Spatio-temporal phenomena in complex systems with time delays. (English) Zbl 1375.37111
The authors present an overview of a class of functional differential equations. They review the general theory developed in this case, describing the main destabilization mechanisms, the use of visualization tools, and commenting on the most important and effective dynamical indicators as well as their properties in different regimes. They show how a suitable approach, based on a comparison with spatio-temporal systems, represents a powerful instrument for disclosing the very basic mechanism of long-delay systems. Various examples from different models and a series of recent experiments are reported.
Reviewer: Anatoly Martynyuk (Kyïv)
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 65P20 | Numerical chaos |
References:
| [1] | Mackey M C and Glass L 1977 Oscillation and chaos in physiological control systems Science197 287 · Zbl 1383.92036 · doi:10.1126/science.267326 |
| [2] | Popovych O V, Hauptmann C and Tass P A 2006 Control of neuronal synchrony by nonlinear delayed feedback. Biol. Cybern.95 69-85 · Zbl 1169.93338 · doi:10.1007/s00422-006-0066-8 |
| [3] | Izhikevich E M 2006 Polychronization: computation with spikes Neural Comput.18 245-82 · Zbl 1090.92006 · doi:10.1162/089976606775093882 |
| [4] | Ikeda K 1979 Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system Opt. Commun.30 257 · doi:10.1016/0030-4018(79)90090-7 |
| [5] | Erneux T 2009 Applied Delay Differential Equations(Surveys and Tutorials in the Applied Mathematical Sciences vol 3) (Berlin: Springer) · Zbl 1201.34002 |
| [6] | Soriano M C, García-Ojalvo J, Mirasso C R and Fischer I 2013 Complex photonics: dynamics and applications of delay-coupled semiconductors lasers Rev. Mod. Phys.85 421-70 · doi:10.1103/RevModPhys.85.421 |
| [7] | Sciamanna M and Shore K A 2015 Physics and applications of laser diode chaos Nat. Photon.9 151-62 · doi:10.1038/nphoton.2014.326 |
| [8] | Kuang Y 1993 Delay Differential Equations with Applications in Population Dynamics(Mathematics in Science and Engineering vol 191) (London: Academic) · Zbl 0777.34002 |
| [9] | Kopylov W, Emary C, Schöll E and Brandes T 2015 Time-delayed feedback control of the Dicke-Hepp-Lieb superradiant quantum phase transition New J. Phys.17 013040 · Zbl 1452.81124 · doi:10.1088/1367-2630/17/1/013040 |
| [10] | Orosz G, Wilson R E and Stepan G 2010 Traffic jams: dynamics and control Phil. Trans. R. Soc. A 368 4455-79 · Zbl 1202.90075 · doi:10.1098/rsta.2010.0205 |
| [11] | Page L 1918 Is a moving mass retarded by the reaction of its own radiation? Phys. Rev.11 376-400 · doi:10.1103/PhysRev.11.376 |
| [12] | Jackson J D 1975 Classical Electrodynamics (New York: Wiley) · Zbl 0997.78500 |
| [13] | De Luca J 1998 Electrodynamics of helium with retardation and self-interaction effects Phys. Rev. Lett.80 680-3 · doi:10.1103/PhysRevLett.80.680 |
| [14] | Lang R and Kobayashi K 1980 External optical feedback effects on semiconductor injection laser properties IEEE J. Quantum Electron.16 347-55 · doi:10.1109/JQE.1980.1070479 |
| [15] | Farmer J D 1982 Chaotic attractors of an infinite-dimensional dynamical system Physica D 4 366-93 · Zbl 1194.37052 · doi:10.1016/0167-2789(82)90042-2 |
