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Fan H, Lai YC, Wang X. Enhancing network synchronization by phase modulation. Phys Rev E 2018; 98:012212. [PMID: 30110721 DOI: 10.1103/physreve.98.012212] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2018] [Indexed: 11/07/2022]
Abstract
Due to time delays in signal transmission and processing, phase lags are inevitable in realistic complex oscillator networks. Conventional wisdom is that phase lags are detrimental to network synchronization. Here we show that judiciously chosen phase lag modulations can result in significantly enhanced network synchronization. We justify our strategy of phase modulation, demonstrate its power in facilitating and enhancing network synchronization with synthetic and empirical network models, and provide an analytic understanding of the underlying mechanism. Our work provides an alternative approach to synchronization optimization in complex networks, with insights into control of complex nonlinear networks.
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Affiliation(s)
- Huawei Fan
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Ying-Cheng Lai
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China.,School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
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2
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Martínez PJ, Euzzor S, Gallas JAC, Meucci R, Chacón R. Identification of minimal parameters for optimal suppression of chaos in dissipative driven systems. Sci Rep 2017; 7:17988. [PMID: 29269736 PMCID: PMC5740119 DOI: 10.1038/s41598-017-17969-9] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2017] [Accepted: 11/20/2017] [Indexed: 11/09/2022] Open
Abstract
Taming chaos arising from dissipative non-autonomous nonlinear systems by applying additional harmonic excitations is a reliable and widely used procedure nowadays. But the suppressory effectiveness of generic non-harmonic periodic excitations continues to be a significant challenge both to our theoretical understanding and in practical applications. Here we show how the effectiveness of generic suppressory excitations is optimally enhanced when the impulse transmitted by them (time integral over two consecutive zeros) is judiciously controlled in a not obvious way. Specifically, the effective amplitude of the suppressory excitation is minimal when the impulse transmitted is maximum. Also, by lowering the impulse transmitted one obtains larger regularization areas in the initial phase difference-amplitude control plane, the price to be paid being the requirement of larger amplitudes. These two remarkable features, which constitute our definition of optimum control, are demonstrated experimentally by means of an analog version of a paradigmatic model, and confirmed numerically by simulations of such a damped driven system including the presence of noise. Our theoretical analysis shows that the controlling effect of varying the impulse is due to a subsequent variation of the energy transmitted by the suppressory excitation.
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Affiliation(s)
- Pedro J Martínez
- Departamento de Física Aplicada, E.I.N.A., Universidad de Zaragoza, E-50018, Zaragoza, Spain. .,Instituto de Ciencia de Materiales de Aragón, CSIC-Universidad de Zaragoza, E-50009, Zaragoza, Spain.
| | - Stefano Euzzor
- Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, Firenze, Italy
| | - Jason A C Gallas
- Departamento de Física, Universidade Federal da Paraíba, 58051-970, Joao Pessoa, Brazil
| | - Riccardo Meucci
- Departamento de Física, Universidade Federal da Paraíba, 58051-970, Joao Pessoa, Brazil
| | - Ricardo Chacón
- Departamento de Física Aplicada, E.I.I., Universidad de Extremadura, Apartado Postal 382, E-06006, Badajoz, Spain.,Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06006, Badajoz, Spain
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3
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da Silva RM, Manchein C, Beims MW. Controlling intermediate dynamics in a family of quadratic maps. CHAOS (WOODBURY, N.Y.) 2017; 27:103101. [PMID: 29092453 DOI: 10.1063/1.4985331] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The intermediate dynamics of composed one-dimensional maps is used to multiply attractors in phase space and create multiple independent bifurcation diagrams which can split apart. Results are shown for the composition of k-paradigmatic quadratic maps with distinct values of parameters generating k-independent bifurcation diagrams with corresponding k orbital points. For specific conditions, the basic mechanism for creating the shifted diagrams is the prohibition of period doubling bifurcations transformed in saddle-node bifurcations.
