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Flunkert V, Yanchuk S, Dahms T, Schöll E. Synchronizability of Networks with Strongly Delayed Links: A Universal Classification. ACTA ACUST UNITED AC 2014. [DOI: 10.1007/s10958-014-2078-6] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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Soriano MC, Flunkert V, Fischer I. Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers. Chaos 2013; 23:043133. [PMID: 24387572 DOI: 10.1063/1.4844335] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We present a systematic approach to identify the similarities and differences between a chaotic system with delayed feedback and two mutually delay-coupled systems. We consider the general case in which the coupled systems are either unsynchronized or in a generally synchronized state, in contrast to the mostly studied case of identical synchronization. We construct a new time-series for each of the two coupling schemes, respectively, and present analytic evidence and numerical confirmation that these two constructed time-series are statistically equivalent. From the construction, it then follows that the distribution of time-series segments that are small compared to the overall delay in the system is independent of the value of the delay and of the coupling scheme. By focusing on numerical simulations of delay-coupled chaotic lasers, we present a practical example of our findings.
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Affiliation(s)
- Miguel C Soriano
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Valentin Flunkert
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Ingo Fischer
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
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Flunkert V, Fischer I, Schöll E. Dynamics, control and information in delay-coupled systems: an overview. Philos Trans A Math Phys Eng Sci 2013; 371:20120465. [PMID: 23960223 PMCID: PMC3758166 DOI: 10.1098/rsta.2012.0465] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Affiliation(s)
- Valentin Flunkert
- IFISC (UIB-CSIC), Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain.
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Zeeb S, Dahms T, Flunkert V, Schöll E, Kanter I, Kinzel W. Discontinuous attractor dimension at the synchronization transition of time-delayed chaotic systems. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 87:042910. [PMID: 23679492 DOI: 10.1103/physreve.87.042910] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2012] [Revised: 01/14/2013] [Indexed: 06/02/2023]
Abstract
The attractor dimension at the transition to complete synchronization in a network of chaotic units with time-delayed couplings is investigated. In particular, we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents for iterated maps and for two coupled semiconductor lasers. We argue that the Kaplan-Yorke dimension must be discontinuous at the transition and compare it to the correlation dimension. For a system of Bernoulli maps, we indeed find a jump in the correlation dimension. The magnitude of the discontinuity in the Kaplan-Yorke dimension is calculated for networks of Bernoulli units as a function of the network size. Furthermore, the scaling of the Kaplan-Yorke dimension as well as of the Kolmogorov entropy with system size and time delay is investigated.
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Affiliation(s)
- Steffen Zeeb
- Institute of Theoretical Physics, University of Würzburg, Am Hubland, D-97074 Würzburg, Germany.
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Heiligenthal S, Dahms T, Yanchuk S, Jüngling T, Flunkert V, Kanter I, Schöll E, Kinzel W. Strong and weak chaos in nonlinear networks with time-delayed couplings. Phys Rev Lett 2011; 107:234102. [PMID: 22182092 DOI: 10.1103/physrevlett.107.234102] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2011] [Indexed: 05/31/2023]
Abstract
We study chaotic synchronization in networks with time-delayed coupling. We introduce the notion of strong and weak chaos, distinguished by the scaling properties of the maximum Lyapunov exponent within the synchronization manifold for large delay times, and relate this to the condition for stable or unstable chaotic synchronization, respectively. In simulations of laser models and experiments with electronic circuits, we identify transitions from weak to strong and back to weak chaos upon monotonically increasing the coupling strength.
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Affiliation(s)
- Sven Heiligenthal
- Institute of Theoretical Physics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany.
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Lehnert J, Hövel P, Flunkert V, Guzenko PY, Fradkov AL, Schöll E. Adaptive tuning of feedback gain in time-delayed feedback control. Chaos 2011; 21:043111. [PMID: 22225348 DOI: 10.1063/1.3647320] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We demonstrate that time-delayed feedback control can be improved by adaptively tuning the feedback gain. This adaptive controller is applied to the stabilization of an unstable fixed point and an unstable periodic orbit embedded in a chaotic attractor. The adaptation algorithm is constructed using the speed-gradient method of control theory. Our computer simulations show that the adaptation algorithm can find an appropriate value of the feedback gain for single and multiple delays. Furthermore, we show that our method is robust to noise and different initial conditions.