| [16] | Myshkis A D 1949 General theory of differential equations with retarded arguments Usp. Mat. Nauk4 99-141 · Zbl 0035.17801 |
| [17] | Bellman R and Cooke K 1963 Differential Difference Equations (New York: Academic) · Zbl 0105.06402 |
| [18] | Kuznetsov S P 1982 Complex dynamics of oscillators with delayed feedback (review) Radiophys. Quantum Electron.25 996-1009 · doi:10.1007/BF01037379 |
| [19] | Hale J K and Lunel S M V 1993 Introduction to Functional Differential Equations (Berlin: Springer) · Zbl 0787.34002 · doi:10.1007/978-1-4612-4342-7 |
| [20] | Diekmann O, van Gils S A, Lunel S M V and Walther H O 1995 Delay Equations: Functional-, Complex-, and Nonlinear Analysis (Berlin: Springer) · Zbl 0826.34002 · doi:10.1007/978-1-4612-4206-2 |
| [21] | Atay F M 2010 Complex Time-Delay Systems (Berlin Heidelberg: Springer) (doi: 10.1007/978-3-642-02329-3) · Zbl 1189.93005 · doi:10.1007/978-3-642-02329-3 |
| [22] | Guo S and Wu J 2013 Bifurcation Theory of Functional Differential Equations(Applied Mathematical Sciences vol 184) (New York: Springer) · Zbl 1316.34003 · doi:10.1007/978-1-4614-6992-6 |
| [23] | Mallet-Paret J 1976 Negatively invariant sets of compact maps and an extension of a theorem of Cartwright J. Differ. Equ.22 331-48 · Zbl 0354.34072 · doi:10.1016/0022-0396(76)90032-2 |
| [24] | Mallet-Paret J and Nussbaum R D 1986 Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation Ann. Mat. Pura Appl.145 33-128 · Zbl 0617.34071 · doi:10.1007/BF01790539 |
| [25] | Grigorieva E V, Kashchenko S A, Loiko N A and Samson A M 1992 Nonlinear dynamics in a laser with a negative delayed feedback Physica D 59 297 · Zbl 0765.34056 · doi:10.1016/0167-2789(92)90072-U |
| [26] | Grigorieva E V and Kashchenko S A 1993 Complex temporal structures in models of a laser with optoelectronic delayed feedback Opt. Commun.102 183 · doi:10.1016/0030-4018(93)90489-R |
| [27] | Grigorieva E V and Kashchenko S 1993 Regular and chaotic pulsations in laser diode with delayed feedback Int. J. Bifurcation Chaos3 1515 · Zbl 0926.70026 · doi:10.1142/S0218127493001197 |
| [28] | Kashchenko S A 1998 The Ginzburg-Landau equation as the normal form for a second-order difference-differential equation with a large delay Comput. Math. Math. Phys.38 443-51 · Zbl 0951.34042 |
| [29] | Kashchenko I and Kaschenko S 2016 Normal and quasinormal forms for systems of difference and differential-difference equations Commun. Nonlinear Sci. Numer. Simul.38 243-56 · Zbl 1471.39004 · doi:10.1016/j.cnsns.2016.02.041 |
| [30] | Vogel T 1965 Theorie des Systemes Evolutifs (Paris: Gauthier-Villars) · Zbl 0132.21701 |
| [31] | Chicone C 2003 Inertial and slow manifolds for delay equations with small delay J. Differ. Equ.190 364-406 · Zbl 1044.34026 · doi:10.1016/S0022-0396(02)00148-1 |
| [32] | Kotomtseva L, Loiko N and Samson A 1985 Development of irregular auto-oscillations in a nonlinear system with lagging Phys. Lett. A 110 339 · doi:10.1016/0375-9601(85)90049-0 |
| [33] | Berre M L, Ressayre E, Tallet A and Gibbs H M 1986 High-dimension chaotic attractors of a nonlinear ring cavity Phys. Rev. Lett.56 274 · doi:10.1103/PhysRevLett.56.274 |
| [34] | Ikeda K and Matsumoto K 1987 High-dimensional chaotic behavior in systems with time-delayed feedback Physica D 29 223-35 · Zbl 0626.58014 · doi:10.1016/0167-2789(87)90058-3 |
