1
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Medeiros ES, Feudel U. Local control for the collective dynamics of self-propelled particles. Phys Rev E 2024; 109:014312. [PMID: 38366537 DOI: 10.1103/physreve.109.014312] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2023] [Accepted: 01/02/2024] [Indexed: 02/18/2024]
Abstract
Utilizing a paradigmatic model for the motion of interacting self-propelled particles, we demonstrate that local accelerations at the level of individual particles can drive transitions between different collective dynamics, leading to a control process. We find that the ability to trigger such transitions is hierarchically distributed among the particles and can form distinctive spatial patterns within the collective. Chaotic dynamics occur during the transitions, which can be attributed to fractal basin boundaries mediating the control process. The particle hierarchies described in this paper offer decentralized capabilities for controlling artificial swarms.
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Affiliation(s)
- Everton S Medeiros
- Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, 26111 Oldenburg, Germany
| | - Ulrike Feudel
- Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, 26111 Oldenburg, Germany
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2
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Hastings A, Abbott KC, Cuddington K, Francis TB, Lai YC, Morozov A, Petrovskii S, Zeeman ML. Effects of stochasticity on the length and behaviour of ecological transients. J R Soc Interface 2021; 18:20210257. [PMID: 34229460 DOI: 10.1098/rsif.2021.0257] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Abstract
There is a growing recognition that ecological systems can spend extended periods of time far away from an asymptotic state, and that ecological understanding will therefore require a deeper appreciation for how long ecological transients arise. Recent work has defined classes of deterministic mechanisms that can lead to long transients. Given the ubiquity of stochasticity in ecological systems, a similar systematic treatment of transients that includes the influence of stochasticity is important. Stochasticity can of course promote the appearance of transient dynamics by preventing systems from settling permanently near their asymptotic state, but stochasticity also interacts with deterministic features to create qualitatively new dynamics. As such, stochasticity may shorten, extend or fundamentally change a system's transient dynamics. Here, we describe a general framework that is developing for understanding the range of possible outcomes when random processes impact the dynamics of ecological systems over realistic time scales. We emphasize that we can understand the ways in which stochasticity can either extend or reduce the lifetime of transients by studying the interactions between the stochastic and deterministic processes present, and we summarize both the current state of knowledge and avenues for future advances.
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Affiliation(s)
- Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA.,Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
| | - Karen C Abbott
- Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA
| | - Kim Cuddington
- Department of Biology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
| | - Tessa B Francis
- Puget Sound Institute, University of Washington Tacoma, Tacoma, WA 98421, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA
| | - Andrew Morozov
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK.,Institute of Ecology and Evolution, Russian Academy of Sciences, Leninsky pr. 33, Moscow 117071, Russia
| | - Sergei Petrovskii
- School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK.,Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., Moscow 117198, Russia
| | - Mary Lou Zeeman
- Department of Mathematics, Bowdoin College, Brunswick, ME 04011, USA
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3
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Meng Y, Lai YC, Grebogi C. Tipping point and noise-induced transients in ecological networks. J R Soc Interface 2020; 17:20200645. [PMID: 33050778 DOI: 10.1098/rsif.2020.0645] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
Abstract
A challenging and outstanding problem in interdisciplinary research is to understand the interplay between transients and stochasticity in high-dimensional dynamical systems. Focusing on the tipping-point dynamics in complex mutualistic networks in ecology constructed from empirical data, we investigate the phenomena of noise-induced collapse and noise-induced recovery. Two types of noise are studied: environmental (Gaussian white) noise and state-dependent demographic noise. The dynamical mechanism responsible for both phenomena is a transition from one stable steady state to another driven by stochastic forcing, mediated by an unstable steady state. Exploiting a generic and effective two-dimensional reduced model for real-world mutualistic networks, we find that the average transient lifetime scales algebraically with the noise amplitude, for both environmental and demographic noise. We develop a physical understanding of the scaling laws through an analysis of the mean first passage time from one steady state to another. The phenomena of noise-induced collapse and recovery and the associated scaling laws have implications for managing high-dimensional ecological systems.
