Targeting fixed-point solutions in nonlinear oscillators through linear augmentation

Sharma, Pradeep; Sharma, Amit; Shrimali, Manish Dev; Prasad, Awadhesh

doi:10.1103/physreve.83.067201

Cited by 55 publications

(41 citation statements)

References 23 publications

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“…Since the m-dimensional linear system has the dynamicsU = −kU, for positive k, and in the absence of coupling to the nonlinear system, this is incapable of having sustained oscillations [77,78]. The additional parameter B in the augmented system thus adaptively drives the X dynamics to the fixed point B.…”

Section: Linear Augmentationmentioning

confidence: 99%

“…16 is quite wide, these can also be stabilized. Of course, this is distinct from targeting; see details in [77].…”

Section: Linear Augmentationmentioning

confidence: 99%

“…Recently Sharma et al [77] have proposed a new strategy through which a nonlinear oscillator, when coupled to a linear system, experiences amplitude death. Consider such a Landau-Stuart oscillatoṙ…”

Section: Linear Augmentation and Other Strategiesmentioning

confidence: 99%

“…A recent scheme [77] (see Sec. 2.6) to stabilize fixed points of nonlinear systems by coupling to a linear dynamical system, U X = F (X) + εU,…”

Section: Linear Augmentationmentioning

confidence: 99%

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Amplitude death: The emergence of stationarity in coupled nonlinear systems

2012

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When nonlinear dynamical systems are coupled, depending on the intrinsic dynamics and the manner in which the coupling is organized, a host of novel phenomena can arise. In this context, an important emergent phenomenon is the complete suppression of oscillations, formally termed amplitude death (AD). Oscillations of the entire system cease as a consequence of the interaction, leading to stationary behavior. The fixed points that the coupling stabilizes can be the otherwise unstable fixed points of the uncoupled system or can correspond to novel stationary points. Such behaviour is of relevance in areas ranging from laser physics to the dynamics of biological systems. In this review we discuss the characteristics of the different coupling strategies and scenarios that lead to AD in a variety of different situations, and draw attention to several open issues and challenging problems for further study.

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Section: Linear Augmentationmentioning

confidence: 99%

“…16 is quite wide, these can also be stabilized. Of course, this is distinct from targeting; see details in [77].…”

Section: Linear Augmentationmentioning

confidence: 99%

Section: Linear Augmentation and Other Strategiesmentioning

confidence: 99%

“…A recent scheme [77] (see Sec. 2.6) to stabilize fixed points of nonlinear systems by coupling to a linear dynamical system, U X = F (X) + εU,…”

Section: Linear Augmentationmentioning

confidence: 99%

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Amplitude death: The emergence of stationarity in coupled nonlinear systems

2012

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“…All these configurations work with linear coupling, and by taking the coupling itself to be nonlinear, one can, in addition, stabilize a targeted or "designed" steady state [21]. AD can also be achieved by coupling a linear and nonlinear system [22], in a configuration termed linear augmentation. When the coupling involves transmission delay, AD follows as a natural consequence [16,17].…”

mentioning

confidence: 99%

Amplitude death: The cessation of oscillations in coupled nonlinear dynamical systems

Saxena¹,

Punetha²,

Prasad³

et al. 2014

AIP Conference Proceedings

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Abstract. Here we extend a recent review (Physics Reports 521, 205 (2012)) of amplitude death, namely the suppression of oscillations due to the coupling interactions between nonlinear dynamical systems. This is an important emergent phenomenon that is operative under a variety of scenarios. We summarize results of recent studies that have significantly added to our understanding of the mechanisms that underlie the process, and also discuss the phase-flip transition, a characteristic and unusual effect that occurs in the transient dynamics as the oscillations die out.Nonlinear systems can show a range of complex dynamics depending on the nature of the equations of motion. When two or more systems are coupled, then there is frequently newer, emergent behavior that depends on the manner in which the systems interact. Synchronization is one such phenomenon, but depending upon the manner in which the coupling is organized, the collective dynamics can be more complex [1,2]. An unusual and unexpected consequence of the coupling is to induce simplicity: in addition to synchrony [4], there can be the suppression of chaos and appearance of periodicity [3], and amplitude death (AD), namely the loss of any oscillatory dynamics when the dynamics is driven to a fixed point [5].In the past few decades, AD has been the subject of extensive study due to potential applications in stabilizing systems to the steady state. Oscillation quenching is often desired as a control mechanism in technology: for suppressing fluctuations in the power output of lasers [6], in thermo-optical oscillators for implementing safety measures [7], in coupled self excited elastic beans [8], or in electrical engineering to stabilize DC grids with constant power loads [9] for instance, and also for medical purposes like treating neuronal disorders [10,11,12]. In other applications, AD has also been proposed as an underlying mechanism for auditory transduction [13] and is also presumed to play an important role in climatology, where the large scale oceanic and atmospheric anomalies are found to be correlated with the zonal coupling of atmospheres of the respective ocean basins [14].A recent review [5] has focussed on AD in different fields. The characteristics of different coupling strategies and scenarios that lead to AD and its occurrence in networks of coupled oscillators and in various experimental situations has been discussed in detail. We summarize these briefly here. Starting with the work by Aronson et al. [15] who showed that mismatched units, when coupled lead to AD, several other scenarios have been proposed. Time-delay coupling [16,17] and conjugate coupling [18] both cause identical coupled systems to show AD.

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