| [35] | Dorizzi B, Grammaticos B, Berre M L, Pomeau Y, Ressayre E and Tallet A 1987 Statistics and dimension of chaos in differential delay systems Phys. Rev. A 35 328 · doi:10.1103/PhysRevA.35.328 |
| [36] | Le Berre M, Ressayre E, Tallet A, Gibbs H M, Kaplan D L and Rose M H 1987 Conjecture on the dimensions of chaotic attractors of delayed-feedback dynamical systems Phys. Rev. A 35 4020-2 · doi:10.1103/PhysRevA.35.4020 |
| [37] | Giacomelli G, Calzavara M and Arecchi F 1989 Instabilities in a semiconductor laser with delayed optoelectronic feedback Opt. Commun.74 97 · doi:10.1016/0030-4018(89)90498-7 |
| [38] | Lepri S, Giacomelli G, Politi A and Arecchi F T 1993 High-dimensional chaos in delayed dynamical systems Physica D 70 235-49 · Zbl 0804.58039 · doi:10.1016/0167-2789(94)90016-7 |
| [39] | Giacomelli G, Lepri S and Politi A 1995 Statistical properties of bidimensional patterns generated from delayed and extended maps Phys. Rev. E 51 3939 · doi:10.1103/PhysRevE.51.3939 |
| [40] | Lee M W, Larger L, Udaltsov V, Genin É and Goedgebuer J-P 2004 Demonstration of a chaos generator with two time delays Opt. Lett.29 325-7 · doi:10.1364/OL.29.000325 |
| [41] | Chembo Kouomou Y, Colet P, Larger L and Gastaud N 2005 Chaotic breathers in delayed electro-optical systems Phys. Rev. Lett.95 203903 · doi:10.1103/PhysRevLett.95.203903 |
| [42] | Lavrov R, Peil M, Jacquot M, Larger L, Udaltsov V and Dudley J 2009 Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization Phys. Rev. E 80 026207 · doi:10.1103/PhysRevE.80.026207 |
| [43] | Arecchi F T, Giacomelli G, Lapucci A and Meucci R 1992 Two-dimensional representation of a delayed dynamical system Phys. Rev. A 45 R4225-8 · doi:10.1103/PhysRevA.45.R4225 |
| [44] | Giacomelli G, Meucci R, Politi A and Arecchi F T 1994 Defects and space-like properties of delayed dynamical systems Phys. Rev. Lett.73 1099-102 · doi:10.1103/PhysRevLett.73.1099 |
| [45] | Giacomelli G and Politi A 1998 Multiple scale analysis of delayed dynamical systems Physica D 117 26-42 · Zbl 0942.34063 · doi:10.1016/S0167-2789(97)00318-7 |
| [46] | Wolfrum M and Yanchuk S 2006 Eckhaus instability in systems with large delay Phys. Rev. Lett.96 220201 · doi:10.1103/PhysRevLett.96.220201 |
| [47] | Yanchuk S and Perlikowski P 2009 Delay and periodicity Phys. Rev. E 79 046221 · doi:10.1103/PhysRevE.79.046221 |
| [48] | Yanchuk S and Wolfrum M 2010 A multiple time scale approach to the stability of external cavity modes in the Lang-Kobayashi system using the limit of large delay SIAM J. Appl. Dyn. Syst.9 519-35 · Zbl 1209.34104 · doi:10.1137/090751335 |
| [49] | Yanchuk S, Lücken L, Wolfrum M and Mielke A 2015 Spectrum and amplitude equations for scalar delay-differential equations with large delay Discrete Contin. Dyn. Syst. A 35 537-53 · Zbl 1312.34100 · doi:10.3934/dcds.2015.35.537 |
| [50] | Buckwar E, Scheutzow M and Pikovsky A 2005 Special issue: Stochastic dynamics with delay and memory. Selected papers based on the presentation at the workshop(Halle/Wittenberg, Germany, 2-5 February 2004) (Singapore: World Scientific) |
