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Hancock F, Rosas FE, Luppi AI, Zhang M, Mediano PAM, Cabral J, Deco G, Kringelbach ML, Breakspear M, Kelso JAS, Turkheimer FE. Metastability demystified - the foundational past, the pragmatic present and the promising future. Nat Rev Neurosci 2024:10.1038/s41583-024-00883-1. [PMID: 39663408 DOI: 10.1038/s41583-024-00883-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 11/01/2024] [Indexed: 12/13/2024]
Abstract
Healthy brain function depends on balancing stable integration between brain areas for effective coordinated functioning, with coexisting segregation that allows subsystems to express their functional specialization. Metastability, a concept from the dynamical systems literature, has been proposed as a key signature that characterizes this balance. Building on this principle, the neuroscience literature has leveraged the phenomenon of metastability to investigate various aspects of brain function in health and disease. However, this body of work often uses the notion of metastability heuristically, and sometimes inaccurately, making it difficult to navigate the vast literature, interpret findings and foster further development of theoretical and experimental methodologies. Here, we provide a comprehensive review of metastability and its applications in neuroscience, covering its scientific and historical foundations and the practical measures used to assess it in empirical data. We also provide a critical analysis of recent theoretical developments, clarifying common misconceptions and paving the road for future developments.
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Affiliation(s)
- Fran Hancock
- Department of Neuroimaging, Institute of Psychiatry, Psychology and Neuroscience, King's College London, London, UK.
| | - Fernando E Rosas
- Department of Informatics, University of Sussex, Brighton, UK.
- Sussex Centre for Consciousness Science, University of Sussex, Brighton, UK.
- Centre for Psychedelic Research, Department of Brain Science, Imperial College London, London, UK.
- Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, UK.
- Sussex AI, University of Sussex, Brighton, UK.
- Centre for Complexity Science, Department of Brain Science, Imperial College London, London, UK.
| | - Andrea I Luppi
- Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, UK
- St John's College, University of Cambridge, Cambridge, UK
- Department of Psychiatry, University of Oxford, Oxford, UK
| | - Mengsen Zhang
- Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA
| | - Pedro A M Mediano
- Department of Computing, Imperial College London, London, UK
- Division of Psychology and Language Sciences, University College London, London, UK
| | - Joana Cabral
- Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, UK
- Life and Health Sciences Research Institute School of Medicine, University of Minho, Braga, Portugal
| | - Gustavo Deco
- Computational Neuroscience Group, Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- Institución Catalana de la Recerca i Estudis Avancats (ICREA), Barcelona, Spain
- Department of Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
- School of Psychological Sciences, Monash University Clayton, Melbourne, Victoria, Australia
| | - Morten L Kringelbach
- Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, UK
- Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark
| | - Michael Breakspear
- School of Psychological Sciences, College of Engineering, Science and the Environment, University of Newcastle, Newcastle, New South Wales, Australia
| | - J A Scott Kelso
- Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, FL, USA
- Intelligent Systems Research Centre, Ulster University, Derry~Londonderry, Northern Ireland
- The Bath Institute for the Augmented Human, University of Bath, Bath, UK
| | - Federico E Turkheimer
- Department of Neuroimaging, Institute of Psychiatry, Psychology and Neuroscience, King's College London, London, UK
- The Institute for Human and Synthetic Minds, King's College London, London, UK
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2
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Meng Y, Lai YC, Grebogi C. The fundamental benefits of multiplexity in ecological networks. J R Soc Interface 2022; 19:20220438. [PMID: 36167085 PMCID: PMC9514891 DOI: 10.1098/rsif.2022.0438] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2022] [Accepted: 09/01/2022] [Indexed: 11/12/2022] Open
Abstract
A tipping point presents perhaps the single most significant threat to an ecological system as it can lead to abrupt species extinction on a massive scale. Climate changes leading to the species decay parameter drifts can drive various ecological systems towards a tipping point. We investigate the tipping-point dynamics in multi-layer ecological networks supported by mutualism. We unveil a natural mechanism by which the occurrence of tipping points can be delayed by multiplexity that broadly describes the diversity of the species abundances, the complexity of the interspecific relationships, and the topology of linkages in ecological networks. For a double-layer system of pollinators and plants, coupling between the network layers occurs when there is dispersal of pollinator species. Multiplexity emerges as the dispersing species establish their presence in the destination layer and have a simultaneous presence in both. We demonstrate that the new mutualistic links induced by the dispersing species with the residence species have fundamental benefits to the well-being of the ecosystem in delaying the tipping point and facilitating species recovery. Articulating and implementing control mechanisms to induce multiplexity can thus help sustain certain types of ecosystems that are in danger of extinction as the result of environmental changes.
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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, AB24 3UE, UK
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, Dresden 01187, Germany
- Center for Systems Biology Dresden, Pfotenhauerstraße 108, Dresden 01307, Germany
| | - 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, AB24 3UE, UK
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3
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Taylor JD, Chauhan AS, Taylor JT, Shilnikov AL, Nogaret A. Noise-activated barrier crossing in multiattractor dissipative neural networks. Phys Rev E 2022; 105:064203. [PMID: 35854623 DOI: 10.1103/physreve.105.064203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2021] [Accepted: 05/17/2022] [Indexed: 06/15/2023]
Abstract
Noise-activated transitions between coexisting attractors are investigated in a chaotic spiking network. At low noise level, attractor hopping consists of discrete bifurcation events that conserve the memory of initial conditions. When the escape probability becomes comparable to the intrabasin hopping probability, the lifetime of attractors is given by a detailed balance where the less coherent attractors act as a sink for the more coherent ones. In this regime, the escape probability follows an activation law allowing us to assign pseudoactivation energies to limit cycle attractors. These pseudoenergies introduce a useful metric for evaluating the resilience of biological rhythms to perturbations.
