The effect of finite response–time in coupled dynamical systems

Saxena, Garima; Prasad, Awadhesh; Ramaswamy, Ramakrishna

doi:10.1007/s12043-011-0179-z

Cited by 3 publications

(6 citation statements)

References 21 publications

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“…Closely related to the case of distributed delays is the situation where systems respond to cumulative signals, namely by integrating information received over an interval in time. This occurs when systems have a finite intrinsic response time and also causes the region of AD to extend indefinitely [56,57].…”

Section: Delay Interactionmentioning

confidence: 99%

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: Delay Interactionmentioning

confidence: 99%

Amplitude death: The emergence of stationarity in coupled nonlinear systems

2012

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“…In the absence of coupling, in the former case the motion is periodic, while in the latter case the dynamics can be (quasi)periodic or chaotic. In both instances we find that the effect of introducing dissipation is to cause the oscillatory dynamics to be damped to a fixed point, namely we find that there is the so-called amplitude death (AD) [6] as has been seen in delay-coupled dissipative dynamical systems [6,15].…”

Section: Introductionmentioning

confidence: 77%

“…In contrast, in non-Hamiltonian systems AD islands are separated by finite range of delay values [6,15] where the coupling function need not vanish. Hence, in those systems the reappearance of oscillations after AD depends both on the coupling strength and on the delay, whereas in coupled Hamiltonian systems we find that this happens only due to delay.…”

Section: Delay Coupled Harmonic Oscillatorsmentioning

confidence: 99%

“…The coupling between dynamical systems can give rise to a number of collective phenomena such as synchronization, phase locking, phase shifting [1], amplitude death [2][3][4][5][6], phase-flip [7][8][9], hysteresis [10], riddling [11] and so on [12]. Since communication between the individual systems is mediated by signals that can have a finite transmission time, many studies account for this by the introduction of time-delay in the coupling [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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Amplitude death phenomena in delay-coupled Hamiltonian systems

2013

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Hamiltonian systems, when coupled via time-delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space. In particular for sufficiently strong coupling there can be amplitude death (AD), namely the stabilization of point attractors and the cessation of oscillatory motion. The approach to the state of AD or oscillation death is also accompanied by a phase-flip in the transient dynamics. A discussion and analysis of the phenomenology is made through an application to the specific cases of harmonic as well as anharmonic coupled oscillators, in particular the Hénon-Heiles system.

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“…Both AD and OD are known to occur in various settings. These are reviewed in [Saxena et al, 2012] and [Koseska et al, 2013], and include mismatched oscillators [Crowley & Field, 1981], [Bar-Eli, 1984], [Bar-Eli, 2011], and [Koseska et al, 2010], delayed interactions [Reddy et al, 1998], [Reddy et al, 1999], , , and [Senthilkumar & Kurths, 2010] (including distributed delays [Atay, 2003] and cumulative signals [Saxena et al, 2010] and [Saxena et al, 2011]), conjugate coupling [Kim, 2005], [Kim et al, 2005], [Karnatak et al, 2010], [Karnatak et al, 2009], and [Zhang et al, 2011], dynamic coupling [Konishi, 2003], nonlinear coupling [Prasad et al, 2010] and [Prasad et al, 2003], linear augmentation [Sharma et al, 2011] and [Resmi et al, 2010],velocity coupling [Saxena et al, 2012], and other schemes.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations and Amplitude Death from Distributed Delays in Coupled Landau-Stuart Oscillators and a Chaotic Parametrically Forced van der Pol-Rayleigh System

Choudhury¹,

Roopnarain²

2020

Preprint

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Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are closed via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter a = a Hopf at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three systems, with oscillations at larger values (corresponding to weaker delay). In the Landau-Stuart system, the Hopf-generated limit cycles for a > a Hopf turn out to be remarkably stable under very large variations of all other system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. This is to be expected as the corresponding undelayed systems are robust oscillators over very wide ranges of their respective parameters. Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as the parameters are pushed far into the post-Hopf regime, and also reveal other features, such as how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractor change as the delay and other parameters are varied. For the chaotic system, very strong delays may still lead to the cessation of oscillations and the onset of AD (even for relatively large values of the system forcing which tends to oppose this phenomenon). Varying of the other important system parameter, the parametric excitation, leads to a rich sequence of dynamical behaviors, with the bifurcations leading from one regime (or type of attractor) into the next being carefully tracked.

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The effect of finite response–time in coupled dynamical systems

Cited by 3 publications

References 21 publications

Amplitude death: The emergence of stationarity in coupled nonlinear systems

Amplitude death: The emergence of stationarity in coupled nonlinear systems

Amplitude death phenomena in delay-coupled Hamiltonian systems

Bifurcations and Amplitude Death from Distributed Delays in Coupled Landau-Stuart Oscillators and a Chaotic Parametrically Forced van der Pol-Rayleigh System

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