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Schneider I, Dai JY. Symmetry groupoids for pattern-selective feedback stabilization of the Chafee-Infante equation. CHAOS (WOODBURY, N.Y.) 2023; 33:073141. [PMID: 37463091 DOI: 10.1063/5.0152662] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/31/2023] [Accepted: 06/27/2023] [Indexed: 07/20/2023]
Abstract
Reaction-diffusion equations are ubiquitous in various scientific domains and their patterns represent a fascinating area of investigation. However, many of these patterns are unstable and, therefore, challenging to observe. To overcome this limitation, we present new noninvasive feedback controls based on symmetry groupoids. As a concrete example, we employ these controls to selectively stabilize unstable equilibria of the Chafee-Infante equation under Dirichlet boundary conditions on the interval. Unlike conventional reflection-based control schemes, our approach incorporates additional symmetries that enable us to design new convolution controls for stabilization. By demonstrating the efficacy of our method, we provide a new tool for investigating and controlling systems with unstable patterns, with potential implications for a wide range of scientific disciplines.
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Affiliation(s)
- I Schneider
- Institut für Mathematik, Universität Rostock, Ulmenstr. 69, 18057 Rostock, Germany
| | - J-Y Dai
- Department of Applied Mathematics, National Chung Hsing University, 145 Xingda Rd., Taichung City, Taiwan
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2
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Mizukami S, Konishi K, Sugitani Y, Kouda T, Hara N. Effects of frequency mismatch on amplitude death in delay-coupled oscillators. Phys Rev E 2021; 104:054207. [PMID: 34942770 DOI: 10.1103/physreve.104.054207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2021] [Accepted: 10/18/2021] [Indexed: 11/07/2022]
Abstract
The present paper analytically reveals the effects of frequency mismatch on the stability of an equilibrium point within a pair of Stuart-Landau oscillators coupled by a delay connection. By analyzing the roots of the characteristic function governing the stability, we find that there exist four types of boundary curves of stability in a coupling parameters space. These four types depend only on the frequency mismatch. The analytical results allow us to design coupling parameters and frequency mismatch such that the equilibrium point is locally stable. We show that, if we choose appropriate frequency mismatches and delay times, then it is possible to induce amplitude death with strong stability, even by weak coupling. In addition, we show that parts of these analytical results are valid for oscillator networks with complete bipartite topologies.
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Affiliation(s)
- Shinsuke Mizukami
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Keiji Konishi
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Yoshiki Sugitani
- Department of Electrical and Electronic Systems Engineering, Ibaraki University, 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan
| | - Takahiro Kouda
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Naoyuki Hara
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
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3
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de Wolff B, Schneider I. Geometric invariance of determining and resonating centers: Odd- and any-number limitations of Pyragas control. CHAOS (WOODBURY, N.Y.) 2021; 31:063125. [PMID: 34241316 DOI: 10.1063/5.0050560] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 06/01/2021] [Indexed: 06/13/2023]
Abstract
In the spirit of the well-known odd-number limitation, we study the failure of Pyragas control of periodic orbits and equilibria. Addressing the periodic orbits first, we derive a fundamental observation on the invariance of the geometric multiplicity of the trivial Floquet multiplier. This observation leads to a clear and unifying understanding of the odd-number limitation, both in the autonomous and the non-autonomous setting. Since the presence of the trivial Floquet multiplier governs the possibility of successful stabilization, we refer to this multiplier as the determining center. The geometric invariance of the determining center also leads to a necessary condition on the gain matrix for the control to be successful. In particular, we exclude scalar gains. The application of Pyragas control on equilibria does not only imply a geometric invariance of the determining center but surprisingly also on centers that resonate with the time delay. Consequently, we formulate odd- and any-number limitations both for real eigenvalues together with an arbitrary time delay as well as for complex conjugated eigenvalue pairs together with a resonating time delay. The very general nature of our results allows for various applications.
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Affiliation(s)
- B de Wolff
- Freie Universität Berlin, Institut für Mathematik, Arnimallee 7, 14195 Berlin, Germany
| | - I Schneider
- Freie Universität Berlin, Institut für Mathematik, Arnimallee 7, 14195 Berlin, Germany
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4
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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] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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5
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Choe CU, Kim RS, Jang H, Hövel P, Schöll E. Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays. ACTA ACUST UNITED AC 2014. [DOI: 10.1007/s40435-013-0049-2] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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6
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Strehober DA, Schöll E, Klapp SHL. Feedback control of flow alignment in sheared liquid crystals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062509. [PMID: 24483470 DOI: 10.1103/physreve.88.062509] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2013] [Indexed: 06/03/2023]
Abstract
Based on a continuum theory, we investigate the manipulation of the nonequilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary "flow alignment," under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the ith component q(i)(t) of the order parameter tensor is supplemented by a control term involving the difference q(i)(t)-q(i)(t-τ). In this diagonal scheme, τ is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength K and τ; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of K are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic.