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Affiliation(s)
- Joseph D Taylor
- Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom
| | - Ashok S Chauhan
- Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom
| | - John T Taylor
- Department of Electronics and Electrical Engineering, University of Bath, Bath BA2 7AY, United Kingdom
| | - Andrey L Shilnikov
- Neuroscience Institute, Georgia State University, Petit Science Center, 100 Piedmont Avenue Atlanta, Georgia 30303, USA
- Department of Mathematics and Statistics, Georgia State University, Petit Science Center, 100 Piedmont Avenue, Atlanta, Georgia 30303, USA
| | - Alain Nogaret
- Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom
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4
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Velocity Multistability vs. Ergodicity Breaking in a Biased Periodic Potential. ENTROPY 2022; 24:e24010098. [PMID: 35052124 PMCID: PMC8774412 DOI: 10.3390/e24010098] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/16/2021] [Revised: 01/04/2022] [Accepted: 01/05/2022] [Indexed: 11/17/2022]
Abstract
Multistability, i.e., the coexistence of several attractors for a given set of system parameters, is one of the most important phenomena occurring in dynamical systems. We consider it in the velocity dynamics of a Brownian particle, driven by thermal fluctuations and moving in a biased periodic potential. In doing so, we focus on the impact of ergodicity-A concept which lies at the core of statistical mechanics. The latter implies that a single trajectory of the system is representative for the whole ensemble and, as a consequence, the initial conditions of the dynamics are fully forgotten. The ergodicity of the deterministic counterpart is strongly broken, and we discuss how the velocity multistability depends on the starting position and velocity of the particle. While for non-zero temperatures the ergodicity is, in principle, restored, in the low temperature regime the velocity dynamics is still affected by initial conditions due to weak ergodicity breaking. For moderate and high temperatures, the multistability is robust with respect to the choice of the starting position and velocity of the particle.
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5
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Boaretto BRR, Budzinski RC, Rossi KL, Manchein C, Prado TL, Feudel U, Lopes SR. Bistability in the synchronization of identical neurons. Phys Rev E 2021; 104:024204. [PMID: 34525513 DOI: 10.1103/physreve.104.024204] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2021] [Accepted: 07/19/2021] [Indexed: 11/07/2022]
Abstract
We investigate the role of bistability in the synchronization of a network of identical bursting neurons coupled through an generic electrical mean-field scheme. These neurons can exhibit distinct multistable states and, in particular, bistable behavior is observed when their sodium conductance is varied. With this, we consider three different initialization compositions: (i) the whole network is in the same periodic state; (ii) half of the network periodic, half chaotic; (iii) half periodic, and half in a different periodic state. We show that (i) and (ii) reach phase synchronization (PS) for all coupling strengths, while for (iii) small coupling regimes do not induce PS, and instead, there is a coexistence of different frequencies. For stronger coupling, case (iii) synchronizes, but after (i) and (ii). Since PS requires all neurons being in the same state (same frequencies), these different behaviors are governed by transitions between the states. We find that, during these transitions, (ii) and (iii) have transient chimera states and that (iii) has breathing chimeras. By studying the stability of each state, we explain the observed transitions. Therefore, bistability of neurons can play a major role in the synchronization of generic networks, with the simple initialization of the system being capable of drastically changing its asymptotic space.
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Affiliation(s)
- B R R Boaretto
- Department of Physics, Universidade Federal do Paraná, 81531-980, Curitiba, PR, Brazil
| | - R C Budzinski
- Department of Physics, Universidade Federal do Paraná, 81531-980, Curitiba, PR, Brazil
| | - K L Rossi
- Department of Physics, Universidade Federal do Paraná, 81531-980, Curitiba, PR, Brazil
| | - C Manchein
- Department of Physics, Universidade do Estado de Santa Catarina, 89219-710 Joinville, SC, Brazil
| | - T L Prado
- Department of Physics, Universidade Federal do Paraná, 81531-980, Curitiba, PR, Brazil
| | - U Feudel
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University Oldenburg, 26111 Oldenburg, Germany
| | - S R Lopes
- Department of Physics, Universidade Federal do Paraná, 81531-980, Curitiba, PR, Brazil
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6
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Spiechowicz J, Łuczka J. Arcsine law and multistable Brownian dynamics in a tilted periodic potential. Phys Rev E 2021; 104:024132. [PMID: 34525677 DOI: 10.1103/physreve.104.024132] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2021] [Accepted: 08/10/2021] [Indexed: 06/13/2023]
Abstract
Multistability is one of the most important phenomena in dynamical systems, e.g., bistability enables the implementation of logic gates and therefore computation. Recently multistability has attracted a greatly renewed interest related to memristors and graphene structures, to name only a few. We investigate tristability in velocity dynamics of a Brownian particle subjected to a tilted periodic potential. It is demonstrated that the origin of this effect is attributed to the arcsine law for the velocity dynamics at the zero temperature limit. We analyze the impact of thermal fluctuations and construct the phase diagram for the stability of the velocity dynamics. It suggests an efficient strategy to control the multistability by changing solely the force acting on the particle or temperature of the system. Our findings for the paradigmatic model of nonequilibrium statistical physics apply to, inter alia, Brownian motors, Josephson junctions, cold atoms dwelling in optical lattices, and colloidal systems.
