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Kemeth FP, Haugland SW, Krischer K. Cluster singularity: The unfolding of clustering behavior in globally coupled Stuart-Landau oscillators. CHAOS (WOODBURY, N.Y.) 2019; 29:023107. [PMID: 30823729 DOI: 10.1063/1.5055839] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2018] [Accepted: 01/17/2019] [Indexed: 06/09/2023]
Abstract
The ubiquitous occurrence of cluster patterns in nature still lacks a comprehensive understanding. It is known that the dynamics of many such natural systems is captured by ensembles of Stuart-Landau oscillators. Here, we investigate clustering dynamics in a mean-coupled ensemble of such limit-cycle oscillators. In particular, we show how clustering occurs in minimal networks and elaborate how the observed 2-cluster states crowd when increasing the number of oscillators. Using persistence, we discuss how this crowding leads to a continuous transition from balanced cluster states to synchronized solutions via the intermediate unbalanced 2-cluster states. These cascade-like transitions emerge from what we call a cluster singularity. At this codimension-2 point, the bifurcations of all 2-cluster states collapse and the stable balanced cluster state bifurcates into the synchronized solution supercritically. We confirm our results using numerical simulations and discuss how our conclusions apply to spatially extended systems.
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Affiliation(s)
- Felix P Kemeth
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Str. 1, D-85748 Garching, Germany
| | - Sindre W Haugland
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Str. 1, D-85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Str. 1, D-85748 Garching, Germany
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Liu Y, Sebek M, Mori F, Kiss IZ. Synchronization of three electrochemical oscillators: From local to global coupling. CHAOS (WOODBURY, N.Y.) 2018; 28:045104. [PMID: 31906643 DOI: 10.1063/1.5012520] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We investigate the formation of synchronization patterns in an oscillatory nickel electrodissolution system in a network obtained by superimposing local and global coupling with three electrodes. We explored the behavior through numerical simulations using kinetic ordinary differential equations, Kuramoto type phase models, and experiments, in which the local to global coupling could be tuned by cross resistances between the three nickel wires. At intermediate coupling strength with predominant global coupling, two of the three oscillators, whose natural frequencies are closer, can synchronize. By adding even a relatively small amount of local coupling (about 9%-25%), a spatially organized partially synchronized state can occur where one of the two synchronized elements is in the center. A formula was derived for predicting the critical coupling strength at which full synchronization will occur independent of the permutation of the natural frequencies of the oscillators over the network. The formula correctly predicts the variation of the critical coupling strength as a function of the global coupling fraction, e.g., with local coupling the critical coupling strength is about twice than that required with global coupling. The results show the importance of the topology of the network on the synchronization properties in a simple three-oscillator setup and could provide guidelines for decrypting coupling topology from identification of synchronization patterns.
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Affiliation(s)
- Yifan Liu
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
| | - Michael Sebek
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
| | - Fumito Mori
- Theoretical Biology Laboratory, RIKEN, Wako 351-0198, Japan
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
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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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Leiser RJ, Rotstein HG. Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity. Phys Rev E 2017; 96:022303. [PMID: 28950537 DOI: 10.1103/physreve.96.022303] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2017] [Indexed: 11/07/2022]
Abstract
Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.
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Affiliation(s)
- Randolph J Leiser
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
| | - Horacio G Rotstein
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA.,Institute for Brain and Neuroscience Research, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
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Biswas A, Das D, Parmananda P. Scaling dependence and synchronization of forced mercury beating heart systems. Phys Rev E 2017; 95:042202. [PMID: 28505734 DOI: 10.1103/physreve.95.042202] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2016] [Indexed: 11/07/2022]
Abstract
We perform experiments on a nonautonomous Mercury beating heart system, which is forced to pulsate using an external square wave potential. At suitable frequencies and volumes, the drop exhibits pulsation with polygonal shapes having n corners. We find the scaling dependence of the forcing frequency ν_{n} on the volume V of the drop and establish the relationship ν_{n}∝n/sqrt[V]. It is shown that the geometrical shape of substrate is important for obtaining closer match to these scaling relationships. Furthermore, we study synchronization of two nonidentical drops driven by the same frequency and establish that synchrony happens when the relationship n_{2}/n_{1}=sqrt[V_{2}/V_{1}] is satisfied.