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Affiliation(s)
- David A Strehober
- Institut für Theoretische Physik, Sekretariat EW 7-1, Technische Universität Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Sekretariat EW 7-1, Technische Universität Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
| | - Sabine H L Klapp
- Institut für Theoretische Physik, Sekretariat EW 7-1, Technische Universität Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
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7
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Kantner M, Yanchuk S. Bifurcation analysis of delay-induced patterns in a ring of Hodgkin-Huxley neurons. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20120470. [PMID: 23960228 DOI: 10.1098/rsta.2012.0470] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Rings of delay-coupled neurons possess a striking capability to produce various stable spiking patterns. In order to reveal the mechanisms of their appearance, we present a bifurcation analysis of the Hodgkin-Huxley (HH) system with delayed feedback as well as a closed loop of HH neurons. We consider mainly the effects of external currents and communication delays. It is shown that typically periodic patterns of different spatial form (wavenumber) appear via Hopf bifurcations as the external current or time delay changes. The Hopf bifurcations are shown to occur in relatively narrow regions of the external current values, which are independent of the delays. Additional patterns, which have the same wavenumbers as the existing ones, appear via saddle-node bifurcations of limit cycles. The obtained bifurcation diagrams are evidence for the important role of communication delays for the emergence of multiple coexistent spiking patterns. The effects of a short-cut, which destroys the rotational symmetry of the ring, are also briefly discussed.
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Affiliation(s)
- Markus Kantner
- Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany
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8
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Gjurchinovski A, Jüngling T, Urumov V, Schöll E. Delayed feedback control of unstable steady states with high-frequency modulation of the delay. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032912. [PMID: 24125330 DOI: 10.1103/physreve.88.032912] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2013] [Indexed: 06/02/2023]
Abstract
We analyze the stabilization of unstable steady states by delayed feedback control with a periodic time-varying delay in the regime of a high-frequency modulation of the delay. The average effect of the delayed feedback term in the control force is equivalent to a distributed delay in the interval of the modulation, and the obtained distribution depends on the type of the modulation. In our analysis we use a simple generic normal form of an unstable focus, and investigate the effects of phase-dependent coupling and the influence of the control loop latency on the controllability. In addition, we have explored the influence of the modulation of the delays in multiple delay feedback schemes consisting of two independent delay lines of Pyragas type. A main advantage of the variable delay is the considerably larger domain of stabilization in parameter space.
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Affiliation(s)
- Aleksandar Gjurchinovski
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Saints Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Macedonia
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9
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Zou W, Senthilkumar DV, Tang Y, Wu Y, Lu J, Kurths J. Amplitude death in nonlinear oscillators with mixed time-delayed coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032916. [PMID: 24125334 DOI: 10.1103/physreve.88.032916] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2013] [Indexed: 06/02/2023]
Abstract
Amplitude death (AD) is an emergent phenomenon whereby two or more autonomously oscillating systems completely lose their oscillations due to coupling. In this work, we study AD in nonlinear oscillators with mixed time-delayed coupling, which is a combination of instantaneous and time-delayed couplings. We find that the mixed time-delayed coupling favors the onset of AD for a larger set of parameters than in the limiting cases of purely instantaneous or completely time-delayed coupling. Coupled identical oscillators experience AD under instantaneous coupling mixed with a small proportion of time-delayed coupling. Our work gives a deeper understanding of delay-induced AD in coupled nonlinear oscillators.
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Affiliation(s)
- Wei Zou
- School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China and Institute of Physics, Humboldt University Berlin, Berlin D-12489, Germany and Potsdam Institute for Climate Impact Research, Telegraphenberg, Potsdam D-14415, Germany
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10
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Zou W, Lu J, Tang Y, Zhang C, Kurths J. Control of delay-induced oscillation death by coupling phase in coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:066208. [PMID: 22304179 DOI: 10.1103/physreve.84.066208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2011] [Revised: 10/10/2011] [Indexed: 05/31/2023]
Abstract
A coupling phase is deemed to be crucial in stabilizing behavior in nonlinear systems. In this paper, we study how the coupling phase influences the delay-induced oscillation death (OD) in coupled oscillators. The OD boundaries are identified analytically even in the presence of the coupling phase. We find that OD only occurs for a coupling phase belonging to a certain interval. The optimal coupling phase, under which the largest OD island forms, is characterized well by a power law scaling with respect to the frequency. The coupling phase turns out to be a key parameter that determines a delay-induced OD. Furthermore, the controlling role of the coupling phase generally is proved to hold fairly for networked delay-coupled oscillators.