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Affiliation(s)
- J Spiechowicz
- Institute of Physics, University of Silesia, 41-500 Chorzów, Poland
| | - J Łuczka
- Institute of Physics, University of Silesia, 41-500 Chorzów, Poland
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7
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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: 16] [Impact Index Per Article: 5.3] [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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8
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Schoenmakers S, Feudel U. A resilience concept based on system functioning: A dynamical systems perspective. CHAOS (WOODBURY, N.Y.) 2021; 31:053126. [PMID: 34240958 DOI: 10.1063/5.0042755] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2021] [Accepted: 04/21/2021] [Indexed: 06/13/2023]
Abstract
We introduce a new framework for resilience, which is traditionally understood as the ability of a system to absorb disturbances and maintain its state, by proposing a shift from a state-based to a system functioning-based approach to resilience, which takes into account that several different coexisting stable states could fulfill the same functioning. As a consequence, not every regime shift, i.e., transition from one stable state to another, is associated with a lack or loss of resilience. We emphasize the importance of flexibility-the ability of a system to shift between different stable states while still maintaining system functioning. Furthermore, we provide a classification of system responses based on the phenomenological properties of possible disturbances, including the role of their timescales. Therefore, we discern fluctuations, shocks, press disturbances, and trends as possible disturbances. We distinguish between two types of mechanisms of resilience: (i) tolerance and flexibility, which are properties of the system, and (ii) adaptation and transformation, which are processes that alter the system's tolerance and flexibility. Furthermore, we discuss quantitative methods to investigate resilience in model systems based on approaches developed in dynamical systems theory.
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Affiliation(s)
- Sarah Schoenmakers
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
| | - Ulrike Feudel
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
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9
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Meng Y, Grebogi C. Control of tipping points in stochastic mutualistic complex networks. CHAOS (WOODBURY, N.Y.) 2021; 31:023118. [PMID: 33653048 DOI: 10.1063/5.0036051] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2020] [Accepted: 01/26/2021] [Indexed: 06/12/2023]
Abstract
Nonlinear stochastic complex networks in ecological systems can exhibit tipping points. They can signify extinction from a survival state and, conversely, a recovery transition from extinction to survival. We investigate a control method that delays the extinction and advances the recovery by controlling the decay rate of pollinators of diverse rankings in a pollinators-plants stochastic mutualistic complex network. Our investigation is grounded on empirical networks occurring in natural habitats. We also address how the control method is affected by both environmental and demographic noises. By comparing the empirical network with the random and scale-free networks, we also study the influence of the topological structure on the control effect. Finally, we carry out a theoretical analysis using a reduced dimensional model. A remarkable result of this work is that the introduction of pollinator species in the habitat, which is immune to environmental deterioration and that is in mutualistic relationship with the collapsed ones, definitely helps in promoting the recovery. This has implications for managing ecological systems.
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Affiliation(s)
- Yu Meng
- Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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10
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Fan H, Kong LW, Wang X, Hastings A, Lai YC. Synchronization within synchronization: transients and intermittency in ecological networks. Natl Sci Rev 2020; 8:nwaa269. [PMID: 34858600 PMCID: PMC8566182 DOI: 10.1093/nsr/nwaa269] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2020] [Revised: 09/28/2020] [Accepted: 09/28/2020] [Indexed: 11/13/2022] Open
Abstract
Transients are fundamental to ecological systems with significant implications to management, conservation and biological control. We uncover a type of transient synchronization behavior in spatial ecological networks whose local dynamics are of the chaotic, predator–prey type. In the parameter regime where there is phase synchronization among all the patches, complete synchronization (i.e. synchronization in both phase and amplitude) can arise in certain pairs of patches as determined by the network symmetry—henceforth the phenomenon of ‘synchronization within synchronization.’ Distinct patterns of complete synchronization coexist but, due to intrinsic instability or noise, each pattern is a transient and there is random, intermittent switching among the patterns in the course of time evolution. The probability distribution of the transient time is found to follow an algebraic scaling law with a divergent average transient lifetime. Based on symmetry considerations, we develop a stability analysis to understand these phenomena. The general principle of symmetry can also be exploited to explain previously discovered, counterintuitive synchronization behaviors in ecological networks.
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Affiliation(s)
- Huawei Fan
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Ling-Wei Kong
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA
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11
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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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12
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Meng Y, Jiang J, Grebogi C, Lai YC. Noise-enabled species recovery in the aftermath of a tipping point. Phys Rev E 2020; 101:012206. [PMID: 32069632 DOI: 10.1103/physreve.101.012206] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2019] [Indexed: 11/07/2022]
Abstract
The beneficial role of noise in promoting species coexistence and preventing extinction has been recognized in theoretical ecology, but previous studies were mostly concerned with low-dimensional systems. We investigate the interplay between noise and nonlinear dynamics in real-world complex mutualistic networks with a focus on species recovery in the aftermath of a tipping point. Particularly, as a critical parameter such as the mutualistic interaction strength passes through a tipping point, the system collapses and approaches an extinction state through a dramatic reduction in the species populations to near-zero values. We demonstrate the striking effect of noise: when the direction of parameter change is reversed through the tipping point, noise enables species recovery which otherwise would not be possible. We uncover an algebraic scaling law between the noise amplitude and the parameter distance from the tipping point to the recovery point and provide a physical understanding through analyzing the nonlinear dynamics based on an effective, reduced-dimension model. Noise, in the form of small population fluctuations, can thus play a positive role in protecting high-dimensional, complex ecological networks.
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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, United Kingdom.,School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Junjie Jiang
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 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, United Kingdom
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA.,Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
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13
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Ray A, Rakshit S, Ghosh D, Dana SK. Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events. CHAOS (WOODBURY, N.Y.) 2019; 29:043131. [PMID: 31042945 DOI: 10.1063/1.5092741] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2019] [Accepted: 04/08/2019] [Indexed: 06/09/2023]
Abstract
We notice signatures of extreme eventslike behavior in a laser based Ikeda map. The trajectory of the system occasionally travels a large distance away from the bounded chaotic region, which appears as intermittent spiking events in the temporal dynamics. The large spiking events satisfy the conditions of extreme events as usually observed in dynamical systems. The probability density function of the large spiking events shows a long-tail distribution consistent with the characteristics of rare events. The interevent intervals obey a Poissonlike distribution. We locate the parameter regions of extreme events in phase diagrams. Furthermore, we study two Ikeda maps to explore how and when extreme events terminate via mutual interaction. A pure diffusion of information exchange is unable to terminate extreme events where synchronous occurrence of extreme events is only possible even for large interaction. On the other hand, a threshold-activated coupling can terminate extreme events above a critical value of mutual interaction.
