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Czak J, Pleimling M. Spatiotemporal patterns emerging from a spatially localized time-delayed feedback scheme. Phys Rev E 2021; 104:064213. [PMID: 35030850 DOI: 10.1103/physreve.104.064213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2021] [Accepted: 11/30/2021] [Indexed: 11/07/2022]
Abstract
In attempts to manage spatiotemporal transient chaos in spatially extended systems, these systems are often subjected to protocols that perturb them as a whole and stabilize globally a new dynamic regime, as, for example, a uniform steady state. In this work we show that selectively perturbing only part of a system can generate space-time patterns that are not observed when controlling the whole system. Depending on the protocol used, these new patterns can emerge either in the perturbed or the unperturbed region. Specifically, we use a spatially localized time-delayed feedback scheme on the one-dimensional Gray-Scott reaction-diffusion system in the transient chaotic regime and demonstrate, through the numerical integration of the resulting kinetic equations, the stabilization of spatially localized space-time patterns that can be perfectly periodic. The mechanism underlying the observed pattern generation is related to diffusion across the interfaces separating the perturbed and unperturbed regions.
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Affiliation(s)
- Jason Czak
- Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA.,Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - Michel Pleimling
- Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA.,Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA.,Academy of Integrated Science, Virginia Tech, Blacksburg, Virginia 24061-0563, USA
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2
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Rombouts J, Gelens L, Erneux T. Travelling fronts in time-delayed reaction-diffusion systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20180127. [PMID: 31329070 PMCID: PMC6661327 DOI: 10.1098/rsta.2018.0127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
We review a series of key travelling front problems in reaction-diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by Erneux et al. (Erneux et al. 2010 Phil. Trans. R. Soc. A 368, 483-493 (doi:10.1098/rsta.2009.0228)) by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.
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Affiliation(s)
- Jan Rombouts
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven (KU Leuven), 3000 Leuven, Belgium
| | - Lendert Gelens
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven (KU Leuven), 3000 Leuven, Belgium
| | - Thomas Erneux
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine C.P. 231, 1050 Bruxelles, Belgium
- e-mail:
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Tarama S, Egelhaaf SU, Löwen H. Traveling band formation in feedback-driven colloids. Phys Rev E 2019; 100:022609. [PMID: 31574772 DOI: 10.1103/physreve.100.022609] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2019] [Indexed: 06/10/2023]
Abstract
Using simulation and theory we study the dynamics of a colloidal suspension in two dimensions subject to a time-delayed repulsive feedback that depends on the positions of the colloidal particles. The colloidal particles experience an additional potential that is a superposition of repulsive potential energies centered around the positions of all the particles a delay time ago. Here we show that such a feedback leads to self-organization of the particles into traveling bands. The width of the bands and their propagation speed can be tuned by the delay time and the range of the imposed repulsive potential. The emerging traveling band behavior is observed in Brownian dynamics computer simulations as well as microscopic dynamic density functional theory. Traveling band formation also persists in systems of finite size leading to rotating traveling waves in the case of circularly confined systems.
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Affiliation(s)
- Sonja Tarama
- Institute for Theoretical Physics II: Soft Matter, Heinrich Heine University Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany
| | - Stefan U Egelhaaf
- Condensed Matter Physics Laboratory, Heinrich Heine University Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany
| | - Hartmut Löwen
- Institute for Theoretical Physics II: Soft Matter, Heinrich Heine University Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany
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Zykov VS. Spiral wave initiation in excitable media. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:rsta.2017.0385. [PMID: 30420544 DOI: 10.1098/rsta.2017.0385] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 07/19/2018] [Indexed: 05/20/2023]
Abstract
Spiral waves represent an important example of dissipative structures observed in many distributed systems in chemistry, biology and physics. By definition, excitable media occupy a stationary resting state in the absence of external perturbations. However, a perturbation exceeding a threshold results in the initiation of an excitation wave propagating through the medium. These waves, in contrast to acoustic and optical ones, disappear at the medium's boundary or after a mutual collision, and the medium returns to the resting state. Nevertheless, an initiation of a rotating spiral wave results in a self-sustained activity. Such activity unexpectedly appearing in cardiac or neuronal tissues usually destroys their dynamics which results in life-threatening diseases. In this context, an understanding of possible scenarios of spiral wave initiation is of great theoretical importance with many practical applications.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.
