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Tokuda IT, Levnajic Z, Ishimura K. A practical method for estimating coupling functions in complex dynamical systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190015. [PMID: 31656141 PMCID: PMC6833996 DOI: 10.1098/rsta.2019.0015] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/02/2019] [Indexed: 06/10/2023]
Abstract
A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
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Affiliation(s)
- Isao T. Tokuda
- Department of Mechanical Engineering, Ritsumeikan University, Kusatsu, Japan
| | - Zoran Levnajic
- Complex Systems and Data Science Lab, Faculty of Information Studies in Novo Mesto, Novo Mesto, Slovenia
| | - Kazuyoshi Ishimura
- Department of Mechanical Engineering, Ritsumeikan University, Kusatsu, Japan
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Sebek M, Kiss IZ. Spatiotemporal Patterns on a Ring Network of Oscillatory Electrochemical Reaction with Negative Global Feedback. Isr J Chem 2018. [DOI: 10.1002/ijch.201700133] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Affiliation(s)
- Michael Sebek
- Department of Chemistry Saint Louis University 3501 Laclede Ave. St. Louis MO 63103 USA
| | - István Z. Kiss
- Department of Chemistry Saint Louis University 3501 Laclede Ave. St. Louis MO 63103 USA
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Ferreira MT, Follmann R, Domingues MO, Macau EEN, Kiss IZ. Experimental phase synchronization detection in non-phase coherent chaotic systems by using the discrete complex wavelet approach. CHAOS (WOODBURY, N.Y.) 2017; 27:083122. [PMID: 28863491 DOI: 10.1063/1.4999908] [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
Phase synchronization may emerge from mutually interacting non-linear oscillators, even under weak coupling, when phase differences are bounded, while amplitudes remain uncorrelated. However, the detection of this phenomenon can be a challenging problem to tackle. In this work, we apply the Discrete Complex Wavelet Approach (DCWA) for phase assignment, considering signals from coupled chaotic systems and experimental data. The DCWA is based on the Dual-Tree Complex Wavelet Transform (DT-CWT), which is a discrete transformation. Due to its multi-scale properties in the context of phase characterization, it is possible to obtain very good results from scalar time series, even with non-phase-coherent chaotic systems without state space reconstruction or pre-processing. The method correctly predicts the phase synchronization for a chemical experiment with three locally coupled, non-phase-coherent chaotic processes. The impact of different time-scales is demonstrated on the synchronization process that outlines the advantages of DCWA for analysis of experimental data.
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Affiliation(s)
- Maria Teodora Ferreira
- Associated Laboratory for Computing and Applied Mathematics (LAC), Brazilian National Institute for Space Research (INPE), São José dos Campos 12227-010, Brazil
| | - Rosangela Follmann
- Associated Laboratory for Computing and Applied Mathematics (LAC), Brazilian National Institute for Space Research (INPE), São José dos Campos 12227-010, Brazil
| | - Margarete O Domingues
- Associated Laboratory for Computing and Applied Mathematics (LAC), Brazilian National Institute for Space Research (INPE), São José dos Campos 12227-010, Brazil
| | - Elbert E N Macau
- Associated Laboratory for Computing and Applied Mathematics (LAC), Brazilian National Institute for Space Research (INPE), São José dos Campos 12227-010, Brazil
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA
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Wang H, Han W, Yang J. Synchronous dynamics in the Kuramoto model with biharmonic interaction and bimodal frequency distribution. Phys Rev E 2017; 96:022202. [PMID: 28950468 DOI: 10.1103/physreve.96.022202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2017] [Indexed: 06/07/2023]
Abstract
In this work, we study the Kuramoto model with biharmonic interaction and bimodal frequency distribution. Rich synchronous dynamics, such as standing wave states, stationary partial synchronous dynamics, and multiplicity of singular synchronous dynamics, are found. Notably, we find a symmetry-breaking synchronous dynamics when the peaks in frequency distribution are not well separated. We present the phase diagrams for two cases: the peaks in the frequency distribution are well separated and the peaks are not well separated. We find that reducing peak distance tends to make the transition between standing wave states and stationary partial synchronous states to be continuous when the multiplicity of singular synchronous state is present or to be discontinuous when the multiplicity is absent.
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Affiliation(s)
- Huobin Wang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Wenchen Han
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
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Burylko O, Kazanovich Y, Borisyuk R. Bifurcation study of phase oscillator systems with attractive and repulsive interaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022911. [PMID: 25215803 DOI: 10.1103/physreve.90.022911] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2014] [Indexed: 06/03/2023]
Abstract
We study a model of globally coupled phase oscillators that contains two groups of oscillators with positive (synchronizing) and negative (desynchronizing) incoming connections for the first and second groups, respectively. This model was previously studied by Hong and Strogatz (the Hong-Strogatz model) in the case of a large number of oscillators. We consider a generalized Hong-Strogatz model with a constant phase shift in coupling. Our approach is based on the study of invariant manifolds and bifurcation analysis of the system. In the case of zero phase shift, various invariant manifolds are analytically described and a new dynamical mode is found. In the case of a nonzero phase shift we obtained a set of bifurcation diagrams for various systems with three or four oscillators. It is shown that in these cases system dynamics can be complex enough and include multistability and chaotic oscillations.