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Affiliation(s)
- Arnob Ray
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Sarbendu Rakshit
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Syamal K Dana
- Department of Mathematics, Jadavpur University, Kolkata 700032, India
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14
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Park J, Do Y, Jang B. Multistability in the cyclic competition system. CHAOS (WOODBURY, N.Y.) 2018; 28:113110. [PMID: 30501221 DOI: 10.1063/1.5045366] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2018] [Accepted: 10/11/2018] [Indexed: 06/09/2023]
Abstract
Cyclically competition models have been successful to gain an insight of biodiversity mechanism in ecosystems. There are, however, still limitations to elucidate complex phenomena arising in real competition. In this paper, we report that a multistability occurs in a simple rock-paper-scissor cyclically competition model by assuming that intraspecific competition depends on the logistic growth of each species density. This complex stability is absent in any cyclically competition model, and we investigate how the proposed intraspecific competition affects biodiversity in the existing society of three species through macroscopic and microscopic approaches. When the system is multistable, we show basins of the asymptotically stable heteroclinic cycle and stable attractors to demonstrate how the survival state is determined by initial densities of three species. Also, we find that the multistability is associated with a subcritical Hopf bifurcation. This surprising finding will give an opportunity to interpret rich dynamical phenomena in ecosystems which may occur in cyclic competition systems with different types of interactions.
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Affiliation(s)
- Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea
| | - Younghae Do
- Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 41566, Republic of Korea
| | - Bongsoo Jang
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea
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15
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16
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Bashkirtseva I. Crises, noise, and tipping in the Hassell population model. CHAOS (WOODBURY, N.Y.) 2018; 28:033603. [PMID: 29604634 DOI: 10.1063/1.4990007] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We consider a problem of the analysis of the noise-induced tipping in population systems. To study this phenomenon, we use Hassell-type system with Allee effect as a conceptual model. A mathematical investigation of the tipping is connected with the analysis of the crisis bifurcations, both boundary and interior. In the parametric study of the abrupt changes in dynamics related to the noise-induced extinction and transition from order to chaos, the stochastic sensitivity function technique and confidence domains are used. The effectiveness of the suggested approach to detect early warnings of critical stochastic transitions is demonstrated.
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Affiliation(s)
- Irina Bashkirtseva
- Institute of Natural Sciences and Mathematics, Ural Federal University, Lenina 51, 620000 Ekaterinburg, Russia
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17
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Wang G, Xu H, Lai YC. Emergence, evolution, and control of multistability in a hybrid topological quantum/classical system. CHAOS (WOODBURY, N.Y.) 2018; 28:033601. [PMID: 29604629 DOI: 10.1063/1.4998244] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We present a novel class of nonlinear dynamical systems-a hybrid of relativistic quantum and classical systems and demonstrate that multistability is ubiquitous. A representative setting is coupled systems of a topological insulator and an insulating ferromagnet, where the former possesses an insulating bulk with topologically protected, dissipationless, and conducting surface electronic states governed by the relativistic quantum Dirac Hamiltonian and the latter is described by the nonlinear classical evolution of its magnetization vector. The interactions between the two are essentially the spin transfer torque from the topological insulator to the ferromagnet and the local proximity induced exchange coupling in the opposite direction. The hybrid system exhibits a rich variety of nonlinear dynamical phenomena besides multistability such as bifurcations, chaos, and phase synchronization. The degree of multistability can be controlled by an external voltage. In the case of two coexisting states, the system is effectively binary, opening a door to exploitation for developing spintronic memory devices. Because of the dissipationless and spin-momentum locking nature of the surface currents of the topological insulator, little power is needed for generating a significant current, making the system appealing for potential applications in next generation of low power memory devices.
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Affiliation(s)
- Guanglei Wang
- School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Hongya Xu
- School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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18
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Chvykov P, England J. Least-rattling feedback from strong time-scale separation. Phys Rev E 2018; 97:032115. [PMID: 29776054 DOI: 10.1103/physreve.97.032115] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2017] [Indexed: 06/08/2023]
Abstract
In most interacting many-body systems associated with some "emergent phenomena," we can identify subgroups of degrees of freedom that relax on dramatically different time scales. Time-scale separation of this kind is particularly helpful in nonequilibrium systems where only the fast variables are subjected to external driving; in such a case, it may be shown through elimination of fast variables that the slow coordinates effectively experience a thermal bath of spatially varying temperature. In this paper, we investigate how such a temperature landscape arises according to how the slow variables affect the character of the driven quasisteady state reached by the fast variables. Brownian motion in the presence of spatial temperature gradients is known to lead to the accumulation of probability density in low-temperature regions. Here, we focus on the implications of attraction to low effective temperature for the long-term evolution of slow variables. After quantitatively deriving the temperature landscape for a general class of overdamped systems using a path-integral technique, we then illustrate in a simple dynamical system how the attraction to low effective temperature has a fine-tuning effect on the slow variable, selecting configurations that bring about exceptionally low force fluctuation in the fast-variable steady state. We furthermore demonstrate that a particularly strong effect of this kind can take place when the slow variable is tuned to bring about orderly, integrable motion in the fast dynamics that avoids thermalizing energy absorbed from the drive. We thus point to a potentially general feedback mechanism in multi-time-scale active systems, that leads to the exploration of slow variable space, as if in search of fine tuning for a "least-rattling" response in the fast coordinates.