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Affiliation(s)
- V S Zykov
- Max Planck Institute for Dynamics and Self-Organization, Goettingen, Germany
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Otto A, Radons G, Bachrathy D, Orosz G. Synchronization in networks with heterogeneous coupling delays. Phys Rev E 2018; 97:012311. [PMID: 29448336 DOI: 10.1103/physreve.97.012311] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2017] [Indexed: 11/07/2022]
Abstract
Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.
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Affiliation(s)
- Andreas Otto
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
| | - Dániel Bachrathy
- Department of Applied Mechanics, Budapest University of Technology and Economics, H-1111, Budapest, Hungary
| | - Gábor Orosz
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA
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González Ochoa HO, Perales GS, Epstein IR, Femat R. Effects of stochastic time-delayed feedback on a dynamical system modeling a chemical oscillator. Phys Rev E 2018; 97:052214. [PMID: 29906855 DOI: 10.1103/physreve.97.052214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2018] [Indexed: 06/08/2023]
Abstract
We examine how stochastic time-delayed negative feedback affects the dynamical behavior of a model oscillatory reaction. We apply constant and stochastic time-delayed negative feedbacks to a point Field-Körös-Noyes photosensitive oscillator and compare their effects. Negative feedback is applied in the form of simulated inhibitory electromagnetic radiation with an intensity proportional to the concentration of oxidized light-sensitive catalyst in the oscillator. We first characterize the system under nondelayed inhibitory feedback; then we explore and compare the effects of constant (deterministic) versus stochastic time-delayed feedback. We find that the oscillatory amplitude, frequency, and waveform are essentially preserved when low-dispersion stochastic delayed feedback is used, whereas small but measurable changes appear when a large dispersion is applied.
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Affiliation(s)
- Héctor O González Ochoa
- Departamento de Electrónica, Universidad de Guadalajara. Av. Revolución 1500, 44430, Guadalajara Jal, México
| | - Gualberto Solís Perales
- Departamento de Electrónica, Universidad de Guadalajara. Av. Revolución 1500, 44430, Guadalajara Jal, México
| | - Irving R Epstein
- Department of Chemistry, Brandeis University, Waltham, Massachusetts 02454-9110, USA
| | - Ricardo Femat
- Instituto Potosino de Investigación Científica y Tecnológica A.C., San Luis Potosí, México
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Garbin B, Javaloyes J, Barland S, Tissoni G. Interactions and collisions of topological solitons in a semiconductor laser with optical injection and feedback. CHAOS (WOODBURY, N.Y.) 2017; 27:114308. [PMID: 29195338 DOI: 10.1063/1.5006751] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We present experimental and numerical results about dynamical interactions of topological solitons in a semiconductor laser with coherent injection and feedback. We show different kind of interactions such as repulsion, annihilation, or formation of soliton bound states, depending on laser parameters. Collisions between single structures and bound states conserve momentum and charge.
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Affiliation(s)
- B Garbin
- The Dodd-Walls Centre for Photonic and Quantum Technologies, and Physics Department, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
| | - J Javaloyes
- Departament de Física, Universitat de les Illes Balears, Cra. De Valldemossa, km 7.5, E-07122 Palma, Spain
| | - S Barland
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
| | - G Tissoni
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
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Tlidi M, Panajotov K, Ferré M, Clerc MG. Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities. CHAOS (WOODBURY, N.Y.) 2017; 27:114312. [PMID: 29195301 DOI: 10.1063/1.5007868] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Time-delayed feedback plays an important role in the dynamics of spatially extended systems. In this contribution, we consider the generic Lugiato-Lefever model with delay feedback that describes Kerr optical frequency comb in all fiber cavities. We show that the delay feedback strongly impacts the spatiotemporal dynamical behavior resulting from modulational instability by (i) reducing the threshold associated with modulational instability and by (ii) decreasing the critical frequency at the onset of this instability. We show that for moderate input intensities it is possible to generate drifting cavity solitons with an asymmetric radiation emitted from the soliton tails. Finally, we characterize the formation of rogue waves induced by the delay feedback.