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Affiliation(s)
- Wei Zou
- School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
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11
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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 (WOODBURY, N.Y.) 2011; 21:043111. [PMID: 22225348 DOI: 10.1063/1.3647320] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [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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12
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Zou W, Zheng X, Zhan M. Insensitive dependence of delay-induced oscillation death on complex networks. CHAOS (WOODBURY, N.Y.) 2011; 21:023130. [PMID: 21721772 DOI: 10.1063/1.3602226] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Oscillation death (also called amplitude death), a phenomenon of coupling induced stabilization of an unstable equilibrium, is studied for an arbitrary symmetric complex network with delay-coupled oscillators, and the critical conditions for its linear stability are explicitly obtained. All cases including one oscillator, a pair of oscillators, regular oscillator networks, and complex oscillator networks with delay feedback coupling, can be treated in a unified form. For an arbitrary symmetric network, we find that the corresponding smallest eigenvalue of the Laplacian λ(N) (0 >λ(N) ≥ -1) completely determines the death island, and as λ(N) is located within the insensitive parameter region for nearly all complex networks, the death island keeps nearly the largest and does not sensitively depend on the complex network structures. This insensitivity effect has been tested for many typical complex networks including Watts-Strogatz (WS) and Newman-Watts (NW) small world networks, general scale-free (SF) networks, Erdos-Renyi (ER) random networks, geographical networks, and networks with community structures and is expected to be helpful for our understanding of dynamics on complex networks.
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Affiliation(s)
- Wei Zou
- School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
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13
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Hinz RC, Hövel P, Schöll E. Transient behavior in systems with time-delayed feedback. CHAOS (WOODBURY, N.Y.) 2011; 21:023114. [PMID: 21721756 DOI: 10.1063/1.3581161] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We investigate the transient times for the onset of control of steady states by time-delayed feedback. The optimization of control by minimizing the transient time before control becomes effective is discussed analytically and numerically, and the competing influences of local and global features are elaborated. We derive an algebraic scaling of the transient time and confirm our findings by numerical simulations in dependence on feedback gain and time delay.
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Affiliation(s)
- Robert C Hinz
- Institut für Theoretische Physik, TU Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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14
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D'Huys O, Fischer I, Danckaert J, Vicente R. Role of delay for the symmetry in the dynamics of networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:046223. [PMID: 21599286 DOI: 10.1103/physreve.83.046223] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2010] [Revised: 12/21/2010] [Indexed: 05/30/2023]
Abstract
The symmetry in a network of oscillators determines the spatiotemporal patterns of activity that can emerge. We study how a delay in the coupling affects symmetry-breaking and -restoring bifurcations. We are able to draw general conclusions in the limit of long delays. For one class of networks we derive a criterion that predicts that delays have a symmetrizing effect. Moreover, we demonstrate that for any network admitting a steady-state solution, a long delay can solely advance the first bifurcation point as compared to the instantaneous-coupling regime.
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Affiliation(s)
- O D'Huys
- Applied Physics Research Group (APHY), Vrije Universiteit Brussel, 1050 Brussel, Belgium
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15
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Flunkert V, Yanchuk S, Dahms T, Schöll E. Synchronizing distant nodes: a universal classification of networks. PHYSICAL REVIEW LETTERS 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] [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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16
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D'Huys O, Vicente R, Danckaert J, Fischer I. Amplitude and phase effects on the synchronization of delay-coupled oscillators. CHAOS (WOODBURY, N.Y.) 2010; 20:043127. [PMID: 21198097 DOI: 10.1063/1.3518363] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
We consider the behavior of Stuart-Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding self-feedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behavior.
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Affiliation(s)
- O D'Huys
- Department of Physics (DNTK), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium.
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17
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Perlikowski P, Yanchuk S, Popovych OV, Tass PA. Periodic patterns in a ring of delay-coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:036208. [PMID: 21230162 DOI: 10.1103/physreve.82.036208] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2010] [Revised: 08/24/2010] [Indexed: 05/30/2023]
Abstract
We describe the appearance and stability of spatiotemporal periodic patterns (rotating waves) in unidirectional rings of coupled oscillators with delayed couplings. We show how delays in the coupling lead to the splitting of each rotating wave into several new ones. The appearance of rotating waves is mediated by the Hopf bifurcations of the symmetric equilibrium. We also conclude that the coupling delays can be effectively replaced by increasing the number of oscillators in the chain. The phenomena are shown for the Stuart-Landau oscillators as well as for the coupled FitzHugh-Nagumo systems modeling an ensemble of spiking neurons interacting via excitatory chemical synapses.