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Affiliation(s)
- Pavel Chvykov
- Physics of Living Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Jeremy England
- Physics of Living Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Feudel U, Pisarchik AN, Showalter K. Multistability and tipping: From mathematics and physics to climate and brain-Minireview and preface to the focus issue. CHAOS (WOODBURY, N.Y.) 2018; 28:033501. [PMID: 29604626 DOI: 10.1063/1.5027718] [Citation(s) in RCA: 36] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Multistability refers to the coexistence of different stable states in nonlinear dynamical systems. This phenomenon has been observed in laboratory experiments and in nature. In this introduction, we briefly introduce the classes of dynamical systems in which this phenomenon has been found and discuss the extension to new system classes. Furthermore, we introduce the concept of critical transitions and discuss approaches to distinguish them according to their characteristics. Finally, we present some specific applications in physics, neuroscience, biology, ecology, and climate science.
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Affiliation(s)
- Ulrike Feudel
- Theoretical Physics/Complex Systems, ICBM, University of Oldenburg, 26129 Oldenburg, Germany
| | - Alexander N Pisarchik
- Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo, 28223 Pozuelo de Alarcon, Madrid, Spain
| | - Kenneth Showalter
- C. Eugene Bennett Department of Chemistry, West Virginia University, Morgantown, West Virginia 26506-6045, USA
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20
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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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Ryashko L. Sensitivity analysis of the noise-induced oscillatory multistability in Higgins model of glycolysis. CHAOS (WOODBURY, N.Y.) 2018; 28:033602. [PMID: 29604640 DOI: 10.1063/1.4989982] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
A phenomenon of the noise-induced oscillatory multistability in glycolysis is studied. As a basic deterministic skeleton, we consider the two-dimensional Higgins model. The noise-induced generation of mixed-mode stochastic oscillations is studied in various parametric zones. Probabilistic mechanisms of the stochastic excitability of equilibria and noise-induced splitting of randomly forced cycles are analysed by the stochastic sensitivity function technique. A parametric zone of supersensitive Canard-type cycles is localized and studied in detail. It is shown that the generation of mixed-mode stochastic oscillations is accompanied by the noise-induced transitions from order to chaos.
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Affiliation(s)
- Lev Ryashko
- Institute of Natural Sciences and Mathematics, Ural Federal University, Lenina, 51, 620000 Ekaterinburg, Russia
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22
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Merkatas C, Kaloudis K, Hatjispyros SJ. A Bayesian nonparametric approach to reconstruction and prediction of random dynamical systems. CHAOS (WOODBURY, N.Y.) 2017; 27:063116. [PMID: 28679231 DOI: 10.1063/1.4990547] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We propose a Bayesian nonparametric mixture model for the reconstruction and prediction from observed time series data, of discretized stochastic dynamical systems, based on Markov Chain Monte Carlo methods. Our results can be used by researchers in physical modeling interested in a fast and accurate estimation of low dimensional stochastic models when the size of the observed time series is small and the noise process (perhaps) is non-Gaussian. The inference procedure is demonstrated specifically in the case of polynomial maps of an arbitrary degree and when a Geometric Stick Breaking mixture process prior over the space of densities, is applied to the additive errors. Our method is parsimonious compared to Bayesian nonparametric techniques based on Dirichlet process mixtures, flexible and general. Simulations based on synthetic time series are presented.
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Affiliation(s)
- Christos Merkatas
- Department of Mathematics, University of the Aegean, Karlovassi 83200, Greece
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23
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Yadav K, Kamal NK, Shrimali MD. Intermittent feedback induces attractor selection. Phys Rev E 2017; 95:042215. [PMID: 28505827 DOI: 10.1103/physreve.95.042215] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2016] [Indexed: 11/07/2022]
Abstract
We present a method for attractor selection in multistable dynamical systems. It involves a feedback term that is active only when the dynamics of the system is in a particular fraction of state space of the attractor. We implement this method first on a simplest symmetric chaotic flow and then on a bistable neuronal system. We find that adding this space-dependent feedback term to the dynamical equations of these systems will drive the dynamics to the desired attractor by annihilating the other. We further demonstrate that the attractor selection due to this feedback term can be used in construction of logic gates, which is one of the practical applications of the proposed method.
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Affiliation(s)
- Kiran Yadav
- Department of Physics, Central University of Rajasthan, Ajmer 305 817 India
| | - Neeraj Kumar Kamal
- Department of Physics, Central University of Rajasthan, Ajmer 305 817 India
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24
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Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles. Sci Rep 2017; 7:42351. [PMID: 28181582 PMCID: PMC5299408 DOI: 10.1038/srep42351] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2016] [Accepted: 01/08/2017] [Indexed: 11/08/2022] Open
Abstract
Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the system's parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a "ghost" of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.
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25
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Bilal S, Singh BK, Prasad A, Michael E. Effects of quasiperiodic forcing in epidemic models. CHAOS (WOODBURY, N.Y.) 2016; 26:093115. [PMID: 27781468 PMCID: PMC7112454 DOI: 10.1063/1.4963174] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/22/2016] [Accepted: 09/09/2016] [Indexed: 06/06/2023]
Abstract
We study changes in the bifurcations of seasonally driven compartmental epidemic models, where the transmission rate is modulated temporally. In the presence of periodic modulation of the transmission rate, the dynamics varies from periodic to chaotic. The route to chaos is typically through period doubling bifurcation. There are coexisting attractors for some sets of parameters. However in the presence of quasiperiodic modulation, tori are created in place of periodic orbits and chaos appears via finite torus doublings. Strange nonchaotic attractors (SNAs) are created at the boundary of chaotic and torus dynamics. Multistability is found to be reduced as a function of quasiperiodic modulation strength. It is argued that occurrence of SNAs gives an opportunity of asymptotic predictability of epidemic growth even when the underlying dynamics is strange.
