Synchronisation and scaling properties of chaotic networks with multiple delays

D'Huys, Otti; Zeeb, Steffen; Jüngling, Thomas; Heiligenthal, Sven; Yanchuk, Serhiy; Kinzel, Wolfgang

doi:10.1209/0295-5075/103/10013

Cited by 20 publications

(23 citation statements)

References 29 publications

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“…Because the time scales have to be well separated, say at least a factor 10 for iterated maps, it is possible to observe this sequential scaling behaviour only for up to three delay times. But, for this case, the hierarchy of strong and consecutive weak chaos has been found in simulations of the iterated functions and semiconductor rate equations, and in analytical calculations for Bernoulli units [22].…”

Section: Multiple Delay Timesmentioning

confidence: 89%

Chaos in networks with time-delayed couplings

Kinzel

2013

Phil. Trans. R. Soc. A.

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Networks of nonlinear units coupled by time-delayed signals can show chaos. In the limit of long delay times, chaos appears in two ways: strong and weak, depending on how the maximal Lyapunov exponent scales with the delay time. Only for weak chaos, a network can synchronize completely, without time shift. The conditions for strong and weak chaos and synchronization in networks with multiple delay times are investigated.

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Section: Multiple Delay Timesmentioning

confidence: 89%

Chaos in networks with time-delayed couplings

Kinzel

2013

Phil. Trans. R. Soc. A.

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“…Bernoulli maps are weakly chaotic if a(1 − ) < 1 holds [7]. In this case the SLE can be approximated by the following expression [28].…”

Section: Synchronization Of Delayed Chaotic Networkmentioning

confidence: 99%

Synchronization of fluctuating delay-coupled chaotic networks

et al. 2017

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We study the synchronization of chaotic units connected through time-delayed fluctuating interactions. Focusing on small-world networks of Bernoulli and Logistic units with a fixed chiral backbone, we compare the synchronization properties of static and fluctuating networks in the regime of large delays. We find that random network switching may enhance the stability of synchronized states. Synchronization appears to be maximally stable when fluctuations are much faster than the time-delay, whereas it disappears for very slow fluctuations. For fluctuation time scales of the order of the time-delay, we report a resynchronizing effect in finite-size networks. Moreover, we observe characteristic oscillations in all regimes, with a periodicity related to the time-delay, as the system approaches or drifts away from the synchronized state.

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“…Although the solutions to (6) are not given explicitly, their approximations can be found using the smallness of ε [11,12,23] (largeness of the delays)…”

Section: Appendix: Destabilization Of the Steady Statementioning

confidence: 99%

Pattern Formation in Systems with Multiple Delayed Feedbacks

Yanchuk

Giacomelli

2014

Phys. Rev. Lett.

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Dynamical systems with complex delayed interactions arise commonly when propagation times are significant, yielding complicated oscillatory instabilities. In this Letter, we introduce a class of systems with multiple, hierarchically long time delays, and using a suitable space-time representation we uncover features otherwise hidden in their temporal dynamics. The behaviour in the case of two delays is shown to "encode" two-dimensional spiral defects and defects turbulence. A multiple scale analysis sets the equivalence to a complex Ginzburg-Landau equation, and a novel criterium for the attainment of the long-delay regime is introduced. We also demonstrate this phenomenon for a semiconductor laser with two delayed optical feedbacks.Systems with time delays are common in many fields, ranging from optics (e.g. laser with feedback [1-4]), vehicle systems [5], to neural networks [6], information processing [7], and many others [8]. A finite propagation velocity of the information introduces in such systems a new relevant scale, which is comparable or higher than the intrinsic timescales. It has been shown that the complexity of such systems, e.g. the dimension of attractors, is finite and it grows linearly with time delay [9]; moreover, the spectrum of Lyapunov exponents approaches a continuous limit for long delay [10][11][12]. As a result, in this case essentially high-dimensional phenomena can occur such as spatio-temporal chaos [13], square waves [8], Eckhaus destabilization [14], or coarsening [3]. In the above mentioned situations, the system involves one long delay, which can be interpreted as the size of a onedimensional, spatially extended system [13,15]. This approach has proven to be instrumental in explaining new phenomena in systems with time delays [16,17].In this Letter, we show that many new challenging problems arise when a system is subject to several delayed feedbacks acting on different scales. In contrast to the single delay situation, essentially new phenomena occur, related to higher spatial dimensions involved in the dynamics, such as spirals or defect turbulence. As an illustration, we consider a specific physical system, namely, a model of a semiconductor laser with two optical feedbacks.A simple paradigmatic setup for the multiple delays case is the following systeṁEq.(1) describes a very general situation: the interplay of the oscillatory instability (Hopf bifurcation) and two delayed feedbacks z τi = z(t − τ i ), that we consider acting on different timescales 1 τ 1 τ 2 . The variable z(t) is complex, and the parameters a, b, and c determine the instantaneous, τ 1 -, and τ 2 -feedback rates, respectively. The instantaneous part of the system (without feedback) is known as the normal form for the Hopf bifurcation.The following basic questions arise: what kind of new phenomena can be observed in systems with several delayed feedbacks? Can one relate the dynamics of such systems to spatially extended systems with several spatial dimensions? In the case of positive answer, under which condi...

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Synchronisation and scaling properties of chaotic networks with multiple delays

Cited by 20 publications

References 29 publications

Chaos in networks with time-delayed couplings

Chaos in networks with time-delayed couplings

Synchronization of fluctuating delay-coupled chaotic networks

Pattern Formation in Systems with Multiple Delayed Feedbacks

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