| [51] | Willeboordse F H 1992 Time-delayed map phenomenological equivalency with a coupled map lattice Int. J. Bifurcation Chaos2 721 · Zbl 0873.58051 · doi:10.1142/S0218127492000847 |
| [52] | Fischer I 1995 Nichtlineare Dynamik und komplexes raum-zeitliches Verhalten von Halbleiterlasern PhD Thesis Philipps-Universität Marburg |
| [53] | Javaloyes J, Ackemann T and Hurtado A 2015 Arrest of domain coarsening via antiperiodic regimes in delay systems Phys. Rev. Lett.115 203901 · doi:10.1103/PhysRevLett.115.203901 |
| [54] | Porte X, D’Huys O, Jüngling T, Brunner D, Soriano M C and Fischer I 2014 Autocorrelation properties of chaotic delay dynamical systems: a study on semiconductor lasers Phys. Rev. E 90 052911 · doi:10.1103/PhysRevE.90.052911 |
| [55] | Giacomelli G, Marin F and Romanelli M 2003 Multi-time-scale dynamics of a laser with polarized optical feedback Phys. Rev. A 67 053809 · doi:10.1103/PhysRevA.67.053809 |
| [56] | Lepri S 1994 Critical phenomena in delayed maps Phys. Lett. A 191 291-5 · doi:10.1016/0375-9601(94)90142-2 |
| [57] | Arecchi F T, Giacomelli G, Lapucci A and Meucci R 1991 Dynamics of a CO2 laser with delayed feedback: the short-delay regime Phys. Rev. A 43 4997 · doi:10.1103/PhysRevA.43.4997 |
| [58] | Kaneko K 1984 Fates of three-torus. I. Double devil’s staircase in lockings Prog. Theor. Phys.71 282-94 · Zbl 1046.37505 · doi:10.1143/PTP.71.282 |
| [59] | Kaneko K 1984 Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Towards a prelude of a ‘field theory of chaos’ Prog. Theor. Phys.72 480-6 · Zbl 1074.37521 · doi:10.1143/PTP.72.480 |
| [60] | Kaneko K 1985 Spatiotemporal intermittency in coupled map lattices Prog. Theor. Phys.74 1033-44 · Zbl 0979.37505 · doi:10.1143/PTP.74.1033 |
| [61] | Yanchuk S and Wolfrum M 2005 Instabilities of equilibria of delay-differential equations with large delay Proc. of ENOC(Eindhoven, Netherlands, 2005) ed D van Campen et al pp 1060-5 |
| [62] | Yanchuk S, Wolfrum M, Hövel P and Schöll E 2006 Control of unstable steady states by long delay feedback Phys. Rev. E 74 026201 · doi:10.1103/PhysRevE.74.026201 |
| [63] | Wolfrum M, Yanchuk S, Hövel P and Schöll E 2010 Complex dynamics in delay-differential equations with large delay Eur. Phys. J. Spec. Top.191 91-103 · doi:10.1140/epjst/e2010-01343-7 |
| [64] | Flunkert V, Yanchuk S, Dahms T and Schöll E 2010 Synchronizing distant nodes: a universal classification of networks Phys. Rev. Lett.105 254101 · doi:10.1103/PhysRevLett.105.254101 |
| [65] | Heiligenthal S, Dahms T, Yanchuk S, Jüngling T, Flunkert V, Kanter I, Schöll E and Kinzel W 2011 Strong and weak chaos in nonlinear networks with time-delayed couplings Phys. Rev. Lett.107 234102 · doi:10.1103/PhysRevLett.107.234102 |
| [66] | Lichtner M, Wolfrum M and Yanchuk S 2011 The spectrum of delay differential equations with large delay SIAM J. Math. Anal.43 788-802 · Zbl 1231.34109 · doi:10.1137/090766796 |
| [67] | D’Huys O, Zeeb S, Jüngling T, Heiligenthal S, Yanchuk S and Kinzel W 2013 Synchronisation and scaling properties of chaotic networks with multiple delays Europhys. Lett.103 10013 · doi:10.1209/0295-5075/103/10013 |
| [68] | Kinzel W 2013 Chaos in networks with time-delayed couplings Phil. Trans. R. Soc. Lond. A 371 20120461 · Zbl 1353.94091 · doi:10.1098/rsta.2012.0461 |