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Affiliation(s)
- Shakir Bilal
- Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India
| | - Brajendra K Singh
- Department of Biological Sciences, University of Notre Dame, Notre Dame, Indiana 46556, USA
| | - Awadhesh Prasad
- Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India
| | - Edwin Michael
- Department of Biological Sciences, University of Notre Dame, Notre Dame, Indiana 46556, USA
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26
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Analysis of the noise-induced regimes in Ricker population model with Allee effect via confidence domains technique. BIOMED RESEARCH INTERNATIONAL 2014; 2014:346239. [PMID: 24982863 PMCID: PMC4058461 DOI: 10.1155/2014/346239] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/26/2014] [Accepted: 05/12/2014] [Indexed: 11/17/2022]
Abstract
We consider a discrete-time Ricker population model with the Allee effect under the random disturbances. It is shown that noise can cause various dynamic regimes, such as stable stochastic oscillations around the equilibrium, noise-induced extinction, and a stochastic trigger. For the parametric analysis of these regimes, we develop a method based on the investigation of the dispersions and arrangement of confidence domains. Using this method, we estimate threshold values of the noise generating such regimes.
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27
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Aoki H, Kaneko K. Slow stochastic switching by collective chaos of fast elements. PHYSICAL REVIEW LETTERS 2013; 111:144102. [PMID: 24138241 DOI: 10.1103/physrevlett.111.144102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2013] [Indexed: 06/02/2023]
Abstract
Coupled dynamical systems with one slow element and many fast elements are analyzed. By averaging over the dynamics of the fast variables, the adiabatic kinetic branch is introduced for the dynamics of the slow variable in the adiabatic limit. The dynamics without the limit are found to be represented by stochastic switching over these branches mediated by the collective chaos of the fast elements, while the switching frequency shows a complicated dependence on the ratio of the two time scales with some resonance structure. The ubiquity of the phenomena in the slow-fast dynamics is also discussed.
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Affiliation(s)
- Hidetoshi Aoki
- Research Center for Complex Systems Biology, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
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28
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Bashkirtseva I, Neiman AB, Ryashko L. Stochastic sensitivity analysis of the noise-induced excitability in a model of a hair bundle. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052711. [PMID: 23767570 DOI: 10.1103/physreve.87.052711] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2013] [Indexed: 06/02/2023]
Abstract
We study effect of weak noise on the dynamics of a hair bundle model near the excitability threshold and near a subcritical Hopf bifurcation. We analyze numerically noise-induced structural changes in the probability density and the power spectral density of the model. In particular, we show that weak noise can induce oscillations with two distinct frequencies in both excitable and limit-cycle regimes. We then applied a recently developed technique of stochastic sensitivity functions which allows us to estimate threshold values of noise intensity corresponding to these transitions.
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Affiliation(s)
- Irina Bashkirtseva
- Department of Mathematics, Ural Federal University, Pr. Lenina 51, Ekaterinburg, Russia.
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29
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Pisarchik AN, Jaimes-Reátegui R, Sevilla-Escoboza R, Huerta-Cuellar G. Multistate intermittency and extreme pulses in a fiber laser. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:056219. [PMID: 23214869 DOI: 10.1103/physreve.86.056219] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2012] [Indexed: 06/01/2023]
Abstract
In our recent Letter [Phys. Rev. Lett. 107, 274101 (2011)], we demonstrated that slow random perturbations of a system parameter were responsible for the emergence of rogue waves in a fiber laser with coexisting attractors. In this paper we investigate how the probability of a particular state to appear in multistate intermittency can be controlled by low-pass noise filtering. We show that the probability of some states depends nonmonotonously on the noise amplitude and cutoff frequency. The conditions for the emergence of extreme pulses in a erbium-doped fiber laser are analyzed numerically and experimentally.
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Affiliation(s)
- A N Pisarchik
- Centro de Investigaciones en Optica, Loma del Bosque 115, 37150 Leon, Guanajuato, Mexico.
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30
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Pisarchik AN, Jaimes-Reátegui R, Sevilla-Escoboza R, Huerta-Cuellar G, Taki M. Rogue waves in a multistable system. PHYSICAL REVIEW LETTERS 2011; 107:274101. [PMID: 22243311 DOI: 10.1103/physrevlett.107.274101] [Citation(s) in RCA: 57] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2011] [Indexed: 05/31/2023]
Abstract
Clear evidence of rogue waves in a multistable system is revealed by experiments with an erbium-doped fiber laser driven by harmonic pump modulation. The mechanism for the rogue wave formation lies in the interplay of stochastic processes with multistable deterministic dynamics. Low-frequency noise applied to a diode pump current induces rare jumps to coexisting subharmonic states with high-amplitude pulses perceived as rogue waves. The probability of these events depends on the noise filtered frequency and grows up when the noise amplitude increases. The probability distribution of spike amplitudes confirms the rogue wave character of the observed phenomenon. The results of numerical simulations are in good agreement with experiments.
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Affiliation(s)
- Alexander N Pisarchik
- Centro de Investigaciones en Optica, Loma del Bosque 115, 37150 Leon, Guanajuato, Mexico.
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31
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Bashkirtseva I, Ryashko L. Stochastic sensitivity analysis of noise-induced excitement in a prey–predator plankton system. FRONTIERS IN LIFE SCIENCE 2011. [DOI: 10.1080/21553769.2012.702666] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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32
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Gan C, Yang S, Lei H. Noisy scattering dynamics in the randomly driven Hénon-Heiles oscillator. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:066204. [PMID: 21230720 DOI: 10.1103/physreve.82.066204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2010] [Indexed: 05/30/2023]
Abstract
Noisy scattering dynamics in the randomly driven Hénon-Heiles oscillator is investigated when the energy is above the threshold to permit particles to escape from the scattering region. First, some basic simulation procedures are briefly introduced and the fractal exit basins appear to be robust when the bounded noisy excitation is imposed on the oscillator. Second, several key fractal characteristics of the sample basin boundaries, such as the delay-time function and the uncertainty dimension, are estimated from which this oscillator is found to be structurally unstable against the bounded noisy excitation. Moreover, the stable and unstable manifolds of some sample chaotic invariant sets are estimated and illustrated in a special two-dimensional Poincaré section. Lastly, several previous methods are developed to identify three arbitrarily chosen noisy scattering time series of the randomly driven Hénon-Heiles oscillator, from which the quasiperiodic-dominant and the chaotic-dominant dynamical behaviors are distinguished.