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Affiliation(s)
- P Perlikowski
- Institute of Mathematics, Humboldt University of Berlin, 10099 Berlin, Germany
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18
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Konishi K, Kokame H, Hara N. Stabilization of a steady state in network oscillators by using diffusive connections with two long time delays. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:016201. [PMID: 20365440 DOI: 10.1103/physreve.81.016201] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2009] [Indexed: 05/29/2023]
Abstract
The present study shows that diffusive connections with two long-time delays can induce the stabilization of a steady state in network oscillators. A linear stability analysis shows that, if the two delay times retain a proportional relation with a certain bias, the stabilization can be achieved independent of the delay times. Furthermore, a simple systematic procedure for designing the coupling strength and the delay times in the connections is proposed. The procedure has the following two advantages: one can employ time delays as long as one wants and the stabilization can be achieved independently of its network topology. Our analytical results are applied to the well-known double-scroll circuit model on a small-world network.
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Affiliation(s)
- Keiji Konishi
- Department of Electrical and Information Systems, Osaka Prefecture University, Sakai, Osaka, Japan
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19
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Gjurchinovski A, Urumov V. Variable-delay feedback control of unstable steady states in retarded time-delayed systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:016209. [PMID: 20365448 DOI: 10.1103/physreve.81.016209] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2009] [Revised: 11/18/2009] [Indexed: 05/29/2023]
Abstract
We study the stability of unstable steady states in scalar retarded time-delayed systems subjected to a variable-delay feedback control. The important aspect of such a control problem is that time-delayed systems are already infinite-dimensional before the delayed feedback control is turned on. When the frequency of the modulation is large compared to the system's dynamics, the analytic approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed-delay system. Otherwise, the stability domains are obtained by a numerical integration of the linearized variable-delay system. The analysis shows that the control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form and the frequency of the delay modulation.
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Affiliation(s)
- A Gjurchinovski
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, Skopje, Macedonia.
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20
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Yanchuk S, Perlikowski P. Delay and periodicity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:046221. [PMID: 19518326 DOI: 10.1103/physreve.79.046221] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2009] [Indexed: 05/27/2023]
Abstract
Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be split into two parts: pseudocontinuous and strongly unstable. The pseudocontinuous part of the spectrum mediates destabilization of periodic solutions.
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Affiliation(s)
- S Yanchuk
- Institute of Mathematics, Humboldt University of Berlin, 10099 Berlin, Germany
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21
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Time-Delayed Feedback Control: From Simple Models to Lasers and Neural Systems. UNDERSTANDING COMPLEX SYSTEMS 2009. [DOI: 10.1007/978-3-642-02329-3_4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
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22
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Guo S, Feng G, Liao X, Liu Q. Hopf bifurcation control in a congestion control model via dynamic delayed feedback. CHAOS (WOODBURY, N.Y.) 2008; 18:043104. [PMID: 19123614 DOI: 10.1063/1.2998220] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
A typical objective of bifurcation control is to delay the onset of undesirable bifurcation. In this paper, the problem of Hopf bifurcation control in a second-order congestion control model is considered. In particular, a suitable Hopf bifurcation is created at a desired location with preferred properties and a dynamic delayed feedback controller is developed for the creation of the Hopf bifurcation. With this controller, one can increase the critical value of the communication delay, and thus guarantee a stationary data sending rate for larger delay. Furthermore, explicit formulae to determine the period and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying perturbation approach. Finally, numerical simulation results are presented to show that the dynamic delayed feedback controller is efficient in controlling Hopf bifurcation.
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Affiliation(s)
- Songtao Guo
- College of Computer Science, Chongqing University, Chongqing 400044, People's Republic of China
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23
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Dahms T, Hövel P, Schöll E. Stabilizing continuous-wave output in semiconductor lasers by time-delayed feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:056213. [PMID: 19113206 DOI: 10.1103/physreve.78.056213] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2008] [Indexed: 05/27/2023]
Abstract
The stabilization of steady states is studied in a modified Lang-Kobayashi model of a semiconductor laser. We show that multiple time-delayed feedback, realized by a Fabry-Perot resonator coupled to the laser, provides a valuable tool for the suppression of unwanted intensity pulsations, and leads to stable continuous-wave operation. The domains of control are calculated in dependence on the feedback strength, delay time (cavity round trip time), memory parameter (mirror reflectivity), latency time, feedback phase, and bandpass filtering. Due to the optical feedback, multistable behavior can also occur in the form of delay-induced intensity pulsations or other modes for certain choices of the control parameters. Control may then still be achieved by slowly ramping the injection current during turn-on.
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Affiliation(s)
- Thomas Dahms
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
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Dahms T, Hövel P, Schöll E. Control of unstable steady states by extended time-delayed feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:056201. [PMID: 18233733 DOI: 10.1103/physreve.76.056201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2007] [Indexed: 05/25/2023]
Abstract
Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar [Phys. Rev. E 50, 3245 (1994)] to an unstable focus. This system represents a generic model of an unstable steady state which can be found, for instance, in Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.
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Affiliation(s)
- Thomas Dahms
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 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. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 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] [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. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 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] [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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