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Affiliation(s)
- Yu Meng
- Institute for Complex Systems and Mathematical Biology, School of Natural and Computing Sciences, King's College, University of Aberdeen, Aberdeen AB24 3UE, UK
| | - Ying-Cheng Lai
- School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA.,Department of Physics, Arizona State University, Tempe, AZ 85287, USA
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, School of Natural and Computing Sciences, King's College, University of Aberdeen, Aberdeen AB24 3UE, UK
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4
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Hegedűs F, Krähling P, Aron M, Lauterborn W, Mettin R, Parlitz U. Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique. CHAOS (WOODBURY, N.Y.) 2020; 30:073123. [PMID: 32752633 DOI: 10.1063/5.0005424] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2020] [Accepted: 06/15/2020] [Indexed: 06/11/2023]
Abstract
A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.
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Affiliation(s)
- F Hegedűs
- Department of Hydrodynamic Systems, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Műegyetem rakpart 3, H-1111 Budapest, Hungary
| | - P Krähling
- Department of Hydrodynamic Systems, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Műegyetem rakpart 3, H-1111 Budapest, Hungary
| | - M Aron
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany and Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
| | - W Lauterborn
- Drittes Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund Platz 1, D-37077 Göttingen, Germany
| | - R Mettin
- Drittes Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund Platz 1, D-37077 Göttingen, Germany
| | - U Parlitz
- Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany and Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
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5
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Denis-le Coarer F, Quirce A, Valle A, Pesquera L, Rodríguez MA, Panajotov K, Sciamanna M. Attractor hopping between polarization dynamical states in a vertical-cavity surface-emitting laser subject to parallel optical injection. Phys Rev E 2018; 97:032201. [PMID: 29776124 DOI: 10.1103/physreve.97.032201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2017] [Indexed: 06/08/2023]
Abstract
We present experimental and theoretical results of noise-induced attractor hopping between dynamical states found in a single transverse mode vertical-cavity surface-emitting laser (VCSEL) subject to parallel optical injection. These transitions involve dynamical states with different polarizations of the light emitted by the VCSEL. We report an experimental map identifying, in the injected power-frequency detuning plane, regions where attractor hopping between two, or even three, different states occur. The transition between these behaviors is characterized by using residence time distributions. We find multistability regions that are characterized by heavy-tailed residence time distributions. These distributions are characterized by a -1.83±0.17 power law. Between these regions we find coherence enhancement of noise-induced attractor hopping in which transitions between states occur regularly. Simulation results show that frequency detuning variations and spontaneous emission noise play a role in causing switching between attractors. We also find attractor hopping between chaotic states with different polarization properties. In this case, simulation results show that spontaneous emission noise inherent to the VCSEL is enough to induce this hopping.
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Affiliation(s)
- Florian Denis-le Coarer
- Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas (CSIC), Universidad de Cantabria, 39005 Santander, Spain
- Chair in Photonics, LMOPS Laboratory, CentraleSupélec, Université de Paris-Saclay and Université de Lorraine, 57070 Metz, France
| | - Ana Quirce
- Vrije Universiteit Brussel, Faculty of Engineering Sciences, Brussels Photonics Team (B-PHOT), Pleinlaan 2, 1050 Brussels, Belgium
| | - Angel Valle
- Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas (CSIC), Universidad de Cantabria, 39005 Santander, Spain
| | - Luis Pesquera
- Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas (CSIC), Universidad de Cantabria, 39005 Santander, Spain
| | - Miguel A Rodríguez
- Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas (CSIC), Universidad de Cantabria, 39005 Santander, Spain
| | - Krassimir Panajotov
- Vrije Universiteit Brussel, Faculty of Engineering Sciences, Brussels Photonics Team (B-PHOT), Pleinlaan 2, 1050 Brussels, Belgium
- Institute of Solid State Physics, 72 Tzarigradsko, Chaussee Blvd., 1784 Sofia, Bulgaria
| | - Marc Sciamanna
- Chair in Photonics, LMOPS Laboratory, CentraleSupélec, Université de Paris-Saclay and Université de Lorraine, 57070 Metz, France
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6
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Sevilla-Escoboza R, Pisarchik AN, Jaimes-Reátegui R, Huerta-Cuellar G. Selective monostability in multi-stable systems. Proc Math Phys Eng Sci 2015. [DOI: 10.1098/rspa.2015.0005] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We propose a robust method that allows a periodic or a chaotic multi-stable system to be transformed to a monostable system at an orbit with dominant frequency of any of the coexisting attractors. Our approach implies the selection of a particular attractor by periodic external modulation with frequency close to the dominant frequency in the power spectrum of a desired orbit and simultaneous annihilation of all other coexisting states by positive feedback, both applied to one of the system parameters. The method does not require any preliminary knowledge of the system dynamics and the phase space structure. The efficiency of the method is demonstrated in both a non-autonomous multi-stable laser with coexisting periodic orbits and an autonomous Rössler-like oscillator with coexisting chaotic attractors. The experiments with an erbium-doped fibre laser provide evidence for the robustness of the proposed method in making the system monostable at an orbit with dominant frequency of any preselected attractor.