| [69] | Heiligenthal S, Jüngling T, D’Huys O, Arroyo-Almanza D A, Soriano M C, Fischer I, Kanter I and Kinzel W 2013 Strong and weak chaos in networks of semiconductor lasers with time-delayed couplings Phys. Rev. E 88 012902 · doi:10.1103/PhysRevE.88.012902 |
| [70] | Oliver N, Jüngling T and Fischer I 2015 Consistency properties of a chaotic semiconductor laser driven by optical feedback Phys. Rev. Lett.114 123902 · doi:10.1103/PhysRevLett.114.123902 |
| [71] | Jüngling T, Soriano M C and Fischer I 2015 Determining the sub-Lyapunov exponent of delay systems from time series Phys. Rev. E 91 062908 · doi:10.1103/PhysRevE.91.062908 |
| [72] | Jüngling T, D’Huys O and Kinzel W 2015 The transition between strong and weak chaos in delay systems: stochastic modeling approach Phys. Rev. E 91 062918 · doi:10.1103/PhysRevE.91.062918 |
| [73] | Dahmen S R, Hinrichsen H and Kinzel W 2008 Space representation of stochastic processes with delay Phys. Rev. E 77 031106 · doi:10.1103/PhysRevE.77.031106 |
| [74] | Zeeb S, Dahms T, Flunkert V, Schöll E, Kanter I and Kinzel W 2013 Discontinuous attractor dimension at the synchronization transition of time-delayed chaotic systems Phys. Rev. E 87 042910 · doi:10.1103/PhysRevE.87.042910 |
| [75] | Giacomelli G and Politi A 1996 Relationship between delayed and spatially extended dynamical systems Phys. Rev. Lett.76 2686-9 · doi:10.1103/PhysRevLett.76.2686 |
| [76] | Pikovsky A and Politi A 2016 Lyapunov Exponents: a Tool to Explore Complex Dynamics (Cambridge: Cambridge University Press) · Zbl 1419.37002 · doi:10.1017/CBO9781139343473 |
| [77] | Hale J K 1977 Theory of Functional Differential Equations (Berlin: Springer) · Zbl 0352.34001 · doi:10.1007/978-1-4612-9892-2 |
| [78] | Yanchuk S 2005 Properties of stationary states of delay equations with large delay and applications to laser dynamics Math. Methods Appl. Sci.28 363-77 · Zbl 1084.34067 · doi:10.1002/mma.584 |
| [79] | Cross M C and Hohenberg P C 1993 Pattern formation outside of equilibrium Rev. Mod. Phys.65 851-1112 · Zbl 1371.37001 · doi:10.1103/RevModPhys.65.851 |
| [80] | Cross M and Greenside H 2009 Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge: Cambridge University Press) · Zbl 1177.82002 · doi:10.1017/CBO9780511627200 |
| [81] | Puzyrev D, Yanchuk S, Vladimirov A and Gurevich S 2014 Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback SIAM J. Appl. Dyn. Syst.13 986-1009 · Zbl 1301.34082 · doi:10.1137/130944643 |
| [82] | Puzyrev D, Vladimirov A, Gurevich S and Yanchuk S 2016 Modulational instability and zigzagging of dissipative solitons induced by delayed feedback Phys. Rev. A 93 041801(R) · doi:10.1103/PhysRevA.93.041801 |
| [83] | Yanchuk S Spectrum in systems with multiple hierarchical time-delays (in preparation) |
| [84] | Yanchuk S and Giacomelli G 2015 Dynamical systems with multiple long delayed feedbacks: multiscale analysis and spatio-temporal equivalence Phys. Rev. E 92 042903 · doi:10.1103/PhysRevE.92.042903 |
| [85] | Yanchuk S and Giacomelli G 2014 Pattern formation in systems with multiple delayed feedbacks Phys. Rev. Lett.112 174103 · doi:10.1103/PhysRevLett.112.174103 |
| [86] | Sieber J, Wolfrum M, Lichtner M and Yanchuk S 2013 On the stability of periodic orbits in delay equations with large delay Discrete Contin. Dyn. Syst. A 33 3109-34 · Zbl 1282.34071 · doi:10.3934/dcds.2013.33.3109 |