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Affiliation(s)
- Rafael M da Silva
- Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, Paraná, Brazil
| | - Cesar Manchein
- Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, Santa Catarina, Brazil
| | - Marcus W Beims
- Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, Paraná, Brazil
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4
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Meucci R, Euzzor S, Pugliese E, Zambrano S, Gallas MR, Gallas JAC. Optimal Phase-Control Strategy for Damped-Driven Duffing Oscillators. PHYSICAL REVIEW LETTERS 2016; 116:044101. [PMID: 26871335 DOI: 10.1103/physrevlett.116.044101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2015] [Indexed: 06/05/2023]
Abstract
Phase-control techniques of chaos aim to extract periodic behaviors from chaotic systems by applying weak harmonic perturbations with a suitably chosen phase. However, little is known about the best strategy for selecting adequate perturbations to reach desired states. Here we use experimental measures and numerical simulations to assess the benefits of controlling individually the three terms of a Duffing oscillator. Using a real-time analog indicator able to discriminate on-the-fly periodic behaviors from chaos, we reconstruct experimentally the phase versus perturbation strength stability areas when periodic perturbations are applied to different terms governing the oscillator. We verify the system to be more sensitive to perturbations applied to the quadratic term of the double-well Duffing oscillator and to the quartic term of the single-well Duffing oscillator.
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Affiliation(s)
- R Meucci
- Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, Firenze, Italy
- Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Brazil
- Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil
| | - S Euzzor
- Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, Firenze, Italy
| | - E Pugliese
- Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, Firenze, Italy
- Dipartimento di Scienze della Terra, Università degli Studi di Firenze, Via G. La Pira 4, Firenze, Italy
| | - S Zambrano
- Università Vita-Salute San Raffaele, Via Olgettina 58, 20132 Milano, Italy
| | - M R Gallas
- Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, Firenze, Italy
- Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Brazil
- Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil
| | - J A C Gallas
- Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, Firenze, Italy
- Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Brazil
- Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil
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5
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Yang W, Lin W, Wang X, Huang L. Synchronization of networked chaotic oscillators under external periodic driving. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032912. [PMID: 25871177 DOI: 10.1103/physreve.91.032912] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2014] [Indexed: 06/04/2023]
Abstract
The dynamical responses of a complex system to external perturbations are of both fundamental interest and practical significance. Here, by the model of networked chaotic oscillators, we investigate how the synchronization behavior of a complex network is influenced by an externally added periodic driving. Interestingly, it is found that by a slight change of the properties of the external driving, e.g., the frequency or phase lag between its intrinsic oscillation and external driving, the network synchronizability could be significantly modified. We demonstrate this phenomenon by different network models and, based on the method of master stability function, give an analysis on the underlying mechanisms. Our studies highlight the importance of external perturbations on the collective behaviors of complex networks, and also provide an alternate approach for controlling network synchronization.
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Affiliation(s)
- Wenchao Yang
- Institute of Computational Physics and Complex Systems and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Weijie Lin
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
- Department of Physics, Zhejiang University, Hangzhou 310027, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Liang Huang
- Institute of Computational Physics and Complex Systems and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
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Sarmah R, Ananthakrishna G. A unified model for the dynamics of driven ribbon with strain and magnetic order parameters. CHAOS (WOODBURY, N.Y.) 2013; 23:013116. [PMID: 23556953 DOI: 10.1063/1.4790845] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
We develop a unified model to explain the dynamics of driven one dimensional ribbon for materials with strain and magnetic order parameters. We show that the model equations in their most general form explain several results on driven magnetostrictive metallic glass ribbons such as the period doubling route to chaos as a function of a dc magnetic field in the presence of a sinusoidal field, the quasiperiodic route to chaos as a function of the sinusoidal field for a fixed dc field, and induced and suppressed chaos in the presence of an additional low amplitude near resonant sinusoidal field. We also investigate the influence of a low amplitude near resonant field on the period doubling route. The model equations also exhibit symmetry restoring crisis with an exponent close to unity. The model can be adopted to explain certain results on magnetoelastic beam and martensitic ribbon under sinusoidal driving conditions. In the latter case, we find interesting dynamics of a periodic one orbit switching between two equivalent wells as a function of an ac magnetic field that eventually makes a direct transition to chaos under resonant driving condition. The model is also applicable to magnetomartensites and materials with two order parameters.