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Affiliation(s)
- R. Sevilla-Escoboza
- Centro Universitario de Los Lagos, Universidad de Guadalajara, Lagos de Moreno, Jalisco 47460, Mexico
| | - A. N. Pisarchik
- Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico
- Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo, 28223 Pozuelo de Alarcon, Madrid, Spain
| | - R. Jaimes-Reátegui
- Centro Universitario de Los Lagos, Universidad de Guadalajara, Lagos de Moreno, Jalisco 47460, Mexico
| | - G. Huerta-Cuellar
- Centro Universitario de Los Lagos, Universidad de Guadalajara, Lagos de Moreno, Jalisco 47460, Mexico
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7
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Rodrigues CS, de Moura APS, Grebogi C. Random fluctuation leads to forbidden escape of particles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:026211. [PMID: 20866897 DOI: 10.1103/physreve.82.026211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/2009] [Revised: 05/25/2010] [Indexed: 05/29/2023]
Abstract
A great number of physical processes are described within the context of Hamiltonian scattering. Previous studies have rather been focused on trajectories starting outside invariant structures, since the ones starting inside are expected to stay trapped there forever. This is true though only for the deterministic case. We show however that, under finitely small random fluctuations of the field, trajectories starting inside Kolmogorov-Arnold-Moser (KAM) islands escape within finite time. The nonhyperbolic dynamics gains then hyperbolic characteristics due to the effect of the random perturbed field. As a consequence, trajectories which are started inside KAM curves escape with hyperboliclike time decay distribution, and the fractal dimension of a set of particles that remain in the scattering region approaches that for hyperbolic systems. We show a universal quadratic power law relating the exponential decay to the amplitude of noise. We present a random walk model to relate this distribution to the amplitude of noise, and investigate these phenomena with a numerical study applying random maps.
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Affiliation(s)
- Christian S Rodrigues
- Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany.
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8
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Goswami BK, Euzzor S, Al Naimee K, Geltrude A, Meucci R, Arecchi FT. Control of stochastic multistable systems: experimental demonstration. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:016211. [PMID: 19658796 DOI: 10.1103/physreve.80.016211] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2009] [Revised: 05/27/2009] [Indexed: 05/28/2023]
Abstract
Stochastic disturbances and spikes (sudden sharp fluctuations of any system parameter), commonly observed among natural and laboratory-scale systems, can perturb the multistable dynamics significantly and become a serious impediment when the device is designed for a certain dynamical behavior. We experimentally demonstrate that suitable periodic modulation of any system parameter may efficiently control such stochastic multistability related problems. The control mechanism is verified individually with two standard models (namely, an analog circuit of Lorenz equations and a cavity-loss modulated CO2 laser), against three externally introduced disturbing signals, (namely, white Gaussian noise, pink noise, and train of spikes). Indeed, with both the systems, it has been observed that the modulation is capable to significantly control untoward jumps to coexisting attractors that otherwise would have occurred due to either of the disturbances. These results establish the robustness and wide applicability of this control mechanism in resolving stochastic multistability related problems.
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Affiliation(s)
- B K Goswami
- Laser and Plasma Technology Division, Bhabha Atomic Research Centre, Mumbai 400085, India
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9
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Huerta-Cuellar G, Pisarchik AN, Barmenkov YO. Experimental characterization of hopping dynamics in a multistable fiber laser. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:035202. [PMID: 18851094 DOI: 10.1103/physreve.78.035202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2008] [Indexed: 05/26/2023]
Abstract
We demonstrate experimental evidence of noise-induced attractor hopping in a multistable fiber laser. Multistate hopping dynamics displays complex statistical properties characterized by nontrivial scalings. When hopping is encountered between two states, the dynamics of the system is characterized by the -32 power law for the probability distribution of periodic windows versus their length, just as in the case of two-state on-off intermittency. A surprising noise saturation effect is found: average output noise in the hopping regime is almost independent of input noise. Such robustness of the system against external noise may be beneficial for some applications: for example, for communications with multistable systems or for designing noise-insensitive detectors.
