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Roy S, Ray A, Chowdhury AR. Kosambi-Cartan-Chern perspective on chaos: Unveiling hidden attractors in nonlinear autonomous systems. Phys Rev E 2024; 109:044205. [PMID: 38755835 DOI: 10.1103/physreve.109.044205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2023] [Accepted: 03/20/2024] [Indexed: 05/18/2024]
Abstract
This article confronts the formidable task of exploring chaos within hidden attractors in nonlinear three-dimensional autonomous systems, highlighting the lack of established analytical and numerical methodologies for such investigations. As the basin of attraction does not touch the unstable manifold, there are no straightforward numerical processes to detect those attractors and one has to implement special numerical and analytical strategies. In this article we present an alternative approach that allows us to predict the basin of attraction associated with hidden attractors, overcoming the existing limitations. The method discussed here is based on the Kosambi-Cartan-Chern theory which enables us to conduct a comprehensive theoretical analysis by means of evaluating geometric invariants and instability exponents, thereby delineating the regions encompassing chaotic and periodic zones. Our analytical predictions are thoroughly validated by numerical results.
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Affiliation(s)
- Somnath Roy
- Department of Physics, Jadavpur University, Kolkata 700075, India
| | - Anirban Ray
- Department of Physics, Gour Mahavidyalaya, Mangalbari, Malda 732142, India
| | - A Roy Chowdhury
- Department of Physics, Jadavpur University, Kolkata 700075, India
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2
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Biswas S, Mandal A, Pal S. Catastrophic and noncatastrophic population crashes in a bitrophic system with dynamic additional food provision to cooperative predators. Phys Rev E 2024; 109:024224. [PMID: 38491580 DOI: 10.1103/physreve.109.024224] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2023] [Accepted: 02/01/2024] [Indexed: 03/18/2024]
Abstract
In this article we contemplate the dynamics of an additional food-provided prey-predator system. We assume that the behavior of cooperative predators induces fear in prey, which radically affects the prey's birth and death rates. We observe that the structural instability imposed by strong cooperative hunting among predators goes away with higher intensities of fear levels affecting the prey's reproductive output and mortality. High levels of prey refuge are not conducive to the survival of predators. In such a situation, adequate supply of high-quality additional food is favorable regarding the persistence and stability of the system. Interestingly, the system potentially exhibits two stable configurations under identical ecological conditions by allowing different bifurcation scenarios, including saddle-node and backward bifurcations, and associated hysteresis effects with prey refuge along with additional food quantity and quality. In the stochastic environment, the system experiences critical transitions through bifurcation-induced tipping events with time-varying additional food for predators. Enhanced disturbance events promote noise-induced switching and tipping events. Finally, our investigation explores whether impending population crashes resulting from the variability of additional food quantity and quality can reliably be predicted using early warning signals in the context of redshifted noise. Overall, our results may provide insights for finding control strategies in the context of community ecology.
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Affiliation(s)
- Saswati Biswas
- Department of Mathematics, School of Natural Sciences, Shiv Nadar Institution of Eminence, Gautam Buddha Nagar, Uttar Pradesh 201314, India
| | - Arindam Mandal
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India
| | - Samares Pal
- Department of Mathematics, University of Kalyani, Kalyani 741235, India
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3
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Huang M, Yang A, Yuan S, Zhang T. Stochastic sensitivity analysis and feedback control of noise-induced transitions in a predator-prey model with anti-predator behavior. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:4219-4242. [PMID: 36899624 DOI: 10.3934/mbe.2023197] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
In this study, we investigate a stochastic predator-prey model with anti-predator behavior. We first analyze the noise-induced transition from a coexistence state to the prey-only equilibrium by using the stochastic sensitive function technique. The critical noise intensity for the occurrence of state switching is estimated by constructing confidence ellipses and confidence bands, respectively, for the coexistence the equilibrium and limit cycle. We then study how to suppress the noise-induced transition by using two different feedback control methods to stabilize the biomass at the attraction region of the coexistence equilibrium and the coexistence limit cycle, respectively. Our research indicates that compared with the prey population, the predators appear more vulnerable and prone to extinction in the presence of environmental noise, but it can be prevented by taking some appropriate feedback control strategies.