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Affiliation(s)
- Ritupan Sarmah
- Materials Research Centre, Indian Institute of Science, Bangalore 560012, India.
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7
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Chacón R, Martínez JA, Miralles JJ. Impulse-induced optimum control of escape from a metastable state by periodic secondary excitations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:066207. [PMID: 23005196 DOI: 10.1103/physreve.85.066207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2012] [Revised: 04/27/2012] [Indexed: 06/01/2023]
Abstract
We characterize the role of the impulse transmitted (time integral over a half-period) by resonant secondary excitations at controlling (suppressing and enhancing) escape from a potential well, which is induced by periodic primary excitations. By using the universal model of a dissipative Helmholtz oscillator, we demonstrate numerically that optimum control of escape occurs when the impulse transmitted by the chaos-controlling excitations is maximum while keeping their amplitude and period fixed. These findings are in complete agreement with analytical predictions from two independent methods: Melnikov analysis and energy-based analysis. Additional numerical results corresponding to other alternative escape-controlling excitations demonstrate the generality of the essential role of the excitation's impulse.
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Affiliation(s)
- R Chacón
- Departamento de Física Aplicada, Escuela de Ingenierías Industriales, Universidad de Extremadura, Apartado Postal 382, E-06006 Badajoz, Spain, EU
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8
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Yao C, Liu Y, Zhan M. Frequency-resonance-enhanced vibrational resonance in bistable systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061122. [PMID: 21797317 DOI: 10.1103/physreve.83.061122] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2011] [Revised: 05/14/2011] [Indexed: 05/31/2023]
Abstract
The dynamics in an overdamped bistable system subject to the action of two periodic forces (assuming their frequencies are ω and Ω, and amplitudes are A and B, respectively) is studied. For the usual vibrational resonance, the nonmonotonic dependence of signal output of the low frequency ω on the change of B for a fixed Ω, the condition Ω≫ω is always assumed in all previous studies. Here, removing this restriction, we find that a resonant behavior can extensively occur with respect to the changes of both the frequency Ω and amplitude B. Especially, the resonance becomes stronger when Ω is chosen such that it is exactly in frequency resonance with ω. This combinative behavior, called frequency-resonance-enhanced vibrational resonance, is of great interest and may shed an improved light on our understanding of the dynamics of nonlinear systems subject to a biharmonic force.
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Affiliation(s)
- Chenggui Yao
- Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China
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9
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Chacón R, Martínez JA. Controlling escape from a potential well by reshaping periodic secondary excitations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:016201. [PMID: 21405754 DOI: 10.1103/physreve.83.016201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2009] [Revised: 10/25/2010] [Indexed: 05/30/2023]
Abstract
The role of the wave form of periodic secondary excitations at controlling (suppressing and enhancing) escape from a potential well is investigated. We demonstrate analytically (by Melnikov analysis) and numerically that a judicious choice of the excitation's wave form greatly improves the effectiveness of the escape-controlling excitations while keeping their amplitude and period fixed. These predictions are confirmed by an energy-based analysis that provides the same optimal values of the escape-controlling parameters. The example of a dissipative Helmholtz oscillator is used to illustrate the accuracy of these results.
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Affiliation(s)
- R Chacón
- Departamento de Física Aplicada, Escuela de Ingenierías Industriales, Universidad de Extremadura, Apartado Postal 382, E-06071 Badajoz, Spain, EU
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10
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Sakthivel G, Rajasekar S. Diffusion dynamics and first passage time in a two-coupled pendulum system. CHAOS (WOODBURY, N.Y.) 2010; 20:033120. [PMID: 20887060 DOI: 10.1063/1.3486074] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
We present the numerical investigation of diffusion process and features of first passage time (FPT) and mean FPT (MFPT) in a two-coupled damped and periodically driven pendulum system. The effect of amplitude of the external periodic force and phase of the force on diffusion constant, distribution of FPT, P(tFPT), and MFPT is analyzed. Normal diffusion is found. Diffusion constant is found to show power-law variation near intermittency and sudden widening crises while linear variation is observed in the quasiperiodic region. In the intermittency crisis the divergence of diffusion constant is similar to the divergence of mean bursting length. P(tFPT) of critical distances of state variable exhibit periodic multiple peaks with decaying amplitude. MFPT of critical distances also follows power-law variation. Diffusion constant and MFPT are sensitive to the phase factor of the periodic force.