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Affiliation(s)
- Mustapha Tlidi
- Faculté des Sciences, Optique Nonlinéaire Théorique, Université libre de Bruxelles (U.L.B.), C.P. 231, Campus Plaine, B-1050 Bruxelles, Belgium
| | - Krassimir Panajotov
- Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussels, Pleinlaan 2, B-1050 Brussels, Belgium
| | - Michel Ferré
- Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Marcel G Clerc
- Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile
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Javaloyes J, Marconi M, Giudici M. Nonlocality Induces Chains of Nested Dissipative Solitons. PHYSICAL REVIEW LETTERS 2017; 119:033904. [PMID: 28777628 DOI: 10.1103/physrevlett.119.033904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2016] [Indexed: 06/07/2023]
Abstract
Dissipative solitons often behave as quasiparticles, and they may form molecules characterized by well-defined bond distances. We show that pointwise nonlocality may lead to a new kind of molecule where bonds are not rigid. The elements of this molecule can shift mutually one with respect to the others while remaining linked together, in a manner similar to interlaced rings in a chain. We report experimental observations of these chains of nested dissipative solitons in a time-delayed laser system.
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Affiliation(s)
- J Javaloyes
- Departament de Física, Universitat de les Illes Baleares, C/Valldemossa km 7.5, 07122 Mallorca, Spain
| | - M Marconi
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, F-06560 Valbonne, France
| | - M Giudici
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, F-06560 Valbonne, France
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Tabbert F, Schelte C, Tlidi M, Gurevich SV. Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model. Phys Rev E 2017; 95:032213. [PMID: 28415377 DOI: 10.1103/physreve.95.032213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2016] [Indexed: 06/07/2023]
Abstract
We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.
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Affiliation(s)
- Felix Tabbert
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany
| | - Christian Schelte
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany
| | - Mustapha Tlidi
- Faculté des Sciences, Université Libre de Bruxelles, Campus Plaine, C.P. 231, Brussels B-1050, Belgium
| | - Svetlana V Gurevich
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany
- Center for Nonlinear Science (CeNoS), University of Münster, Corrensstrasse 2, D-48149 Münster, Germany
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Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay. ENTROPY 2016. [DOI: 10.3390/e18030064] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Gurevich SV. Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:rsta.2014.0014. [PMID: 25246671 DOI: 10.1098/rsta.2014.0014] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
The dynamics of a single breathing localized structure in a three-component reaction-diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov-Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner.
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Affiliation(s)
- Svetlana V Gurevich
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
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Vladimirov AG, Pimenov A, Gurevich SV, Panajotov K, Averlant E, Tlidi M. Cavity solitons in vertical-cavity surface-emitting lasers. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:rsta.2014.0013. [PMID: 25246674 DOI: 10.1098/rsta.2014.0013] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We investigate a control of the motion of localized structures (LSs) of light by means of delay feedback in the transverse section of a broad area nonlinear optical system. The delayed feedback is found to induce a spontaneous motion of a solitary LS that is stationary and stable in the absence of feedback. We focus our analysis on an experimentally relevant system, namely the vertical-cavity surface-emitting laser (VCSEL). We first present an experimental demonstration of the appearance of LSs in a 80 μm aperture VCSEL. Then, we theoretically investigate the self-mobility properties of the LSs in the presence of a time-delayed optical feedback and analyse the effect of the feedback phase and the carrier lifetime on the delay-induced spontaneous drift instability of these structures. We show that these two parameters affect strongly the space-time dynamics of two-dimensional LSs. We derive an analytical formula for the threshold associated with drift instability of LSs and a normal form equation describing the slow time evolution of the speed of the moving structure.
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Affiliation(s)
- A G Vladimirov
- Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany Saint Petersburg State University, Faculty of Physics, St Petersburg, Russia
| | - A Pimenov
- Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany
| | - S V Gurevich
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
| | - K Panajotov
- Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium Institute of Solid State Physics, 72 Tzarigradsko Chaussee Boulevard, 1784 Sofia, Bulgaria
| | - E Averlant
- Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium Université Libre de Bruxelles (U.L.B.), Faculté des Sciences, CP. 231, Campus Plaine, 1050 Bruxelles, Belgium
| | - M Tlidi
- Université Libre de Bruxelles (U.L.B.), Faculté des Sciences, CP. 231, Campus Plaine, 1050 Bruxelles, Belgium
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