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Affiliation(s)
- Chunbiao Gan
- Department of Mechanical Engineering, Zhejiang University, Hangzhou, China
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33
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Rodrigues CS, Grebogi C, de Moura APS. Escape from attracting sets in randomly perturbed systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:046217. [PMID: 21230375 DOI: 10.1103/physreve.82.046217] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2010] [Revised: 08/20/2010] [Indexed: 05/30/2023]
Abstract
The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed to have Gaussian distribution. In this paper, we address the problem of escape induced by bounded noise. We show that the dynamics of escape from an attractor's basin is equivalent to that of a closed system with an appropriately chosen "hole." Using this equivalence, we show that there is a minimum noise amplitude above which escape takes place, and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition. We verify our analytical predictions through numerical simulations of two well-known two-dimensional maps with noise.
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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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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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35
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Huerta-Cuellar G, Pisarchik AN, Kir'yanov AV, Barmenkov YO, Del Valle Hernández J. Prebifurcation noise amplification in a fiber laser. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:036204. [PMID: 19392032 DOI: 10.1103/physreve.79.036204] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2008] [Indexed: 05/27/2023]
Abstract
We report on the experimental evidence of noise amplification in an erbium-doped fiber laser in the vicinity of saddle-node, period-doubling, and crisis bifurcations. We demonstrate this interesting phenomenon by analyzing the laser bifurcation diagrams and power spectra. Numerical simulations on the base of an advanced laser model display good agreement with the experimental results.
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Affiliation(s)
- G Huerta-Cuellar
- Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico
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36
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Do Y, Lai YC. Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems. CHAOS (WOODBURY, N.Y.) 2008; 18:043107. [PMID: 19123617 DOI: 10.1063/1.2985853] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.
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Affiliation(s)
- Younghae Do
- Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
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Goswami BK. Control of multistate hopping intermittency. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:066208. [PMID: 19256926 DOI: 10.1103/physreve.78.066208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2008] [Indexed: 05/27/2023]
Abstract
In multistable regimes, noise can create "multistate hopping intermittency," i.e., intermittent transitions among coexisting stable attractors. We demonstrate that a small periodic perturbation can significantly control such hopping intermittency. By "control" we imply a qualitative change in the probability distribution of occupation in the phase space around the stable attractors. In other words, if the uncontrolled system exhibits a preference to stay around a given attractor (say " A ") in comparison to another attractor (say " B "), the control perturbation creates a contrasting scenario so that attractor B is most frequently visited and consequently, the occupation probability becomes maximum around B instead of A . The control perturbation works in the following way: It destroys attractor A by boundary crisis while attractor B remains stable. As a result, even if the system is pushed by noise into the erstwhile basin of attractor A , the system does not remain there for long and therefore stays longer around attractor B . Significantly, such a change in the intermittent scenario can be obtained by a small-amplitude and slow-periodic perturbation. The control is theoretically demonstrated with two standard models, namely, Lorenz equations (for autonomous systems), and the periodically driven, damped Toda oscillator (for nonautonomous systems). Recent experiments with a cavity-loss modulated CO2 laser and an analog circuit of Lorenz equations have validated our theoretical demonstrations excellently.
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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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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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Billings L, Schwartz IB. Identifying almost invariant sets in stochastic dynamical systems. CHAOS (WOODBURY, N.Y.) 2008; 18:023122. [PMID: 18601489 DOI: 10.1063/1.2929748] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
We consider the approximation of fluctuation induced almost invariant sets arising from stochastic dynamical systems. The dynamical evolution of densities is derived from the stochastic Frobenius-Perron operator. Given a stochastic kernel with a known distribution, approximate almost invariant sets are found by translating the problem into an eigenvalue problem derived from reversible Markov processes. Analytic and computational examples of the methods are used to illustrate the technique, and are shown to reveal the probability transport between almost invariant sets in nonlinear stochastic systems. Both small and large noise cases are considered.
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Affiliation(s)
- Lora Billings
- Department of Mathematical Sciences, Montclair State University, Montclair, New Jersey 07043, USA
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40
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Seoane JM, Sanjuán MAF, Lai YC. Fractal dimension in dissipative chaotic scattering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:016208. [PMID: 17677544 DOI: 10.1103/physreve.76.016208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2006] [Revised: 02/06/2007] [Indexed: 05/16/2023]
Abstract
The effect of weak dissipation on chaotic scattering is relevant to situations of physical interest. We investigate how the fractal dimension of the set of singularities in a scattering function varies as the system becomes progressively more dissipative. A crossover phenomenon is uncovered where the dimension decreases relatively more rapidly as a dissipation parameter is increased from zero and then exhibits a much slower rate of decrease. We provide a heuristic theory and numerical support from both discrete-time and continuous-time scattering systems to establish the generality of this phenomenon. Our result is expected to be important for physical phenomena such as the advection of inertial particles in open chaotic flows, among others.
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Affiliation(s)
- Jesús M Seoane
- Nonlinear Dynamics and Chaos Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain.
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41
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Safonov LA, Yamamoto Y. Noise-driven switching between limit cycles and adaptability in a small-dimensional excitable network with balanced coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:031914. [PMID: 16605565 DOI: 10.1103/physreve.73.031914] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2005] [Revised: 01/03/2006] [Indexed: 05/08/2023]
Abstract
We study a system of globally coupled FitzHugh-Nagumo equations. Each unit is either excitatory or inhibitory. If the numbers of units of both types are in a specific ratio, we observe the presence of multistable oscillatory states with different excitation or firing rates. In the presence of noise, there is noise-driven switching between these states and the resultant firing pattern is long-range correlated. The choice between higher and lower frequency oscillations depends on the input, which results in increasing adaptability of the system's output to the periodic input.