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Affiliation(s)
- G Sakthivel
- School of Physics, Bharathidasan University, Tiruchirapalli 620 024, India
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11
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Seoane JM, Zambrano S, Euzzor S, Meucci R, Arecchi FT, Sanjuán MAF. Avoiding escapes in open dynamical systems using phase control. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:016205. [PMID: 18764033 DOI: 10.1103/physreve.78.016205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2007] [Revised: 05/30/2008] [Indexed: 05/26/2023]
Abstract
In this paper we study how to avoid escapes in open dynamical systems in the presence of dissipation and forcing, as it occurs in realistic physical situations. We use as a prototype model the Helmholtz oscillator, which is the simplest nonlinear oscillator with escapes. For some parameter values, this oscillator presents a critical value of the forcing for which all particles escape from its single well. By using the phase control technique, weakly changing the shape of the potential via a periodic perturbation of suitable phase varphi , we avoid the escapes in different regions of the phase space. We provide numerical evidence, heuristic arguments, and an experimental implementation in an electronic circuit of this phenomenon. Finally, we expect that this method might be useful for avoiding escapes in more complicated physical situations.
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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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12
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Zambrano S, Mariño IP, Salvadori F, Meucci R, Sanjuán MAF, Arecchi FT. Phase control of intermittency in dynamical systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:016202. [PMID: 16907172 DOI: 10.1103/physreve.74.016202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2006] [Indexed: 05/11/2023]
Abstract
We present a nonfeedback method to tame or enhance crisis-induced intermittency in dynamical systems. By adding a small harmonic perturbation to a parameter of the system, the intermittent behavior can be suppressed or enhanced depending on the value of the phase difference between the main driving and the perturbation. The validity of the method is shown both in the model and in an experiment with a CO2 laser. An analysis of this scheme applied to the quadratic map near crisis illustrates the role of phase control in nonlinear dynamical systems.
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Affiliation(s)
- Samuel Zambrano
- Nonlinear Dynamics and Chaos Group, Departamento de Ciencias de la Naturaleza y Física Aplicada, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
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13
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Wang X, Lai YC, Lai CH. Effect of resonant-frequency mismatch on attractors. CHAOS (WOODBURY, N.Y.) 2006; 16:023127. [PMID: 16822030 DOI: 10.1063/1.2208566] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
Resonant perturbations are effective for harnessing nonlinear oscillators for various applications such as controlling chaos and inducing chaos. Of physical interest is the effect of small frequency mismatch on the attractors of the underlying dynamical systems. By utilizing a prototype of nonlinear oscillators, the periodically forced Duffing oscillator and its variant, we find a phenomenon: resonant-frequency mismatch can result in attractors that are nonchaotic but are apparently strange in the sense that they possess a negative Lyapunov exponent but its information dimension measured using finite numerics assumes a fractional value. We call such attractors pseudo-strange. The transition to pesudo-strange attractors as a system parameter changes can be understood analytically by regarding the system as nonstationary and using the Melnikov function. Our results imply that pseudo-strange attractors are common in nonstationary dynamical systems.
