, such as tunneling through external potential barriers (or in multistable external potentials) can be easily demonstrated with our retarded potential by introducing an external Duffing potential in replacement of the harmonic well used here [50]. A similar situation occurs when studying the flow of electrons through potential barriers, where this paradoxical phenomenon becomes explained when interpreted in terms of the quantum potential, which appears in the Hamilton-Jacobi equation of the quantum system, and which is frequently disregarded when interpreting physical phenomena [51].…”

Section: Discussionmentioning

confidence: 85%

“…In other words, we are suggesting that the switch between different wells leading to an intermittent behavior can be interpreted in terms of the robust intermittency phenomenon. This dynamics is due to the nonlinear resonances that allow the particle to jump back and forth over the potential barrier [50].…”

Section: Discussionmentioning

confidence: 99%

Orbit quantization in a retarded harmonic oscillator

López¹

2023

Preprint

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We study the dynamics of a damped harmonic oscillator in the presence of a retarded potential with state-dependent time-delayed feedback. In the limit of small time-delays, we show that the oscillator is equivalent to a Liénard system. This allows us to analytically predict the value of the first Hopf bifurcation, unleashing a self-oscillatory motion. We compute bifurcation diagrams for several model parameter values and analyse multistable domains in detail. Using the Lyapunov energy function, two well-resolved energy levels represented by two coexisting stable limit cycles are discerned. Further exploration of the parameter space reveals the existence of a superposition limit cycle, encompassing two degenerate coexisting limit cycles at the fundamental energy level.When the system is driven very far from equilibrium, a multiscale strange attractor displaying intrinsic and robust intermittency is uncovered.

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“…The effect of the time-delay in the periodically driven damped Duffing oscillator was analyzed in [Ravichandran et al, 2012] using a perturbation theory. A new resonance for the time-delayed overdamped Duffing oscillator in the presence of a Bogdanov-Takens bifurcation leading to the phenomenon called Bogdanov-Takens resonance was analyzed in [Coccolo et al, 2018].…”

Section: Introductionmentioning

confidence: 99%

Delay-Induced Resonance in the Time-Delayed Duffing Oscillator

Cantisán

Coccolo

Seoane

et al. 2020

Int. J. Bifurcation Chaos

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The phenomenon of delay-induced resonance implies that in a nonlinear system a time-delay term may be used as an effective enhancer of the oscillations caused by an external forcing maintaining the same frequency. This is possible for the parameters for which the time-delay induces sustained oscillations. Here, we study this type of resonance in the overdamped and underdamped time-delayed Duffing oscillators, and we explore some new features. One of them is the conjugate phenomenon: the oscillations caused by the time-delay may be enhanced by means of the forcing without modifying their frequency. The resonance takes place when the frequency of the oscillations induced by the time-delay matches the ones caused by the forcing and vice versa. This is an interesting result as the nature of both perturbations is different. Even for the parameters for which the time-delay does not induce sustained oscillations, we show that a resonance may appear following a different mechanism.

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Bogdanov–Takens resonance in time-delayed systems

Cited by 10 publications

References 27 publications

A stroboscopic averaging algorithm for highly oscillatory delay problems

A stroboscopic averaging algorithm for highly oscillatory delay problems

Orbit quantization in a retarded harmonic oscillator

Delay-Induced Resonance in the Time-Delayed Duffing Oscillator

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