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Affiliation(s)
- Mengya Huang
- College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
| | - Anji Yang
- College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
| | - Sanling Yuan
- College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
| | - Tonghua Zhang
- Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
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4
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Kirkow V, Wang H, Garcia PV, Ahmed S, Heggerud CM. Impacts of a changing environment on a stoichiometric producer-grazer system: a stochastic modelling approach. Ecol Modell 2022. [DOI: 10.1016/j.ecolmodel.2022.109971] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
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5
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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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6
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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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7
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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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8
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Alkhayuon H, Ashwin P. Weak tracking in nonautonomous chaotic systems. Phys Rev E 2020; 102:052210. [PMID: 33327197 DOI: 10.1103/physreve.102.052210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2020] [Accepted: 10/22/2020] [Indexed: 06/12/2023]
Abstract
Previous studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to "parameter shifts" between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous Rössler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.
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Affiliation(s)
- Hassan Alkhayuon
- School of Mathematical Sciences, University College Cork, Cork T12 XF62, Ireland
| | - Peter Ashwin
- Centre for Systems, Dynamics and Control, Department of Mathematics, University of Exeter, Exeter EX4 4QF, United Kingdom
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9
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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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10
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Klinshov V, Shchapin D, D'Huys O. Mode Hopping in Oscillating Systems with Stochastic Delays. PHYSICAL REVIEW LETTERS 2020; 125:034101. [PMID: 32745403 DOI: 10.1103/physrevlett.125.034101] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2020] [Revised: 05/06/2020] [Accepted: 06/08/2020] [Indexed: 06/11/2023]
Abstract
We study a noisy oscillator with pulse delayed feedback, theoretically and in an electronic experimental implementation. Without noise, this system has multiple stable periodic regimes. We consider two types of noise: (i) phase noise acting on the oscillator state variable and (ii) stochastic fluctuations of the coupling delay. For both types of stochastic perturbations the system hops between the deterministic regimes, but it shows dramatically different scaling properties for different types of noise. The robustness to conventional phase noise increases with coupling strength. However for stochastic variations in the coupling delay, the lifetimes decrease exponentially with the coupling strength. We provide an analytic explanation for these scaling properties in a linearized model. Our findings thus indicate that the robustness of a system to stochastic perturbations strongly depends on the nature of these perturbations.
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Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Dmitry Shchapin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Otti D'Huys
- Department of Mathematics, Aston University, B4 7ET Birmingham, United Kingdom
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11
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Yuan S, Wu D, Lan G, Wang H. Noise-Induced Transitions in a Nonsmooth Producer-Grazer Model with Stoichiometric Constraints. Bull Math Biol 2020; 82:55. [PMID: 32350614 PMCID: PMC7190610 DOI: 10.1007/s11538-020-00733-y] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2019] [Accepted: 04/09/2020] [Indexed: 12/03/2022]
Abstract
Stoichiometric producer-grazer models are nonsmooth due to the Liebig's Law of Minimum and can generate new dynamics such as bistability for producer-grazer interactions. Environmental noises can be extremely important and change dynamical behaviors of a stoichiometric producer-grazer model. In this paper, we consider a stochastically forced producer-grazer model and study the phenomena of noise-induced state switching between two stochastic attractors in the bistable zone. Namely, there is a frequent random hopping of phase trajectories between attracting basins of the attractors. In addition, by applying the stochastic sensitivity function technique, we construct the confidence ellipse and confidence band to find the configurational arrangement of equilibria and a limit cycle, respectively.