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Affiliation(s)
- Xingang Wang
- Department of Physics, National University of Singapore, Singapore 117542
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14
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Zambrano S, Allaria E, Brugioni S, Leyva I, Meucci R, Sanjuán MAF, Arecchi FT. Numerical and experimental exploration of phase control of chaos. CHAOS (WOODBURY, N.Y.) 2006; 16:013111. [PMID: 16599742 DOI: 10.1063/1.2161437] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
A well-known method to suppress chaos in a periodically forced chaotic system is to add a harmonic perturbation. The phase control of chaos scheme uses the phase difference between a small added harmonic perturbation and the main driving to suppress chaos, leading the system to different periodic orbits. Using the Duffing oscillator as a paradigm, we present here an in-depth study of this technique. A thorough numerical exploration has been made focused in the important role played by the phase, from which new interesting patterns in parameter space have appeared. On the other hand, our novel experimental implementation of phase control in an electronic circuit confirms both the well-known features of this method and the new ones detected numerically. All this may help in future implementations of phase control of chaos, which is globally confirmed here to be robust and easy to implement experimentally.
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Affiliation(s)
- Samuel Zambrano
- Nonlinear Dynamics and Chaos Group, Departamento de Matemáticas y Física Aplicadas y Ciencias de la Naturaleza, Universidad Rey Juan Carlos, Tulipán, s/n, 28933 Móstoles, Madrid, Spain
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15
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Kouomou YC, Colet P, Larger L, Gastaud N. Chaotic breathers in delayed electro-optical systems. PHYSICAL REVIEW LETTERS 2005; 95:203903. [PMID: 16384058 DOI: 10.1103/physrevlett.95.203903] [Citation(s) in RCA: 60] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2004] [Indexed: 05/05/2023]
Abstract
We show that in integro-differential delayed dynamical systems, a hybrid state of simultaneous fast-scale chaos and slow-scale periodicity can emerge subsequently to a sequence of Hopf bifurcations. The resulting time trace thereby consists in chaotic oscillations "breathing" periodically at a significantly lower frequency. Experimental evidence of this type of dynamics in delayed dynamical systems is achieved with a Mach-Zehnder modulator optically fed by a semiconductor laser and is subjected to a delayed nonlinear electro-optical feedback. We also propose a theoretical understanding of the phenomenon.
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Affiliation(s)
- Y Chembo Kouomou
- Instituto Mediterráneo de Estudios Avanzados IMEDEA (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain.
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16
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Lai YC, Kandangath A, Krishnamoorthy S, Gaudet JA, de Moura APS. Inducing chaos by resonant perturbations: theory and experiment. PHYSICAL REVIEW LETTERS 2005; 94:214101. [PMID: 16090322 DOI: 10.1103/physrevlett.94.214101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2004] [Indexed: 05/03/2023]
Abstract
We propose a scheme to induce chaos in nonlinear oscillators that either are by themselves incapable of exhibiting chaos or are far away from parameter regions of chaotic behaviors. Our idea is to make use of small, judiciously chosen perturbations in the form of weak periodic signals with time-varying frequency and phase, and to drive the system into a hierarchy of nonlinear resonant states and eventually into chaos. We demonstrate this method by using numerical examples and a laboratory experiment with a Duffing type of electronic circuit driven by a phase-locked loop. The phase-locked loop can track the instantaneous frequency and phase of the Duffing circuit and deliver resonant perturbations to generate robust chaos.
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Affiliation(s)
- Ying-Cheng Lai
- Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA
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17
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Chizhevsky VN, Corbalán R. Phase scaling properties of perturbation-induced multistability in a driven nonlinear system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:016201. [PMID: 12241455 DOI: 10.1103/physreve.66.016201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2001] [Indexed: 05/23/2023]
Abstract
Phase scaling relations for the onsets of both saddle-node bifurcations and boundary crises induced by resonant periodic perturbations at subharmonic frequencies are found in a period-doubled system from the results of numerical simulation. These phase dependences determine the domains of existence of induced attractors in (bifurcation parameter, perturbation phase) parameter space. The overlapping of these domains leads to the formation of zones with different numbers of coexisting attractors. The numerical evidence was obtained on the basis of single-mode rate equations of a laser with parameters corresponding to a realistic loss-modulated CO2 laser.