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Affiliation(s)
- Oleksandr Burylko
- Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Strasse 3, 01601 Kyiv, Ukraine
| | - Yakov Kazanovich
- Institute of Mathematical Problems of Biology, Russian Academy of Sciences, 142290 Pushchino, Russia
| | - Roman Borisyuk
- Institute of Mathematical Problems of Biology, Russian Academy of Sciences, 142290 Pushchino, Russia and School of Computing and Mathematics, Plymouth University, Plymouth PL4 8AA, United Kingdom
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Ju P, Dai Q, Cheng H, Yang J. Dynamics in the Sakaguchi-Kuramoto model with two subpopulations [corrected]. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012903. [PMID: 25122356 DOI: 10.1103/physreve.90.012903] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2014] [Indexed: 06/03/2023]
Abstract
The dynamics in a variant of globally coupled Sakaguchi-Kuramoto [corrected]. phase oscillators is studied. The model consists of two subpopulations, each with a different phase lag and interaction strength. Using Ott-Antonson ansatz, we analyze the dynamics in the model and present the numerical results. There exist stationary synchronous states which are generalized π states and two types of traveling wave states. We find that the traveling wave states are the dominant dynamics in comparison with the stationary states. Particularly, we find that the stationary and traveling wave states can be smoothly connected through the properly chosen parameter paths.
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Affiliation(s)
- Ping Ju
- School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People's Republic of China
| | - Qionglin Dai
- School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People's Republic of China
| | - Hongyan Cheng
- School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People's Republic of China
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People's Republic of China
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Li K, Ma S, Li H, Yang J. Transition to synchronization in a Kuramoto model with the first- and second-order interaction terms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032917. [PMID: 24730923 DOI: 10.1103/physreve.89.032917] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2013] [Indexed: 06/03/2023]
Abstract
We investigate a Kuramoto model incorporated with the first-order and the second-order interaction terms. We show that the model displays the coexistence of multiattractors and different attractors may be characterized by the phase distributions of oscillators. By investigating the transition diagrams in both forward continuation and backward continuation, we find that the synchronous state with unimodal phase distribution is the most stable one while the state in cluster synchrony with evenly distributed bimodal phase distribution is the least stable one. We also present the phase diagram of the model in the parameter space.
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Affiliation(s)
- Keren Li
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Shen Ma
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Haihong Li
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
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Yuan D, Zhang M, Yang J. Dynamics of the Kuramoto model in the presence of correlation between distributions of frequencies and coupling strengths. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012910. [PMID: 24580300 DOI: 10.1103/physreve.89.012910] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2013] [Indexed: 06/03/2023]
Abstract
As a paradigmatic model, the Kuramoto model has provided a platform for investigating synchronization among nonidentical oscillators. In this work, we consider the Kuramoto model consisting of conformists with positive coupling strength and contrarians with negative coupling strength. We introduce the correlation between the distributions of natural frequencies and the coupling strengths of oscillators. Three different types of correlations are considered. We find rich dynamics result from the correlation such as different types of traveling wave states and, most interestingly, another type of nonstationary state: an oscillating π state.
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Affiliation(s)
- Di Yuan
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China and School of Physics and Electrical Engineering, Anyang Normal University, Anyang 455000, People's Republic of China
| | - Mei Zhang
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
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Astakhov S, Fujiwara N, Gulay A, Tsukamoto N, Kurths J. Hopf bifurcation and multistability in a system of phase oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032908. [PMID: 24125326 DOI: 10.1103/physreve.88.032908] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2012] [Revised: 04/26/2013] [Indexed: 06/02/2023]
Abstract
We study the phase reduction of two coupled van der Pol oscillators with asymmetric repulsive coupling under an external harmonic force. We show that the system of two phase oscillators undergoes a Hopf bifurcation and possesses multistability on a 2π-periodic phase plane. We describe the bifurcation mechanisms of formation of multistability in the phase-reduced system and show that the Andronov-Hopf bifurcation in the phase-reduced system is not an artifact of the reduction approach but, indeed, has its prototype in the nonreduced system. The bifurcational mechanisms presented in the paper enable one to describe synchronization effects in a wide class of interacting systems with repulsive coupling e.g., genetic oscillators.
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Affiliation(s)
- Sergey Astakhov
- Physics Department, Chair of Radiophysics and Nonlinear Dynamics, Saratov State University, 410012 Saratov, Russia
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Kralemann B, Pikovsky A, Rosenblum M. Detecting triplet locking by triplet synchronization indices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052904. [PMID: 23767595 DOI: 10.1103/physreve.87.052904] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2013] [Indexed: 06/02/2023]
Abstract
We discuss the effect of triplet synchrony in oscillatory networks. In this state the phases and the frequencies of three coupled oscillators fulfill the conditions of a triplet locking, whereas every pair of systems remains asynchronous. We suggest an easy to compute measure, a triplet synchronization index, which can be used to detect such states from experimental data.