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Affiliation(s)
- Sanling Yuan
- College of Science, University of Shanghai for Science and Technology, Shanghai, 200093 China
| | - Dongmei Wu
- College of Science, University of Shanghai for Science and Technology, Shanghai, 200093 China
| | - Guijie Lan
- College of Science, University of Shanghai for Science and Technology, Shanghai, 200093 China
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 Canada
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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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Kaminker V, Wackerbauer R. Alternating activity patterns and a chimeralike state in a network of globally coupled excitable Morris-Lecar neurons. CHAOS (WOODBURY, N.Y.) 2019; 29:053121. [PMID: 31154794 DOI: 10.1063/1.5093483] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2019] [Accepted: 04/30/2019] [Indexed: 06/09/2023]
Abstract
Spatiotemporal chaos collapses to either a rest state or a propagating pulse in a ring network of diffusively coupled, excitable Morris-Lecar neurons. Adding global varying synaptic coupling to the ring network reveals complex transient behavior. Spatiotemporal chaos collapses into a transient pulse that reinitiates spatiotemporal chaos to allow sequential pattern switching until a collapse to the rest state. A domain of irregular neuron activity coexists with a domain of inactive neurons forming a transient chimeralike state. Transient spatial localization of the chimeralike state is observed for stronger synapses.
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Affiliation(s)
- Vitaliy Kaminker
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
| | - Renate Wackerbauer
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
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14
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Lucarini V, Bódai T. Transitions across Melancholia States in a Climate Model: Reconciling the Deterministic and Stochastic Points of View. PHYSICAL REVIEW LETTERS 2019; 122:158701. [PMID: 31050495 DOI: 10.1103/physrevlett.122.158701] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2018] [Indexed: 06/09/2023]
Abstract
The Earth is well known to be, in the current astronomical configuration, in a regime where two asymptotic states can be realized. The warm state we live in is in competition with the ice-covered snowball state. The bistability exists as a result of the positive ice-albedo feedback. In a previous investigation performed on a intermediate complexity climate model we identified the unstable climate states (melancholia states) separating the coexisting climates, and studied their dynamical and geometrical properties. The melancholia states are ice covered up to the midlatitudes and attract trajectories initialized on the basin boundary. In this Letter, we study how stochastically perturbing the parameter controlling the intensity of the incoming solar radiation impacts the stability of the climate. We detect transitions between the warm and the snowball state and analyze in detail the properties of the noise-induced escapes from the corresponding basins of attraction. We determine the most probable paths for the transitions and find evidence that the melancholia states act as gateways, similarly to saddle points in an energy landscape.
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Affiliation(s)
- Valerio Lucarini
- Centre for the Mathematics of Planet Earth, University of Reading, Reading, RG66AX United Kingdom
- Department of Mathematics and Statistics, University of Reading, Reading, RG66AX United Kingdom
- CEN, University of Hamburg, Hamburg, 20144 Germany
| | - Tamás Bódai
- Centre for the Mathematics of Planet Earth, University of Reading, Reading, RG66AX United Kingdom
- Department of Mathematics and Statistics, University of Reading, Reading, RG66AX United Kingdom
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15
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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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16
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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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17
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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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Hartle H, Wackerbauer R. Transient chaos and associated system-intrinsic switching of spacetime patterns in two synaptically coupled layers of Morris-Lecar neurons. Phys Rev E 2018; 96:032223. [PMID: 29347029 DOI: 10.1103/physreve.96.032223] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2017] [Indexed: 11/07/2022]
Abstract
Spatiotemporal chaos collapses to either a rest state or a propagating pulse solution in a single layer of diffusively coupled, excitable Morris-Lecar neurons. Weak synaptic coupling of two such layers reveals system intrinsic switching of spatiotemporal activity patterns within and between the layers at irregular times. Within a layer, switching sequences include spatiotemporal chaos, erratic and regular pulse propagation, spontaneous network wide neuron activity, and rest state. A momentary substantial reduction in neuron activity in one layer can reinitiate transient spatiotemporal chaos in the other layer, which can induce a swap of spatiotemporal chaos with a pulse state between the layers. Presynaptic input maximizes the distance between propagating pulses, in contrast to pulse merging in the absence of synapses.