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Affiliation(s)
- Leonid A Safonov
- Educational Physiology Laboratory, Graduate School of Education, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
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42
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Namikawa J. Chaotic itinerancy and power-law residence time distribution in stochastic dynamical systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:026204. [PMID: 16196681 DOI: 10.1103/physreve.72.026204] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2004] [Indexed: 05/04/2023]
Abstract
Chaotic itinerant motion among varieties of ordered states is described by a stochastic model based on the mechanism of chaotic itinerancy. The model consists of a random walk on a half-line and a Markov chain with a transition probability matrix. The stability of attractor ruin in the model is investigated by analyzing the residence time distribution of orbits at attractor ruins. It is shown that the residence time distribution averaged over all attractor ruins can be described by the superposition of (truncated) power-law distributions if the basin of attraction for each attractor ruin has a zero measure. This result is confirmed by simulation of models exhibiting chaotic itinerancy. Chaotic itinerancy is also shown to be absent in coupled Milnor attractor systems if the transition probability among attractor ruins can be represented as a Markov chain.
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Affiliation(s)
- Jun Namikawa
- School of Knowledge Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa 923-1292, Japan.
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43
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Willeboordse FH, Kaneko K. Externally controlled attractor selection in a high-dimensional system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:026207. [PMID: 16196684 DOI: 10.1103/physreve.72.026207] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2005] [Indexed: 05/04/2023]
Abstract
A high-dimensional dynamical system with global couplings that can serve as a prototype for systems with very large numbers of attractors like memory is investigated and shown to be controllable by external inputs. By changing the duration that noise is added, final attractors are selected as the number of degrees of freedom of the nonsynchronized elements decreases one by one over time. Furthermore, it is found that this selection of attractors is also possible by controlling the sweeping speed of a parameter. The mechanism for this controlled selection is explained and shown to be rather general. Applications to attractor switching are given.
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Billings L, Schwartz IB, Morgan DS, Bollt EM, Meucci R, Allaria E. Stochastic bifurcation in a driven laser system: experiment and theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:026220. [PMID: 15447578 DOI: 10.1103/physreve.70.026220] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2003] [Indexed: 05/24/2023]
Abstract
We analyze the effects of stochastic perturbations in a physical example occurring as a higher-dimensional dynamical system. The physical model is that of a class- B laser, which is perturbed stochastically with finite noise. The effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one of chaoslike behavior, or noise-induced chaos. To analyze the qualitative change, we apply the technique of the stochastic Frobenius-Perron operator [L. Billings et al., Phys. Rev. Lett. 88, 234101 (2002)] to a model of the experimental system. Our main result is the identification of a global mechanism to induce chaoslike behavior by adding stochastic perturbations in a realistic model system of an optics experiment. In quantifying the stochastic bifurcation, we have computed a transition matrix describing the probability of transport from one region of phase space to another, which approximates the stochastic Frobenius-Perron operator. This mechanism depends on both the standard deviation of the noise and the global topology of the system. Our result pinpoints regions of stochastic transport whereby topological deterministic dynamics subjected to sufficient noise results in noise-induced chaos in both theory and experiment.
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Affiliation(s)
- Lora Billings
- Department of Mathematical Sciences, Montclair State University, Montclair, New Jersey 07043, USA
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45
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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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Abstract
Chaotic itinerancy is universal dynamics in high-dimensional dynamical systems, showing itinerant motion among varieties of low-dimensional ordered states through high-dimensional chaos. Discovery, basic features, characterization, examples, and significance of chaotic itinerancy are surveyed.
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Affiliation(s)
- Kunihiko Kaneko
- Department of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
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47
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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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48
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Masoller C. Noise-induced resonance in delayed feedback systems. PHYSICAL REVIEW LETTERS 2002; 88:034102. [PMID: 11801062 DOI: 10.1103/physrevlett.88.034102] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2001] [Indexed: 05/23/2023]
Abstract
We study the influence of noise in the dynamics of a laser with optical feedback. For appropriate choices of the feedback parameters, several attractors coexist, and large enough noise induces jumps among the attractors. Based on the residence times probability density, it is shown that with increasing noise the dynamics of attractor jumping exhibits a resonant behavior, which is due to the interplay of noise and delayed feedback. It is also shown that this type of resonance is not specific to the model equations used, since it also occurs in other delay differential equations.
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Affiliation(s)
- C Masoller
- Instituto de Física, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo 11400, Uruguay
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Soskin SM, Mannella R, Arrayás M, Silchenko AN. Strong enhancement of noise-induced escape by nonadiabatic periodic driving due to transient chaos. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:051111. [PMID: 11414891 DOI: 10.1103/physreve.63.051111] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2000] [Revised: 09/18/2000] [Indexed: 05/23/2023]
Abstract
We have found a mechanism by which a moderately weak nonadiabatic periodic driving may significantly facilitate noise-induced interwell transitions in an underdamped multiwell system. The mechanism is associated with the onset of a homoclinic tangle in the noise-free system: if the ratio of the driving amplitude A to the damping gamma exceeds a critical value approximately 1, then the basins of attraction of the linear responses related to different wells are mixed in a complex manner in some layer associated with the separatrix of the undriven nondissipative system, and the minimal energy in such layer is lower than the top of the barrier. Thus the energy to which the system needs to be activated by the noise, to be able to make a transition, is lower than the top of the barrier.
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Affiliation(s)
- S M Soskin
- Institute of Semiconductor Physics, Ukrainian National Academy of Sciences, Kiev, Ukraine
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50
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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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