| [87] | Pesin Y 1997 Dimension Theory in Dynamical Systems: Contemporary Views and Applications(Chicago Lectures in Mathematics) (Chicago, IL: University of Chicago Press) · doi:10.7208/chicago/9780226662237.001.0001 |
| [88] | Kaplan J and Yorke J 1979 Chapter Chaotic behavior of multidimensional difference equations Functional Differential Equations, Approximation of Fixed Points vol 730 (Berlin: Springer) pp 204-27 · Zbl 0448.58020 · doi:10.1007/BFb0064319 |
| [89] | Yanchuk S and Sieber J 2013 Relative equilibria and relative periodic solutions in systems with time-delay and S1 symmetry (arXiv:1306.3327) |
| [90] | Kuznetsov Y 1995 Elements of Applied Bifurcation Theory(Applied Mathematical Sciences vol 112) (Berlin: Springer) · Zbl 0829.58029 · doi:10.1007/978-1-4757-2421-9 |
| [91] | Chaté H and Manneville P 1996 Phase diagram of the two-dimensional complex Ginzburg-Landau equation Phys. A: Stat. Mech. Appl.224 348-68 · doi:10.1016/0378-4371(95)00361-4 |
| [92] | Aranson I and Kramer L 2002 The world of the complex Ginzburg-Landau equation Rev. Mod. Phys.74 99-143 · Zbl 1205.35299 · doi:10.1103/RevModPhys.74.99 |
| [93] | Deissler R J and Kaneko K 1987 Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems Phys. Lett. A 119 397-402 · doi:10.1016/0375-9601(87)90581-0 |
| [94] | Lepri S, Politi A and Torcini A 1996 Chronotopic Lyapunov analysis. I. A detailed characterization of 1D systems J. Stat. Phys.82 1429-52 · Zbl 1260.37059 · doi:10.1007/BF02183390 |
| [95] | Lepri S, Politi A and Torcini A 1997 Chronotopic Lyapunov analysis: II. Toward a unified approach J. Stat. Phys.88 31-45 · Zbl 0920.58030 · doi:10.1007/BF02508463 |
| [96] | Briggs R J 1964 Electron-Stream Interaction with Plasmas (Cambridge, MA: MIT Press) |
| [97] | Giacomelli G, Hegger R, Politi A and Vassalli M 2000 Convective Lyapunov exponents and propagation of correlations Phys. Rev. Lett.85 3616 · doi:10.1103/PhysRevLett.85.3616 |
| [98] | Hegger R, Bünner M J, Kantz H and Giaquinta A 1998 Identifying and modeling delay feedback systems Phys. Rev. Lett.81 558 · doi:10.1103/PhysRevLett.81.558 |
| [99] | Flunkert V and Schöll E 2009 Pydelay—a python tool for solving delay differential equations (arXiv:0911.1633) |
| [100] | Giacomelli G, Marino F, Zaks M A and Yanchuk S 2012 Coarsening in a bistable system with long-delayed feedback Europhys. Lett.99 58005 · doi:10.1209/0295-5075/99/58005 |
| [101] | Giacomelli G, Marino F, Zaks M A and Yanchuk S 2013 Nucleation in bistable dynamical systems with long delay Phys. Rev. E 88 062920 · doi:10.1103/PhysRevE.88.062920 |
| [102] | Engelborghs K, Luzyanina T and Roose D 2002 Numerical bifurcation analysis of delay differential equations using DDE-Biftool ACM Trans. Math. Soft.28 1-21 · Zbl 1070.65556 · doi:10.1145/513001.513002 |
| [103] | Eckhaus W 1965 Studies in Non-Linear Stability Theory(Springer Tracts in Natural Philosophy vol 6) (New York: Springer) · Zbl 0125.33101 · doi:10.1007/978-3-642-88317-0 |
| [104] | Tuckerman L S and Barkley D 1990 Bifurcation analysis of the Eckhaus instability Physica D 46 57-86 · Zbl 0721.35008 · doi:10.1016/0167-2789(90)90113-4 |