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Affiliation(s)
- J Lehnert
- Institut für Theoretische Physik, TU Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
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Flunkert V, Schöll E. Towards easier realization of time-delayed feedback control of odd-number orbits. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 84:016214. [PMID: 21867280 DOI: 10.1103/physreve.84.016214] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2011] [Indexed: 05/31/2023]
Abstract
We develop generalized time-delayed feedback schemes for the stabilization of periodic orbits with an odd number of positive Floquet exponents, which are particularly well suited for experimental realization. We construct the parameter regimes of successful control and validate these by numerical simulations and numerical continuation methods. In particular, it is shown how periodic orbits can be stabilized with symmetric feedback matrices by introducing an additional latency time in the control loop. Finally, we show using normal form analysis and numerical simulations how our results could be implemented in a laser setup using optoelectronic feedback.
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Affiliation(s)
- V Flunkert
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany.
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Hicke K, D'Huys O, Flunkert V, Schöll E, Danckaert J, Fischer I. Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 83:056211. [PMID: 21728634 DOI: 10.1103/physreve.83.056211] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2010] [Indexed: 05/31/2023]
Abstract
We study the synchronization properties of the delay dynamics of two identical semiconductor lasers coupled through a semitransparent mirror. Via an analytical and numerical approach, we investigate the influence of asymmetries, in particular mismatches of self- and cross-coupling strength and differences in self- and cross-coupling delay. We show that the former mismatch affects the stability of the zero-lag state but not the dynamics within the synchronization manifold, while the latter mismatch does not affect the quality of synchronization but alters the dynamics significantly. Our results are extended to different unidirectional coupling schemes. This is highly relevant for communication schemes utilizing chaotic dynamics. Finally, the influence of nonlinear gain saturation on the dynamics and stability of synchronization is discussed.
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Affiliation(s)
- K Hicke
- Instituto de Fisica Interdisciplinar y Sistemas Complejos (IFISC), UIB-CSIC, Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
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Flunkert V, Yanchuk S, Dahms T, Schöll E. Synchronizing distant nodes: a universal classification of networks. Phys Rev Lett 2010; 105:254101. [PMID: 21231592 DOI: 10.1103/physrevlett.105.254101] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2010] [Indexed: 05/26/2023]
Abstract
Stability of synchronization in delay-coupled networks of identical units generally depends in a complicated way on the coupling topology. We show that for large coupling delays synchronizability relates in a simple way to the spectral properties of the network topology. The master stability function used to determine the stability of synchronous solutions has a universal structure in the limit of large delay: It is rotationally symmetric around the origin and increases monotonically with the radius in the complex plane. This allows a universal classification of networks with respect to their synchronization properties and solves the problem of complete synchronization in networks with strongly delayed coupling.
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Affiliation(s)
- V Flunkert
- Institut für Theoretische Physik, TU Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany.
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Fiedler B, Flunkert V, Hövel P, Schöll E. Delay stabilization of periodic orbits in coupled oscillator systems. Philos Trans A Math Phys Eng Sci 2010; 368:319-341. [PMID: 20008404 DOI: 10.1098/rsta.2009.0232] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
We study diffusively coupled oscillators in Hopf normal form. By introducing a non-invasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super- and subcritical cases, we state a condition on the oscillator's nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of odd-number orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.
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Affiliation(s)
- B Fiedler
- Institut für Mathematik I, Free Universität Berlin, Arnimallee 2-6, 14195 Berlin, Germany
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Flunkert V, D'Huys O, Danckaert J, Fischer I, Schöll E. Bubbling in delay-coupled lasers. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 79:065201. [PMID: 19658547 DOI: 10.1103/physreve.79.065201] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2009] [Indexed: 05/28/2023]
Abstract
We theoretically study chaos synchronization of two lasers which are delay coupled via an active or a passive relay. While the lasers are synchronized, their dynamics is identical to a single laser with delayed feedback for a passive relay and identical to two delay-coupled lasers for an active relay. Depending on the coupling parameters the system exhibits bubbling, i.e., noise-induced desynchronization, or on-off intermittency. We associate the desynchronization dynamics in the coherence collapse and low-frequency fluctuation regimes with the transverse instability of some of the compound cavity's antimodes. Finally, we demonstrate how, by using an active relay, bubbling can be suppressed.