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Affiliation(s)
- Guillermo Huerta-Cuellar
- Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico
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10
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Baptista MS, de Almeida LOB, Slaets JFW, Köberle R, Grebogi C. A complex biological system: the fly's visual module. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2008; 366:345-57. [PMID: 17673416 DOI: 10.1098/rsta.2007.2093] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
Is the characterization of biological systems as complex systems in the mathematical sense a fruitful assertion? In this paper we argue in the affirmative, although obviously we do not attempt to confront all the issues raised by this question. We use the fly's visual system as an example and analyse our experimental results of one particular neuron in the fly's visual system from this point of view. We find that the motion-sensitive 'H1' neuron, which converts incoming signals into a sequence of identical pulses or 'spikes', encodes the information contained in the stimulus into an alphabet composed of a few letters. This encoding occurs on multilayered sets, one of the features attributed to complex systems. The conversion of intervals between consecutive occurrences of spikes into an alphabet requires us to construct a generating partition. This entails a one-to-one correspondence between sequences of spike intervals and words written in the alphabet. The alphabet dynamics is multifractal both with and without stimulus, though the multifractality increases with the stimulus entropy. This is in sharp contrast to models generating independent spike intervals, such as models using Poisson statistics, whose dynamics is monofractal. We embed the support of the probability measure, which describes the distribution of words written in this alphabet, in a two-dimensional space, whose topology can be reproduced by an M-shaped map. This map has positive Lyapunov exponents, indicating a chaotic-like encoding.
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Affiliation(s)
- Murilo S Baptista
- Institut für Physik Am Neuen Palais 10, Universität Potsdam, 14469 Potsdam, Germany
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11
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Aguirre J, D'Ovidio F, Sanjuán MAF. Controlling chaotic transients: Yorke's game of survival. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:016203. [PMID: 14995689 DOI: 10.1103/physreve.69.016203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2003] [Indexed: 05/24/2023]
Abstract
We consider the tent map as the prototype of a chaotic system with escapes. We show analytically that a small, bounded, but carefully chosen perturbation added to the system can trap forever an orbit close to the chaotic saddle, even in presence of noise of larger, although bounded, amplitude. This problem is focused as a two-person, mathematical game between two players called "the protagonist" and "the adversary." The protagonist's goal is to survive. He can lose but cannot win; the best he can do is survive to play another round, struggling ad infinitum. In the absence of actions by either player, the dynamics diverge, leaving a relatively safe region, and we say the protagonist loses. What makes survival difficult is that the adversary is allowed stronger "actions" than the protagonist. What makes survival possible is (i) the background dynamics (the tent map here) are chaotic and (ii) the protagonist knows the action of the adversary in choosing his response and is permitted to choose the initial point x(0) of the game. We use the "slope 3" tent map in an example of this problem. We show that it is possible for the protagonist to survive.
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Affiliation(s)
- Jacobo Aguirre
- Nonlinear Dynamics and Chaos Group, Departamento de Matemáticas, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain.
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12
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Pisarchik AN, Barmenkov YO, Kir'yanov AV. Experimental demonstration of attractor annihilation in a multistable fiber laser. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:066211. [PMID: 14754301 DOI: 10.1103/physreve.68.066211] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2003] [Indexed: 05/24/2023]
Abstract
We report on the experimental open-loop control of generalized multistability in a system with coexisting attractors. The experimental system is an erbium-doped fiber laser with pump modulation of the diode laser. We demonstrate that additional weak harmonic modulation of the diode current annihilates one or two stable limit cycles in the laser. The ability of the method to select a desired state is illustrated through a codimension-two bifurcation diagram in the parameter space of the frequency and amplitude of the control modulation. We identify main resonances on the bifurcation lines (annihilation curves) and evaluate conditions for attractor annihilation.