| [105] | Chow S-N, Hale J K and Huang W 1992 From sine waves to square waves in delay equations Proc. R. Soc. Edinburgh A 120 223-9 · Zbl 0764.34048 · doi:10.1017/S0308210500032108 |
| [106] | Nizette M 2004 Stability of square oscillations in a delayed-feedback system Phys. Rev. E 70 056204 · doi:10.1103/PhysRevE.70.056204 |
| [107] | Nizette M 2003 Front dynamics in a delayed-feedback system with external forcing Physica D 183 220-44 · Zbl 1085.34057 · doi:10.1016/S0167-2789(03)00175-1 |
| [108] | Weicker L, Erneux T, D’Huys O, Danckaert J, Jacquot M, Chembo Y and Larger L 2012 Strongly asymmetric square waves in a time-delayed system Phys. Rev. E 86 055201 · doi:10.1103/physreve.86.055201 |
| [109] | Larger L, Lacourt P-A, Poinsot S and Hanna M 2005 From flow to map in an experimental high-dimensional electro-optic nonlinear delay oscillator Phys. Rev. Lett.95 043903 · doi:10.1103/PhysRevLett.95.043903 |
| [110] | Weicker L, Erneux T, Jacquot M, Chembo Y and Larger L 2012 Crenelated fast oscillatory outputs of a two-delay electro-optic oscillator Phys. Rev. E 85 026206 · doi:10.1103/PhysRevE.85.026206 |
| [111] | Dee G and van Saarloos W 1988 Phys. Rev. Lett.60 2641 · doi:10.1103/PhysRevLett.60.2641 |
| [112] | Bray A J 1994 Theory of phase-ordering kinetics Adv. Phys.43 357 · doi:10.1080/00018739400101505 |
| [113] | Landau L D and Lifshitz E M 1980 Statistical Physics(Course of Theoretical Physics vol 5) (Amsterdam: Elsevier) |
| [114] | Bär M, Zülicke C, Eiswirth M and Ertl G 1992 Theoretical modeling of spatiotemporal selforganization in a surface catalyzed reaction exhibiting bistable kinetics J. Chem. Phys.96 8595 · doi:10.1063/1.462312 |
| [115] | Gomila D, Colet P, Oppo G-L and Miguel M S 2004 Stable droplets and nucleation in asymmetric bistable nonlinear optic systems J. Opt. B: Quantum Semiclass. Opt.6 S265-70 · doi:10.1088/1464-4266/6/5/014 |
| [116] | Argentina M, Coullet P and Mahadevan L 1997 Colliding waves in the model excitable medium: preservation, annihilation and bifurcation Phys. Rev. Lett.79 2803 · doi:10.1103/PhysRevLett.79.2803 |
| [117] | Smith H 2011 An Introduction to Delay Differential Equations with Applications to the Life Sciences (New York: Springer) · Zbl 1227.34001 · doi:10.1007/978-1-4419-7646-8 |
| [118] | Vladimirov A G and Turaev D 2005 Model for passive mode locking in semiconductor lasers Phys. Rev. A 72 033808 · doi:10.1103/PhysRevA.72.033808 |
| [119] | Nizette M, Rachinskii D, Vladimirov A and Wolfrum M 2006 Pulse interaction via gain and loss dynamics in passive mode locking Physica D 218 95-104 · Zbl 1276.34075 · doi:10.1016/j.physd.2006.04.013 |
| [120] | Garbin B, Javaloyes J, Tissoni G and Barland S 2015 Topological solitons as addressable phase bits in a driven laser Nat. Commun.6 5915 · doi:10.1038/ncomms6915 |
| [121] | Marconi M, Javaloyes J, Barland S, Balle S and Giudici M 2015 Vectorial dissipative solitons in vertical-cavity surface-emitting lasers with delays Nat. Photon.9 450-5 · doi:10.1038/nphoton.2015.92 |
| [122] | Pyragas K 1992 Continuous control of chaos by self-controlling feedback Phys. Lett. A 170 421-8 · doi:10.1016/0375-9601(92)90745-8 |