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Affiliation(s)
- Animesh Biswas
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
| | - Dibyendu Das
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
| | - P Parmananda
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
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Stich M, Casal A, Beta C. Stabilization of standing waves through time-delay feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042910. [PMID: 24229252 DOI: 10.1103/physreve.88.042910] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2013] [Indexed: 06/02/2023]
Abstract
Standing waves are studied as solutions of a complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms. The onset is described as an instability of the uniform oscillations with respect to spatially periodic perturbations. The solution of the standing wave pattern is given analytically and studied through simulations.
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Affiliation(s)
- Michael Stich
- Departamento de Matemática Aplicada, ETSAM, Universidad Politécnica de Madrid, Avenida Juan de Herrera 4, 28040 Madrid, Spain
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Rotstein HG, Wu H. Swing, release, and escape mechanisms contribute to the generation of phase-locked cluster patterns in a globally coupled FitzHugh-Nagumo model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:066207. [PMID: 23368024 DOI: 10.1103/physreve.86.066207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2012] [Revised: 08/31/2012] [Indexed: 06/01/2023]
Abstract
We investigate the mechanism of generation of phase-locked cluster patterns in a globally coupled FitzhHugh-Nagumo model where the fast variable (activator) receives global feedback from the slow variable (inhibitor). We identify three qualitatively different mechanisms (swing-and-release, hold-and-release, and escape-and-release) that contribute to the generation of these patterns. We describe these mechanisms and use this framework to explain under what circumstances two initially out-of-phase oscillatory clusters reach steady phase-locked and in-phase synchronized solutions, and how the phase difference between these steady state cluster patterns depends on the clusters relative size, the global coupling intensity, and other model parameters.
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Affiliation(s)
- Horacio G Rotstein
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA.
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García-Morales V, Orlov A, Krischer K. Subharmonic phase clusters in the complex Ginzburg-Landau equation with nonlinear global coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:065202. [PMID: 21230696 DOI: 10.1103/physreve.82.065202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2010] [Indexed: 05/30/2023]
Abstract
A wide variety of subharmonic n -phase cluster patterns was observed in experiments with spatially extended chemical and electrochemical oscillators. These patterns cannot be captured with a phase model. We demonstrate that the introduction of a nonlinear global coupling (NGC) in the complex Ginzburg-Landau equation has subharmonic cluster pattern solutions in wide parameter ranges. The NGC introduces a conservation law for the oscillatory state of the homogeneous mode, which describes the strong oscillations of the mean field in the experiments. We show that the NGC causes a pronounced 2:1 self-resonance on any spatial inhomogeneity, leading to two-phase subharmonic clustering, as well as additional higher resonances. Nonequilibrium Ising-Bloch transitions occur as the coupling strength is varied.
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Seidel YE, Jusys Z, Lindström RW, Stenfeldt M, Kasemo B, Krischer K. Oscillatory behaviour in Galvanostatic Formaldehyde Oxidation on Nanostructured Pt/Glassy Carbon Model Electrodes. Chemphyschem 2010; 11:1405-15. [DOI: 10.1002/cphc.200901029] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Miethe I, García-Morales V, Krischer K. Irregular subharmonic cluster patterns in an autonomous photoelectrochemical oscillator. PHYSICAL REVIEW LETTERS 2009; 102:194101. [PMID: 19518955 DOI: 10.1103/physrevlett.102.194101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2009] [Indexed: 05/27/2023]
Abstract
Unusual subharmonic cluster patterns are observed during the oscillatory electro-oxidation of n-Si(111) under illumination. 2D in situ imaging of the electrode by means of an ellipsometric setup allows local variations in the oxide layer thickness to be monitored. The local oscillators exhibit an irregular distribution of the amplitude with the extrema locked to the constant base frequency of the total current. In addition, Ising 2-phase clustering occurs at half the base frequency. This intrinsic dynamics is described by means of a modified complex Ginzburg-Landau equation.
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Affiliation(s)
- Iljana Miethe
- Physik-Department E19, Technische Universität München, James-Franck-Strasse 1, D-85748 Garching, Germany
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