Amplitude death: The emergence of stationarity in coupled nonlinear systems

Saxena, Garima; Prasad, Awadhesh; Ramaswamy, Ramakrishna

doi:10.1016/j.physrep.2012.09.003

Cited by 352 publications

(246 citation statements)

References 127 publications

Supporting

Mentioning

239

Contrasting

Unclassified

Order By: Relevance

“…It is also known that coupling limit cycle oscillators together can eliminate oscillations [13,14]. However, we show that these phenomena can occur in a planar limit cycle system when each component is subjected to additive white noise.…”

mentioning

confidence: 65%

Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability

Newby

Schwemmer

2014

Phys. Rev. Lett.

View full text Add to dashboard Cite

The effects of noise on the dynamics of nonlinear systems is known to lead to many counterintuitive behaviors. Using simple planar limit cycle oscillators, we show that the addition of moderate noise leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in the opposite direction of the deterministic limit cycle, or cease oscillating altogether. Utilizing standard techniques from stochastic calculus and recently developed stochastic phase reduction methods, we elucidate the mechanisms underlying the different dynamics and verify our analysis with the use of numerical simulations. Lastly, we show that similar bistable behavior is found when moderate noise is applied to the more biologically realistic FitzHugh-Nagumo model.Limit cycle oscillators have been widely used to model various natural phenomena [1][2][3]. As such, they have been the subject of extensive study in the field of nonlinear science. In recent years, the effects of noise on the dynamics of limit cycle oscillators has received much interest [4][5][6][7][8]. When the noise is weak, formal reduction methods can be performed to reduce the dimensionality of the system to a so-called phase equation [7,8]. In this case, one can analytically show that both the magnitude (and correlation time for colored noise) of the added noise shift the mean frequency of oscillations away from the natural frequency of the limit cycle. On the other hand, when the noise is large, the trajectories of the system appear completely random and bear no resemblance to the deterministic limit cycle behavior.The case of moderate noise applied to a limit cycle oscillator has received less attention. It is known that moderate noise can cause stochastic resonance in systems close to a bifurcation to limit cycle oscillations [9,10], and in systems where limit cycles arise as the result of periodic forcing [4]. However, an exploration of how moderate noise interacts with the underlying deterministic dynamics of a system displaying limit cycle behavior has not yet been undertaken and is the purpose of the current Letter. We find that an oscillator subject to moderate noise can display numerous interesting and counterintuitive behaviors. In some cases, the addition of noise causes the phase to behave like a bistable switch, while in other cases noise can act to completely eliminate oscillations, and even cause the trajectories to rotate in the opposite direction of the deterministic limit cycle.Bistability in the amplitude of the limit cycle has been shown to occur in the Stuart-Landau (SL) system when it is subjected to periodic forcing [11], or specially constructed stochastic forcing [12]. It is also known that coupling limit cycle oscillators together can eliminate oscillations [13,14]. However, we show that these phenomena can occur in a planar limit cycle system when each component is subjected to additive white noise.For simplicity, we consider a radially symmetric deterministic system. One such symmetric oscillator that we employ is the ...

show abstract

mentioning

confidence: 65%

Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability

Newby

Schwemmer

2014

Phys. Rev. Lett.

View full text Add to dashboard Cite

show abstract

“…[19,20,21,22,23,24,25]. The occurrence of amplitude death is reported in coupled oscillators as due to various mechanisms like dynamic coupling [26], time delay coupling [27], nonlinear coupling [28], conjugate coupling or coupling via dissimilar variables in identical oscillators, parameter mismatch and distributed frequencies [29,30,31] in coupling systems [32].…”

Section: Introductionmentioning

confidence: 99%

Suppression of dynamics and frequency synchronization in coupled slow and fast dynamical systems

Gupta

Ambika

2016

Eur. Phys. J. B

View full text Add to dashboard Cite

Abstract. We present our study on the emergent states of two interacting nonlinear systems with differing dynamical time scales. We find that the inability of the interacting systems to fall in step leads to difference in phase as well as change in amplitude. If the mismatch is small, the systems settle to a frequency synchronized state with constant phase difference. But as mismatch in time scale increases, the systems have to compromise to a state of no oscillations. We illustrate this for standard nonlinear systems and identify the regions of quenched dynamics in the parameter plane. The transition curves to this state are studied analytically and confirmed by direct numerical simulations. As an important special case, we revisit the well-known model of coupled ocean atmosphere system used in climate studies for the interactive dynamics of a fast oscillating atmosphere and slowly changing ocean. Our study in this context indicates occurrence of multi stable periodic states and steady states of convection coexisting in the system, with a complex basin structure.

show abstract

“…[1,2,3,4,5,6,7,8]. There are numerous studies on the dynamics of the single pendulum and coupled pendulums, but mostly devoted to in-plane oscillations [9,10,11].…”

Section: Introductionmentioning

confidence: 99%

The dynamics of co- and counter rotating coupled spherical pendula

Witkowski

Perlikowski

Prasad

et al. 2014

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

Abstract. The dynamics of co-and counter-rotating coupled spherical pendulums (two lower pendulums are mounted at the end of the upper pendulum) is considered. Linear mode analysis shows the existence of three rotating modes. Starting from linear modes allow we calculate the nonlinear normal modes, which are and present them in frequencyenergy plots. With the increase of energy in one mode we observe a symmetry breaking pitchfork bifurcation. In the second part of the paper we consider energy transfer between pendulums having different energies. The results for co-rotating (all pendulums rotate in the same direction) and counter-rotating motion (one of lower pendulums rotates in the opposite direction) are presented. In general, the energy fluctuations in counter-rotating pendulums are found to be higher than in the co-rotating case.

show abstract

Amplitude death: The emergence of stationarity in coupled nonlinear systems

Cited by 352 publications

References 127 publications

Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability

Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability

Suppression of dynamics and frequency synchronization in coupled slow and fast dynamical systems

The dynamics of co- and counter rotating coupled spherical pendula

Contact Info

Product

Resources

About