| [123] | Hikihara T and Ueda Y 1999 An expansion of system with time delayed feedback control into spatio-temporal state space Chaos9 887-92 · Zbl 1056.93544 · doi:10.1063/1.166461 |
| [124] | Franz A L, Roy R, Shaw L B and Schwartz I B 2008 Effect of multiple time delays on intensity fluctuation dynamics in fiber ring lasers Phys. Rev. E 78 016208 · doi:10.1103/PhysRevE.78.016208 |
| [125] | Nicolis G and Prigogine I 1977 Self-Organization in Non Equilibrium Systems (New York: Wiley) · Zbl 0363.93005 |
| [126] | Marino F, Giacomelli G and Barland S 2014 Front pinning and localized state analogues in long-delayed bistable systems Phys. Rev. Lett.112 103901 · doi:10.1103/PhysRevLett.112.103901 |
| [127] | Porte X, Soriano M C and Fischer I 2014 Similarity properties in the dynamics of delayed-feedback semiconductor lasers Phys. Rev. A 89 023822 · doi:10.1103/PhysRevA.89.023822 |
| [128] | Appeltant L, Soriano M C, Van der Sande G, Danckaert J, Massar S, Dambre J, Schrauwen B, Mirasso C R and Fischer I 2011 Information processing using a single dynamical node as complex system Nat. Commun.2 468 · doi:10.1038/ncomms1476 |
| [129] | Larger L, Soriano M C, Brunner D, Appeltant L, Gutierrez J M, Pesquera L, Mirasso C R and Fischer I 2012 Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing Opt. Express20 3241-9 · doi:10.1364/OE.20.003241 |
| [130] | Martinenghi R, Rybalko S, Jacquot M, Chembo Y K and Larger L 2012 Photonic nonlinear transient computing with multiple-delay wavelength dynamics Phys. Rev. Lett.108 244101 · doi:10.1103/PhysRevLett.108.244101 |
| [131] | Kuramoto Y and Battogtokh D 2002 Coexistence of coherence and incoherence in nonlocally coupled phase oscillators Nonlinear Phenom. Complex Syst.5 380-5 |
| [132] | Larger L, Penkovsky B and Maistrenko Y 2013 Virtual chimera states for delayed-feedback systems Phys. Rev. Lett.111 054103 · doi:10.1103/PhysRevLett.111.054103 |
| [133] | Semenov V, Zakharova A, Maistrenko Y and Schöll E 2016 Delayed-feedback chimera states: Forced multiclusters and stochastic resonance Europhys. Lett.115 10005 · doi:10.1209/0295-5075/115/10005 |
| [134] | Larger L, Penkovsky B and Maistrenko Y 2015 Laser chimeras as a paradigm for multistable patterns in complex systems Nat. Commun.6 7752 · doi:10.1038/ncomms8752 |
| [135] | Turitsyna E G, Smirnov S V, Sugavanam S, Tarasov N, Shu X, Babin S A, Podivilov E V, Churkin D V, Falkovich G and Turitsyn S K 2013 The laminar-turbulent transition in a fibre laser Nat. Photon.7 783-6 · doi:10.1038/nphoton.2013.246 |
| [136] | Jang J K, Erkintalo M, Murdoch S G and Coen S 2013 Ultraweak long-range interactions of solitons observed over astronomical distances Nat. Photon.7 657-63 · doi:10.1038/nphoton.2013.157 |
| [137] | Lugiato L A and Lefever R 1987 Spatial dissipative structures in passive optical systems Phys. Rev. Lett.58 2209-11 · doi:10.1103/PhysRevLett.58.2209 |
| [138] | Marconi M, Javaloyes J, Balle S and Giudici M 2014 How lasing localized structures evolve out of passive mode locking Phys. Rev. Lett.112 223901 · doi:10.1103/PhysRevLett.112.223901 |
| [139] | Romeira B, Avó R, Figueiredo J M L, Barland S and Javaloyes J 2016 Regenerative memory in time-delayed neuromorphic photonic resonators Sci. Rep.6 19510 · doi:10.1038/srep19510 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