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Affiliation(s)
- Harrison Hartle
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
| | - Renate Wackerbauer
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
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19
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Choi D, Wishon MJ, Chang CY, Citrin DS, Locquet A. Multistate intermittency on the route to chaos of a semiconductor laser subjected to optical feedback from a long external cavity. CHAOS (WOODBURY, N.Y.) 2018; 28:011102. [PMID: 29390638 DOI: 10.1063/1.5013332] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We observe experimentally two regimes of intermittency on the route to chaos of a semiconductor laser subjected to optical feedback from a long external cavity as the feedback level is increased. The first regime encountered corresponds to multistate intermittency involving two or three states composed of several combinations of periodic, quasiperiodic, and subharmonic dynamics. The second regime is observed for larger feedback levels and involves intermittency between period-doubled and chaotic regimes. This latter type of intermittency displays statistical properties similar to those of on-off intermittency.
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Affiliation(s)
- Daeyoung Choi
- Georgia Tech-CNRS UMI 2958, Georgia Tech Lorraine, 2 Rue Marconi, F-57070 Metz, France
| | - Michael J Wishon
- Georgia Tech-CNRS UMI 2958, Georgia Tech Lorraine, 2 Rue Marconi, F-57070 Metz, France
| | - C Y Chang
- Georgia Tech-CNRS UMI 2958, Georgia Tech Lorraine, 2 Rue Marconi, F-57070 Metz, France
| | - D S Citrin
- Georgia Tech-CNRS UMI 2958, Georgia Tech Lorraine, 2 Rue Marconi, F-57070 Metz, France
| | - A Locquet
- Georgia Tech-CNRS UMI 2958, Georgia Tech Lorraine, 2 Rue Marconi, F-57070 Metz, France
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20
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Hramov AE, Koronovskii AA, Moskalenko OI, Zhuravlev MO, Jaimes-Reategui R, Pisarchik AN. Separation of coexisting dynamical regimes in multistate intermittency based on wavelet spectrum energies in an erbium-doped fiber laser. Phys Rev E 2016; 93:052218. [PMID: 27300891 DOI: 10.1103/physreve.93.052218] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2015] [Indexed: 06/06/2023]
Abstract
We propose a method for the detection and localization of different types of coexisting oscillatory regimes that alternate with each other leading to multistate intermittency. Our approach is based on consideration of wavelet spectrum energies. The proposed technique is tested in an erbium-doped fiber laser with four coexisting periodic orbits, where external noise induces intermittent switches between the coexisting states. Statistical characteristics of multistate intermittency, such as the mean duration of the phases for every oscillation type, are examined with the help of the developed method. We demonstrate strong advantages of the proposed technique over previously used amplitude methods.
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Affiliation(s)
- Alexander E Hramov
- Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia and Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054, Russia
| | - Alexey A Koronovskii
- Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia and Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054, Russia
| | - Olga I Moskalenko
- Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia and Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054, Russia
| | - Maksim O Zhuravlev
- Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia and Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054, Russia
| | - Rider Jaimes-Reategui
- Universidad de Guadalajara, Centro Universitario de los Lagos, Enrique Díaz de León 1144, Paseos de la Montaña, 47460, Lagos de Moreno, Jalisco, Mexico
| | - Alexander N Pisarchik
- Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo, 28223 Pozuelo de Alarcon, Madrid, Spain and Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico
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21
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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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22
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Bashkirtseva I, Chen G, Ryashko L. Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system. CHAOS (WOODBURY, N.Y.) 2012; 22:033104. [PMID: 23020443 DOI: 10.1063/1.4732543] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noise-induced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noise-induced hopping which generates chaos are demonstrated on the stochastic Chen system.
