An experimental study of slow passage through Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator

Premraj, D.; Suresh, K.; Banerjee, Tanmoy; Thamilmaran, K.

doi:10.1016/j.cnsns.2016.01.012

Cited by 47 publications

(11 citation statements)

References 27 publications

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“…This quasi-steady picture changes if the bifurcation parameter varies at a finite rate (figure 1b): bifurcation delay occurs, and it is a function of the rate (e.g. [40][41][42]). For a quasi-steady variation of ν in the presence of stochastic forcing (figure 1c), the hysteresis is suppressed in a statistical sense.…”

Section: Introductionmentioning

confidence: 99%

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

et al. 2018

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Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.

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Section: Introductionmentioning

confidence: 99%

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

et al. 2018

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“…In order to see if system (31) has a unique solution, we have to determine whether the previous determinant is zero or not. To do so, we consider the characteristic polynomials of system (29) in each region, which are given by where f stands the derivative of f (x) in the corresponding region j ∈ {L, M }. Since the determinant is zero when λ L = λ M , this leads to p L (λ L ) − p M (λ L ) = 0.…”

Section: Application To the Doi-kumagai Neural Burstermentioning

confidence: 99%

“…Since we maintain the equilibrium in the middle strip {|x| ≤ 1}, the expected behavior of the slow passage is the one described in section 3.1 about the presence of a buffer point, in particular in theorem 3. Therefore, let us consider system (29), where we rewrite f as…”

Section: Application To the Doi-kumagai Neural Burstermentioning

confidence: 99%

“…The phenomenon of slow passage through a Hopf bifurcation (delayed loss of stability) occurs in models coming from a vast range of application areas; for instance, in Neuroscience [1], Physics [29], Population Biology [24] or Chemistry [17], to name but a few. Slow-fast dynamical systems in which a delayed loss of stability happens often display complex oscillatory behaviors such as bursting [18,19,32,4] or mixed-mode oscillations (MMOs) [1,11,16,20].…”

Section: Introductionmentioning

confidence: 99%

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Slow passage through a Hopf-like bifurcation in piecewise linear systems: application to elliptic bursting

Penalva¹,

Desroches²,

Teruel³

et al. 2021

Preprint

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The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly-varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of a Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems. In the present work, we study for the first time this phenomenon in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slowfast system to exhibit a slow passage through a Hopf-like bifurcation, in link with the number of linearity zones considered in the system and possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi-Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.

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“…In this case, not only the adjustment timescale of the system but also the forcing timescale can be important. File generated with AMS Word template 1.0 Dependencies of bifurcations on the forcing timescale have been reported in other scientific fields such as acoustics, laser physics, engineering, and neuroscience (Mandel and Erneux 1987;Mannella et al 1987;Baer and Gaekel 2008;Bergeot et al 2014;Premraj et al 2016;Bonciolini et al 2018). They showed that time-varying forcing can postpone bifurcation of a dynamical system so that the bifurcation requires a larger parameter change beyond the static bifurcation point-the parameter value at which the bifurcation would occur if the quasi-static equilibrium is satisfied.…”

Section: Introductionmentioning

confidence: 98%

Feedback processes modulating the sensitivity of Atlantic thermohaline circulation to freshwater forcing timescales

Kim

et al. 2021

Journal of Climate

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Paleo proxy records indicate that abrupt changes in thermohaline circulation (THC) were induced by rapid meltwater discharge from retreating ice sheets. Such abrupt changes in the THC have been understood as a hysteresis behavior of nonlinear system. Previous studies, however, primarily focused on a near-static hysteresis under fixed or slowly varying freshwater forcing (FWF), reflecting the equilibrated response of the THC. This study aims to improve the current understanding of transient THC responses under rapidly varying forcing and its dependency on forcing timescales. The results simulated by an Earth system model suggest that the bifurcation is delayed as the forcing timescale is shorter, causing the Atlantic meridional overturning circulation collapse (recovery) to occur at higher (lower) FWF values. The delayed shutdown/recovery occurs because bifurcation is determined not by the FWF value at the time but by the total amount of freshwater remaining over the THC convection region. The remaining freshwater amount is primarily determined by the forcing accumulation (i.e., time-integrated FWF), which is modulated by the freshwater/salt advection by ocean circulations and freshwater flux by the atmospheric hydrological cycle. In general, the latter is overwhelmed by the former. When the forced freshwater amount is the same, the modulation effect is stronger under slowly varying forcing because more time is provided for the feedback processes.

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An experimental study of slow passage through Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator

Cited by 47 publications

References 27 publications

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Slow passage through a Hopf-like bifurcation in piecewise linear systems: application to elliptic bursting

Feedback processes modulating the sensitivity of Atlantic thermohaline circulation to freshwater forcing timescales

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