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Affiliation(s)
- A N Pisarchik
- Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico
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13
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McGuire PC, Bohr H, Clark JW, Haschke R, Pershing CL, Rafelski J. Threshold disorder as a source of diverse and complex behavior in random nets. Neural Netw 2002; 15:1243-58. [PMID: 12425441 DOI: 10.1016/s0893-6080(02)00087-4] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Abstract
We study the diversity of complex spatio-temporal patterns in the behavior of random synchronous asymmetric neural networks (RSANNs). Special attention is given to the impact of disordered threshold values on limit-cycle diversity and limit-cycle complexity in RSANNs which have 'normal' thresholds by default. Surprisingly, RSANNs exhibit only a small repertoire of rather complex limit-cycle patterns when all parameters are fixed. This repertoire of complex patterns is also rather stable with respect to small parameter changes. These two unexpected results may generalize to the study of other complex systems. In order to reach beyond this seemingly disabling 'stable and small' aspect of the limit-cycle repertoire of RSANNs, we have found that if an RSANN has threshold disorder above a critical level, then there is a rapid increase of the size of the repertoire of patterns. The repertoire size initially follows a power-law function of the magnitude of the threshold disorder. As the disorder increases further, the limit-cycle patterns themselves become simpler until at a second critical level most of the limit cycles become simple fixed points. Nonetheless, for moderate changes in the threshold parameters, RSANNs are found to display specific features of behavior desired for rapidly responding processing systems: accessibility to a large set of complex patterns.
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14
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Goswami BK, Basu S. Transforming complex multistability to controlled monostability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026214. [PMID: 12241274 DOI: 10.1103/physreve.66.026214] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2001] [Revised: 05/21/2002] [Indexed: 05/23/2023]
Abstract
Multistability, a commonly observed feature among nonlinear systems, could be inconvenient under various circumstances. We demonstrate that a control in the form of slow and weak periodic parameter modulation can be effectively applied to transform a complex multistable system to a controlled monostable one. For the representative of a nonlinear system, we choose the Hénon map as the standard model. The number of coexisting stable states is known to increase as the dissipativity reduces. We show that even in the low dissipative limit, when the number of coexisting states could be arbitrarily large, the periodic parameter modulation can destroy the states coexisting with stable period 1. Thus, the system can be brought from any other branch to period-1 branch, leading to controlled monostability. This method works in the presence of noise as well.
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Affiliation(s)
- Binoy Krishna Goswami
- Laser and Plasma Technology Division, Bhabha Atomic Research Centre, Mumbai 400085, India.
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15
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Henson SM, Costantino RF, Desharnais RA, Cushing JM, Dennis B. Basins of attraction: population dynamics with two stable 4-cycles. OIKOS 2002. [DOI: 10.1034/j.1600-0706.2002.980102.x] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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16
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Kraut S, Feudel U. Multistability, noise, and attractor hopping: the crucial role of chaotic saddles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:015207. [PMID: 12241417 DOI: 10.1103/physreve.66.015207] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2001] [Revised: 05/03/2002] [Indexed: 05/23/2023]
Abstract
We investigate the hopping dynamics between different attractors in a multistable system under the influence of noise. Using symbolic dynamics we find a sudden increase of dynamical entropies, when a system parameter is varied. This effect is explained by a bifurcation involving two chaotic saddles. We also demonstrate that the transient lifetimes on the saddle obey a scaling law in analogy to crisis.
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Affiliation(s)
- Suso Kraut
- Institut für Physik, Universität Potsdam, Postfach 601553, D-14415 Potsdam, Germany
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17
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Bolotin YL, Gonchar VY, Krokhin AA, Hernández-Tejeda PH, Tur A, Yanovsky VV. Local and global control of high-period unstable orbits in reversible maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:026218. [PMID: 11497688 DOI: 10.1103/physreve.64.026218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2000] [Revised: 05/07/2001] [Indexed: 05/23/2023]
Abstract
We study the nonlinear dynamics of a complex system, described by a two-dimensional reversible map. The phase space of this map exhibits elements typical of Hamiltonian systems (stability islands) as well as of dissipative systems (attractor). Due to the interaction between the stability islands and the attractor, the transition to chaos in this system occurs through the collapse of the stability island and stochastization of the limiting-cycles orbits. We show how to apply the method of discrete parametric control to stabilize unstable high-period orbits. To achieve highly efficient control we introduce the concepts of local and global control. These concepts are useful in situations where there are "dangerous" points on the target orbit, i.e., the points where the probability of breakdown of control is high. As a result, the dangerous points turn out to be much more sensitive to external noise than other points on the orbit, and only the dangerous points determine how effective the control is.