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Affiliation(s)
- Irina Bashkirtseva
- Department of Mathematics, Ural State University, Lenina, 51, Ekaterinburg, Russia
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23
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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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24
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Tél T, Lai YC. Quasipotential approach to critical scaling in noise-induced chaos. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:056208. [PMID: 20866308 DOI: 10.1103/physreve.81.056208] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2009] [Revised: 04/12/2010] [Indexed: 05/29/2023]
Abstract
When a dynamical system exhibits transient chaos and a nonchaotic attractor, as in a periodic window, noise can induce a chaotic attractor. In particular, when the noise amplitude exceeds a critical value, the largest Lyapunov exponent of the attractor of the system starts to increase from zero. While a scaling law for the variation of the Lyapunov exponent with noise was uncovered previously, it is mostly based on numerical evidence and a heuristic analysis. This paper presents a more general approach to the scaling law, one based on the concept of quasipotentials. Besides providing deeper insights into the problem of noise-induced chaos, the quasipotential approach enables previously unresolved issues to be addressed. The fractal properties of noise-induced chaotic attractors and applications to biological systems are also discussed.
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Affiliation(s)
- Tamás Tél
- Institute for Theoretical Physics, Eötvös University, Pázmány P. s. 1/A, Budapest H-1117, Hungary
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25
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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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26
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Gelens L, Beri S, Van der Sande G, Mezosi G, Sorel M, Danckaert J, Verschaffelt G. Exploring multistability in semiconductor ring lasers: theory and experiment. PHYSICAL REVIEW LETTERS 2009; 102:193904. [PMID: 19518954 DOI: 10.1103/physrevlett.102.193904] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2009] [Indexed: 05/27/2023]
Abstract
We report the first experimental observation of multistable states in a single-longitudinal mode semiconductor ring laser. We show how the operation of the device can be steered to either monostable, bistable, or multistable dynamical regimes in a controlled way. We observe that the dynamical regimes are organized in well-reproducible sequences that match the bifurcation diagrams of a two-dimensional model. By analyzing the phase space in this model, we predict how the stochastic transitions between multistable states take place and confirm it experimentally.
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Affiliation(s)
- L Gelens
- Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
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27
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Bashkirtseva I, Ryashko L. Constructive analysis of noise-induced transitions for coexisting periodic attractors of the Lorenz model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:041106. [PMID: 19518172 DOI: 10.1103/physreve.79.041106] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2008] [Indexed: 05/27/2023]
Abstract
We study the stochastically forced Lorenz model in the parameter zone admitting two coexisting limit cycles under the transition to chaos via period-doubling bifurcations. Noise-induced transitions between both different parts of the single attractor and two coexisting separate attractors are demonstrated. The effects of structural stabilization and noise symmetrization are discussed. We suggest a stochastic sensitivity function technique for the analysis of noise-induced transitions between two coexisting limit cycles. This approach allows us to construct the dispersion ellipses of random trajectories for any Poincare sections. Possibilities of our descriptive-geometric method for a detailed analysis of noise-induced transitions between two periodic attractors of Lorenz model are demonstrated.
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28
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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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29
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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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30
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Breban R, Vardavas R, Blower S. Mean-field analysis of an inductive reasoning game: application to influenza vaccination. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:031127. [PMID: 17930219 DOI: 10.1103/physreve.76.031127] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2007] [Revised: 06/08/2007] [Indexed: 05/23/2023]
Abstract
Recently we have introduced an inductive reasoning game of voluntary yearly vaccination to establish whether or not a population of individuals acting in their own self-interest would be able to prevent influenza epidemics. Here, we analyze our model to describe the dynamics of the collective yearly vaccination uptake. We discuss the mean-field equations of our model and first order effects of fluctuations. We explain why our model predicts that severe epidemics are periodically expected even without the introduction of pandemic strains. We find that fluctuations in the collective yearly vaccination uptake induce severe epidemics with an expected periodicity that depends on the number of independent decision makers in the population. The mean-field dynamics also reveal that there are conditions for which the dynamics become robust to the fluctuations. However, the transition between fluctuation-sensitive and fluctuation-robust dynamics occurs for biologically implausible parameters. We also analyze our model when incentive-based vaccination programs are offered. When a family-based incentive is offered, the expected periodicity of severe epidemics is increased. This results from the fact that the number of independent decision makers is reduced, increasing the effect of the fluctuations. However, incentives based on the number of years of prepayment of vaccination may yield fluctuation-robust dynamics where severe epidemics are prevented. In this case, depending on prepayment, the transition between fluctuation-sensitive and fluctuation-robust dynamics may occur for biologically plausible parameters. Our analysis provides a practical method for identifying how many years of free vaccination should be provided in order to successfully ameliorate influenza epidemics.