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Affiliation(s)
- V Flunkert
- Institut für Theoretische Physik, TU Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
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Fiedler B, Yanchuk S, Flunkert V, Hövel P, Wünsche HJ, Schöll E. Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 77:066207. [PMID: 18643350 DOI: 10.1103/physreve.77.066207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2008] [Indexed: 05/26/2023]
Abstract
We consider the delayed feedback control method for stabilization of unstable rotating waves near a fold bifurcation. Theoretical analysis of a generic model and numerical bifurcation analysis of the rate-equations model demonstrate that such orbits can always be stabilized by a proper choice of control parameters. Our paper confirms the recently discovered invalidity of the so-called "odd-number limitation" of delayed feedback control. Previous results have been restricted to the vicinity of a subcritical Hopf bifurcation. We now refute such a limitation for rotating waves near a fold bifurcation. We include an application to all-optical realization of the control in three-section semiconductor lasers.
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Affiliation(s)
- B Fiedler
- Institut für Mathematik I, FU Berlin, Arnimallee 2-6, D-14195 Berlin, Germany
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Flunkert V, Schöll E. Suppressing noise-induced intensity pulsations in semiconductor lasers by means of time-delayed feedback. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 76:066202. [PMID: 18233899 DOI: 10.1103/physreve.76.066202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2007] [Indexed: 05/25/2023]
Abstract
We investigate the possibility to suppress noise-induced intensity pulsations (relaxation oscillations) in semiconductor lasers by means of a time-delayed feedback control scheme. This idea is first studied in a generic normal-form model, where we derive an analytic expression for the mean amplitude of the oscillations and demonstrate that it can be strongly modulated by varying the delay time. We then investigate the control scheme analytically and numerically in a laser model of Lang-Kobayashi type and show that relaxation oscillations excited by noise can be very efficiently suppressed via feedback from a Fabry-Perot resonator.
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Affiliation(s)
- Valentin Flunkert
- Institut für Theoretische Physik, Technische Universität Berlin, Germany
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Just W, Fiedler B, Georgi M, Flunkert V, Hövel P, Schöll E. Beyond the odd number limitation: a bifurcation analysis of time-delayed feedback control. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 76:026210. [PMID: 17930124 DOI: 10.1103/physreve.76.026210] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2007] [Indexed: 05/25/2023]
Abstract
We investigate the normal form of a subcritical Hopf bifurcation subjected to time-delayed feedback control. Bifurcation diagrams which cover time-dependent states as well are obtained by analytical means. The computations show that unstable limit cycles with an odd number of positive Floquet exponents can be stabilized by time-delayed feedback control, contrary to incorrect claims in the literature. The model system constitutes one of the few examples where a nonlinear time delay system can be treated entirely by analytical means.
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Affiliation(s)
- W Just
- School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom.
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Choe CU, Flunkert V, Hövel P, Benner H, Schöll E. Conversion of stability in systems close to a Hopf bifurcation by time-delayed coupling. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 75:046206. [PMID: 17500977 DOI: 10.1103/physreve.75.046206] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2006] [Indexed: 05/15/2023]
Abstract
We propose a control method with time delayed coupling which makes it possible to convert the stability features of systems close to a Hopf bifurcation. We consider two delay-coupled normal forms for Hopf bifurcation and demonstrate the conversion of stability, i.e., an interchange between the sub- and supercritical Hopf bifurcation. The control system provides us with an unified method for stabilizing both the unstable periodic orbit and the unstable steady state and reveals typical effects like amplitude death and phase locking. The main method and the results are applicable to a wide class of systems showing Hopf bifurcations, for example, the Van der Pol oscillator. The analytical theory is supported by numerical simulations of two delay-coupled Van der Pol oscillators, which show good agreement with the theory.
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Affiliation(s)
- Chol-Ung Choe
- Department of Physics, University of Science, Unjong-District, Pyongyang, DPR Korea
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Fiedler B, Flunkert V, Georgi M, Hövel P, Schöll E. Refuting the odd-number limitation of time-delayed feedback control. Phys Rev Lett 2007; 98:114101. [PMID: 17501057 DOI: 10.1103/physrevlett.98.114101] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/22/2006] [Indexed: 05/15/2023]
Abstract
We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology, and life sciences, where subcritical Hopf bifurcations occur.
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Affiliation(s)
- B Fiedler
- Institut für Mathematik I, FU Berlin, Arnimallee 2-6, D-14195 Berlin, Germany
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