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Affiliation(s)
- Y L Bolotin
- National Science Center, Kharkov Institute of Physics and Technology, Ukraine
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18
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Gong PL, Xu JX. Global dynamics and stochastic resonance of the forced FitzHugh-Nagumo neuron model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:031906. [PMID: 11308677 DOI: 10.1103/physreve.63.031906] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/1999] [Revised: 11/28/2000] [Indexed: 05/23/2023]
Abstract
We have analyzed the responses of an excitable FitzHugh-Nagumo neuron model to a weak periodic signal with and without noise. In contrast to previous studies which have dealt with stochastic resonance in the excitable model when the model with periodic input has only one stable attractor, we have focused our attention on the relationship between the global dynamics of the forced excitable neuron model and stochastic resonance. Our results show that for some parameters the forced FitzHugh-Nagumo neuron model has two attractors: the small-amplitude subthreshold periodic oscillation and the large-amplitude suprathreshold periodic oscillation. Random transitions between these two periodic oscillations are the essential mechanism underlying stochastic resonance in this model. Differences of such stochastic resonance to that in a classical bistable system and the excitable system are discussed. We also report that the state of the basin of attraction has a significant effect on the stability of neuronal firings, in the sense that the fractal basin boundary of the system enhances the noise-induced transitions.
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Affiliation(s)
- P L Gong
- School of Civil Engineering and Mechanics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
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Gadaleta S, Dangelmayr G. Learning to control a complex multistable system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:036217. [PMID: 11308751 DOI: 10.1103/physreve.63.036217] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2000] [Revised: 09/19/2000] [Indexed: 05/23/2023]
Abstract
In this paper the control of a periodically kicked mechanical rotor without gravity in the presence of noise is investigated. In recent work it was demonstrated that this system possesses many competing attracting states and thus shows the characteristics of a complex multistable system. We demonstrate that it is possible to stabilize the system at a desired attracting state even in the presence of high noise level. The control method is based on a recently developed algorithm [S. Gadaleta and G. Dangelmayr, Chaos 9, 775 (1999)] for the control of chaotic systems and applies reinforcement learning to find a global optimal control policy directing the system from any initial state towards the desired state in a minimum number of iterations. Being data-based, the method does not require any information about governing dynamical equations.
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Affiliation(s)
- S Gadaleta
- Department of Mathematics, Colorado State University, Weber Building, Fort Collins, Colorado 80523, USA.
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Kraut S, Feudel U, Grebogi C. Preference of attractors in noisy multistable systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 59:5253-60. [PMID: 11969483 DOI: 10.1103/physreve.59.5253] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/1998] [Indexed: 04/18/2023]
Abstract
A model system exhibiting a large number of attractors is investigated under the influence of noise. Several methods for discriminating two qualitatively different regions of the noise intensity are presented, and the phenomenon of noise-induced preference of attractors is reported. Finally, the relevance of our findings for detection of multiple stable states of systems occurring in nature or in the laboratory is pointed out.
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Affiliation(s)
- S Kraut
- Institut für Physik, Universität Potsdam, Postfach 601553, D-14415 Potsdam, Germany
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Feudel U, Grebogi C. Multistability and the control of complexity. CHAOS (WOODBURY, N.Y.) 1997; 7:597-604. [PMID: 12779685 DOI: 10.1063/1.166259] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We show how multistability arises in nonlinear dynamics and discuss the properties of such a behavior. In particular, we show that most attractors are periodic in multistable systems, meaning that chaotic attractors are rare in such systems. After arguing that multistable systems have the general traits expected from a complex system, we pass to control them. Our controlling complexity ideas allow for both the stabilization and destabilization of any one of the coexisting states. The control of complexity differs from the standard control of chaos approach, an approach that makes use of the unstable periodic orbits embedded in an extended chaotic attractor. (c) 1997 American Institute of Physics.
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Affiliation(s)
- Ulrike Feudel
- Institut fur Physik, Universitat Potsdam, PF 601553, D-14415 Potsdam, Germany
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Lai YC, Grebogi C. Complexity in Hamiltonian-driven dissipative chaotic dynamical systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4667-4675. [PMID: 9965644 DOI: 10.1103/physreve.54.4667] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Feudel U, Grebogi C, Hunt BR, Yorke JA. Map with more than 100 coexisting low-period periodic attractors. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:71-81. [PMID: 9965048 DOI: 10.1103/physreve.54.71] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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