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Affiliation(s)
- Romulus Breban
- Semel Institute for Neuroscience and Human Behavior, David Geffen School of Medicine, University of California, Los Angeles, California 90095-1555, USA
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31
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Guan S, Lai CH, Wei GW. Bistable chaos without symmetry in generalized synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:036209. [PMID: 15903548 DOI: 10.1103/physreve.71.036209] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2004] [Revised: 06/23/2004] [Indexed: 05/02/2023]
Abstract
Frequently, multistable chaos is found in dynamical systems with symmetry. We demonstrate a rare example of bistable chaos in generalized synchronization (GS) in coupled chaotic systems without symmetry. Bistable chaos in GS refers to two chaotic attractors in the response system which both synchronize with the driving dynamics in the sense of GS. By choosing appropriate coupling, the coupled system could be symmetric or asymmetric. Interestingly, it is found that the response system exhibits bistability in both cases. Three different types of bistable chaos have been identified. The crisis bifurcations which lead to the bistability are explored, and the relation between the bistable attractors is analyzed. The basin of attraction of the bistable attractors is extensively studied in both parameter space and initial condition space. The fractal basin boundary and the riddled basin are observed and they are characterized in terms of the uncertainty exponent.
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Affiliation(s)
- Shuguang Guan
- Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, 117508 Singapore
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32
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Rech PC, Beims MW, Gallas JAC. Basin size evolution between dissipative and conservative limits. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:017202. [PMID: 15697773 DOI: 10.1103/physreve.71.017202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2004] [Revised: 10/07/2004] [Indexed: 05/24/2023]
Abstract
Recent methods for stabilizing systems like, e.g., loss-modulated CO2 lasers, involve inducing controlled monostability via slow parameter modulations. However, such stabilization methods presuppose detailed knowledge of the structure and size of basins of attraction. In this Brief Report, we numerically investigate basin size evolution when parameters are varied between dissipative and conservative limits. Basin volumes shrink fast as the conservative limit is approached, being well approximated by Gaussian profiles, independently of the period. Basin shrinkage and vanishing is due to the absence of bounded motions in the Hamiltonian limit. In addition, we find basin volume to remain essentially constant along a peculiar parameter path along which it is possible to recover the dissipation rate solely from metric properties of self-similar structures in phase-space.
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Affiliation(s)
- Paulo Cesar Rech
- Departamento de Física, Universidade do Estado de Santa Catarina, 89223-100 Joinville, Brazil.
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33
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Kraut S, Grebogi C. Escaping from nonhyperbolic chaotic attractors. PHYSICAL REVIEW LETTERS 2004; 92:234101. [PMID: 15245159 DOI: 10.1103/physrevlett.92.234101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2004] [Indexed: 05/24/2023]
Abstract
The noise-induced escape process from a nonhyperbolic chaotic attractor is of physical and fundamental importance. We address this problem by uncovering the general mechanism of escape in the relevant low noise limit using the Hamiltonian theory of large fluctuations and by establishing the crucial role of the primary homoclinic tangency closest to the basin boundary in the dynamical process. In order to demonstrate that, we provide an unambiguous solution of the variational equations from the Hamiltonian theory. Our results are substantiated with the help of physical and dynamical paradigms, such as the Hénon and the Ikeda maps. It is further pointed out that our findings should be valid for driven flow systems and for experimental data.
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Affiliation(s)
- Suso Kraut
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 Sao Paulo, Brazil
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