Chaos in time delay systems, an educational review

Wernecke, Hendrik; Sándor, Bulcsú; Gros, Claudius

doi:10.1016/j.physrep.2019.08.001

Cited by 42 publications

(29 citation statements)

References 194 publications

(383 reference statements)

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“…For the system considered in the present work, however, the sub-LE is in good approximation a fixed property of the dynamical node and independent on the feedback signals. Moreover, the sub-LE is always negative, which corresponds to weak chaos and is a necessary condition for laminar chaos [27,28]. In weak chaos, the response of the dynamical node exhibits a functional dependency to its time-delayed drive, similar to the map F above, whereas strong chaos generates a level of inconsistency in the mapping between delay cycles [31][32][33].…”

Section: Theoretical Backgroundmentioning

confidence: 97%

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Laminar chaos in nonlinear electronic circuits with delay clock modulation

2020

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We study laminar chaos in an electronic experiment. A two-diode nonlinear circuit with delayed feedback shows chaotic dynamics similar to the Mackey-Glass or Ikeda delay systems. Clock modulation of a single delay line leads to a conservative variable delay, which with a second delay line is augmented to dissipative delays, leading to laminar chaotic regimes. We discuss the properties of this particular delay modulation and demonstrate experimental aspects of laminar chaos in terms of power spectra and return maps.

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Section: Theoretical Backgroundmentioning

confidence: 97%

“…Within dissipative delays, the same authors have subsequently discovered the phenomenon of laminar chaos [27]. Chaotic dynamics is common in delay systems and has been studied extensively for the case of large delays [28]. A distinction between strong and weak chaos has been made based on the sub-Lyapunov exponent of the delay system [29].…”

Section: Introductionmentioning

confidence: 99%

Laminar chaos in nonlinear electronic circuits with delay clock modulation

2020

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“…The time-average in Eq. (15) was taken over at least 5000 delay periods. Since noise is always inherent to experimental systems in the absence of external noise (ζ = 0), the autocorrelation C(θ) in this case also slightly deviates from the theoretical behavior given by Eq.…”

Section: How To Scan Parameters For Laminar Chaosmentioning

confidence: 99%

Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos

et al. 2020

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Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional Turbulent Chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional Laminar Chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [J. D. Hart, R. Roy, D. Müller-Bender, A. Otto, and G. Radons, PRL 123 154101 (2019)], where it is demonstrated that Laminar Chaos is a robust phenomenon, which can be observed in experimental systems. We provide a time series analysis toolbox for the detection of robust features of Laminar Chaos. We benchmark our toolbox by experimental time series and time series of a model system which is described by a nonlinear Langevin equation with time-varying delay. The benchmark is done for different noise strengths for both the experimental system and the model system, where Laminar Chaos can be detected, even if it is hard to distinguish from Turbulent Chaos by a visual analysis of the trajectory.

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“…As such, time delay systems can provide a higher level of computational security against embedding reconstruction. (ii) TD systems provide hyperchaos with multiple positive Lyapunov exponents (LEs) [17]. Due to these reasons, a number of simple and well characterized TD systems have been designed to produce chaos and hyperchaos [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Enhancing Ikeda Time Delay System by Breaking the Symmetry of Sine Nonlinearity

Gao

2019

Complexity

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In the present contribution, an asymmetric central contraction mutation (ACCM) model is proposed to enhance the Ikeda time delay system. The modified Ikeda system model is designed by introducing a superimposed tanh function term into the sine nonlinearity term. Stability and Hopf bifurcation characteristics of the system are analyzed theoretically. Numerical simulations, carried out in terms of bifurcation diagrams, Lyapunov exponents spectrum, phase portraits, and two-parameter (2D) largest Lyapunov exponent diagrams are employed to highlight the complex dynamical behaviors exhibited by the enhanced system. The results indicate that the modified system has rich dynamical behaviors including limit cycle, multiscroll hyperchaos, chaos, and hyperchaos. Moreover, as a major outcome of this paper, considering the fragile chaos phenomenon, the ACCM-Ikeda time delay system has better dynamical complexity and larger connected chaotic parameter spaces (connectedness means that there is no stripe corresponding to nonchaotic dynamics embedded in the chaos regions).

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Chaos in time delay systems, an educational review

Cited by 42 publications

References 194 publications

Laminar chaos in nonlinear electronic circuits with delay clock modulation

Laminar chaos in nonlinear electronic circuits with delay clock modulation

Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos

Enhancing Ikeda Time Delay System by Breaking the Symmetry of Sine Nonlinearity

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