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Affiliation(s)
- Björn Kralemann
- Institut für Pädagogik, Christian-Albrechts-Universität zu Kiel, Olshausenstrasse 75, 24118 Kiel, Germany
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Hong H, Strogatz SH. Mean-field behavior in coupled oscillators with attractive and repulsive interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:056210. [PMID: 23004846 DOI: 10.1103/physreve.85.056210] [Citation(s) in RCA: 36] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2011] [Indexed: 05/20/2023]
Abstract
We consider a variant of the Kuramoto model of coupled oscillators in which both attractive and repulsive pairwise interactions are allowed. The sign of the coupling is assumed to be a characteristic of a given oscillator. Specifically, some oscillators repel all the others, thus favoring an antiphase relationship with them. Other oscillators attract all the others, thus favoring an in-phase relationship. The Ott-Antonsen ansatz is used to derive the exact low-dimensional dynamics governing the system's long-term macroscopic behavior. The resulting analytical predictions agree with simulations of the full system. We explore the effects of changing various parameters, such as the width of the distribution of natural frequencies and the relative strengths and proportions of the positive and negative interactions. For the particular model studied here we find, unexpectedly, that the mixed interactions produce no new effects. The system exhibits conventional mean-field behavior and displays a second-order phase transition like that found in the original Kuramoto model. In contrast to our recent study of a different model with mixed interactions [Phys. Rev. Lett. 106, 054102 (2011)], the π state and traveling-wave state do not appear for the coupling type considered here.
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Affiliation(s)
- Hyunsuk Hong
- Department of Physics, Chonbuk National University, Jeonju, Korea
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Astakhov S, Feoktistov A, Anishchenko VS, Kurths J. Synchronization of multi-frequency noise-induced oscillations. CHAOS (WOODBURY, N.Y.) 2011; 21:047513. [PMID: 22225387 DOI: 10.1063/1.3659281] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.
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Lehnertz K. Assessing directed interactions from neurophysiological signals--an overview. Physiol Meas 2011; 32:1715-24. [PMID: 22027099 DOI: 10.1088/0967-3334/32/11/r01] [Citation(s) in RCA: 47] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/25/2022]
Abstract
The study of synchronization phenomena in coupled dynamical systems is an active field of research in many scientific disciplines including the neurosciences. Over the last decades, a number of time series analysis techniques have been proposed to capture both linear and nonlinear aspects of interactions. While most of these techniques allow one to quantify the strength of interactions, developments that resulted from advances in nonlinear dynamics and in information and synchronization theory aim at assessing directed interactions. Most of these techniques, however, assume the underlying systems to be at least approximately stationary and require a large number of data points to robustly assess directed interactions. Recent extensions allow assessing directed interactions from short and transient signals and are particularly suited for the analysis of evoked and event-related activity.
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Affiliation(s)
- Klaus Lehnertz
- Department of Epileptology, University of Bonn, Sigmund-Freud-Strasse 25, Bonn, Germany.
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Hong H, Strogatz SH. Conformists and contrarians in a Kuramoto model with identical natural frequencies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:046202. [PMID: 22181240 DOI: 10.1103/physreve.84.046202] [Citation(s) in RCA: 68] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2011] [Indexed: 05/20/2023]
Abstract
We consider a variant of the Kuramoto model in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These contrarian oscillators tend to align in antiphase with the mean field, whereas, the positively coupled conformist oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.
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Affiliation(s)
- Hyunsuk Hong
- Department of Physics, Research Institute of Physics and Chemistry, Chonbuk National University, Jeonju 561-756, Korea
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Wickramasinghe M, Mrugacz EM, Kiss IZ. Dynamics of electrochemical oscillators with electrode size disparity: asymmetrical coupling and anomalous phase synchronization. Phys Chem Chem Phys 2011; 13:15483-91. [PMID: 21808800 DOI: 10.1039/c1cp21429b] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Experiments are carried out in dual electrode oscillatory Ni electrodissolution in which the two electrodes have different surface areas. The transition to phase synchronization is analyzed as asymmetrical coupling strength, induced by placing a cross resistance between the electrodes, is varied. It is shown that because of nonisochronicity (phase shear, i.e., strong dependence of period on amplitude) of the oscillators, anomalous phase synchronization effects can be observed: advanced/delayed synchronization and, to a lesser extent, frequency difference enhancement. The type of synchronization is strongly affected by the underlying heterogeneities of the oscillators; in the experiments with a slow driver (large surface area) electrode the synchronization is advanced, with a fast driver electrode the synchronization is delayed with respect to symmetrical coupling. The findings thus reveal that the interplay of asymmetrical coupling with the types of inherent heterogeneities plays an important role for the interpretation of size effects in the dynamical behavior of a nonlinear chemical reaction.
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Affiliation(s)
- Mahesh Wickramasinghe
- Department of Chemistry, Saint Louis University, 3501 Laclede Ave., St. Louis, MO 63103, USA
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