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Affiliation(s)
- V N Chizhevsky
- B. I. Stepanov Institute of Physics, National Academy of Science of Belarus, 220072 Minsk, Belarus.
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18
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Xu D, Li Z, Bishop SR. Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems. CHAOS (WOODBURY, N.Y.) 2001; 11:439-442. [PMID: 12779480 DOI: 10.1063/1.1380370] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Scaling factor characterizes the synchronized dynamics of projective synchronization in partially linear chaotic systems but it is difficult to be estimated. To manipulate projective synchronization of chaotic systems in a favored way, a control algorithm is introduced to direct the scaling factor onto a desired value. The control approach is derived from the Lyapunov stability theory. It allows us to arbitrarily amplify or reduce the scale of the response of the slave system via a feedback control on the master system. In numerical experiments, we illustrate the application to the Lorenz system. (c) 2001 American Institute of Physics.
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Affiliation(s)
- Daolin Xu
- Center for Advanced Numerical Engineering Simulation, School of Mechanical and Production Engineering, Nanyang Technological University, 639798 Singapore, Singapore
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19
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Zhan M, Hu G, Zhang Y, He D. Generalized splay state in coupled chaotic oscillators induced by weak mutual resonant interactions. PHYSICAL REVIEW LETTERS 2001; 86:1510-1513. [PMID: 11290180 DOI: 10.1103/physrevlett.86.1510] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2000] [Indexed: 05/23/2023]
Abstract
Dynamic behavior of coupled chaotic oscillators is investigated. A transition from high-dimensional hyperchaos to a generalized periodic splay state is found for extremely weak coupling. Chaotic nature of a single oscillator and mutual resonant interactions are regarded to be responsible for this self-organized ordering. The functional phase distribution of the generalized splay state, which is essentially different from the equal-phase-separation distribution of the conventional splay states, can be well predicted by analyzing a single periodically forced oscillator.
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Affiliation(s)
- M Zhan
- Department of Physics, Beijing Normal University, China
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20
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Mirus KA, Sprott JC. Controlling chaos in low- and high-dimensional systems with periodic parametric perturbations. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 59:5313-24. [PMID: 11969491 DOI: 10.1103/physreve.59.5313] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/1998] [Indexed: 04/18/2023]
Abstract
The effect of applying a periodic perturbation to an accessible parameter of various chaotic systems is examined. Numerical results indicate that perturbation frequencies near the natural frequencies of the unstable periodic orbits of the chaotic systems can result in limit cycles for relatively small perturbations. Such perturbations can also control or significantly reduce the dimension of high-dimensional systems. Initial application to the control of fluctuations in a prototypical magnetic fusion plasma device will be reviewed.
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Affiliation(s)
- K A Mirus
- Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
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Chizhevsky VN, Corbalán R. Experimental observation of perturbation-induced intermittency in the dynamics of a loss-modulated CO2 laser. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4576-4579. [PMID: 9965631 DOI: 10.1103/physreve.54.4576] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Xie R, Xu G. Dynamical features of a classically chaotic quantum system: Symmetry breaking and the disappearance of the squeezing effect. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:1402-1405. [PMID: 9965210 DOI: 10.1103/physreve.54.1402] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Li HJ, Chern JL. Goal-oriented scheme for taming chaos with a weak periodic perturbation: Experiment in a diode resonator. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:2118-2121. [PMID: 9965294 DOI: 10.1103/physreve.54.2118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Chacón R. Geometrical Resonance as a Chaos Eliminating Mechanism. PHYSICAL REVIEW LETTERS 1996; 77:482-485. [PMID: 10062822 DOI: 10.1103/physrevlett.77.482] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Vilaseca R, Kul'minskii A, Corbalán R. Tracking unstable steady states by large periodic modulation of a control parameter in a nonlinear system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:82-85. [PMID: 9965049 DOI: 10.1103/physreve.54.82] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Colet P, Braiman Y. Control of chaos in multimode solid state lasers by the use of small periodic perturbations. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:200-206. [PMID: 9964249 DOI: 10.1103/physreve.53.200] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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