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Awal NM, Epstein IR. Period-doubling route to mixed-mode chaos. Phys Rev E 2021; 104:024211. [PMID: 34525595 DOI: 10.1103/physreve.104.024211] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2020] [Accepted: 07/16/2021] [Indexed: 06/13/2023]
Abstract
Mixed-mode oscillations (MMOs) are a complex dynamical behavior in which each cycle of oscillation consists of one or more large amplitude spikes followed by one or more small amplitude peaks. MMOs typically undergo period-adding bifurcations under parameter variation. We demonstrate here, in a set of three identical, linearly coupled van der Pol oscillators, a scenario in which MMOs exhibit a period-doubling sequence to chaos that preserves the MMO structure, as well as period-adding bifurcations. We characterize the chaotic nature of the MMOs and attribute their existence to a master-slave-like forcing of the inner oscillator by the outer two with a sufficient phase difference between them. Simulations of a single nonautonomous oscillator forced by two sine functions support this interpretation and suggest that the MMO period-doubling scenario may be more general.
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Affiliation(s)
- Naziru M Awal
- Department of Chemistry, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Irving R Epstein
- Department of Chemistry, Brandeis University, Waltham, Massachusetts 02453, USA
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2
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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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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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Siebert J, Alonso S, Bär M, Schöll E. Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052909. [PMID: 25353863 DOI: 10.1103/physreve.89.052909] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2014] [Indexed: 06/04/2023]
Abstract
A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.
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Affiliation(s)
- Julien Siebert
- Technische Universität, Institut für Theoretische Physik, Hardenberstrasse 36, 10623 Berlin, Germany
| | - Sergio Alonso
- Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany
| | - Markus Bär
- Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany
| | - Eckehard Schöll
- Technische Universität, Institut für Theoretische Physik, Hardenberstrasse 36, 10623 Berlin, Germany
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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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Rotstein HG, Wu H. Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system. J Chem Phys 2012; 137:104908. [PMID: 22979891 DOI: 10.1063/1.4749792] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized oscillatory cluster patterns in a globally coupled Oregonator model where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the oscillation amplitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition exerted by the global feedback term. The transition from phase-locked to localized cluster patterns occurs as the intensity of global feedback increases. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models. The methods we use can be extended to the investigation of oscillatory chemical reactions with other types of non-local coupling.
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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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Marino F, Ciszak M, Abdalah SF, Al-Naimee K, Meucci R, Arecchi FT. Mixed-mode oscillations via canard explosions in light-emitting diodes with optoelectronic feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:047201. [PMID: 22181318 DOI: 10.1103/physreve.84.047201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2011] [Revised: 09/19/2011] [Indexed: 05/31/2023]
Abstract
Chaotically spiking attractors in semiconductor lasers with optoelectronic feedback have been recently observed to be the result of canard phenomena in three-dimensional phase space (incomplete homoclinic scenarios). Since light-emitting diodes display the same dynamics and are much more easily controllable, we use one of these systems to complete the attractor analysis demonstrating experimentally and theoretically the occurrence of complex sequences of periodic mixed-mode oscillations. In particular, we investigate the transition between periodic and chaotic mixed-mode states and analyze the effects of the unavoidable experimental noise on these transitions.
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Affiliation(s)
- F Marino
- Dipartimento di Fisica, Università di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino, Firenze, Italy
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Boubendir Y, Méndez V, Rotstein HG. Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:036601. [PMID: 21230197 DOI: 10.1103/physreve.82.036601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2010] [Indexed: 05/23/2023]
Abstract
We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.
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Affiliation(s)
- Yassine Boubendir
- Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102, USA
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Bauer PR, Bonnefont A, Krischer K. Spatially Resolved ATR-FTIRS Study of the Formation of Macroscopic Domains and Microislands during CO Electrooxidation on Pt. Chemphyschem 2010; 11:3002-10. [PMID: 20715270 DOI: 10.1002/cphc.201000301] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Affiliation(s)
- Philipp R Bauer
- Physik-Department E19a, Technische Universität München, James-Franck-Str. 1, 85748 Garching, Germany
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Sadeghi S, Thompson M. Temporal patterns and oscillatory voltage perturbation during an electrochemical process. Phys Chem Chem Phys 2010; 12:6795-809. [DOI: 10.1039/c000966k] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Bonnefont A, Morschl R, Bauer P, Krischer K. Electrochemical impedance spectroscopy of patterned steady states on electrode surfaces. Electrochim Acta 2009. [DOI: 10.1016/j.electacta.2009.03.054] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Brons M, Kaper TJ, Rotstein HG. Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis. CHAOS (WOODBURY, N.Y.) 2008; 18:015101. [PMID: 18377082 DOI: 10.1063/1.2903177] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.
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Affiliation(s)
- Morten Brons
- Department of Mathematics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
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Sundarram S, Luss D. Dynamics of Transversal Hot Zones in a Shallow Packed Bed Reactor during Oxidation of Mixtures of C3H6 and CO. Ind Eng Chem Res 2007. [DOI: 10.1021/ie060597c] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Sandhya Sundarram
- Department of Chemical and Biomolecular Engineering, University of Houston, Houston, Texas 77204
| | - Dan Luss
- Department of Chemical and Biomolecular Engineering, University of Houston, Houston, Texas 77204
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Sundarram S, Marwaha B, Luss D. Global-coupling induced temperature patterns on top of packed-bed reactors. Chem Eng Sci 2005. [DOI: 10.1016/j.ces.2005.04.066] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Marwaha B, Sundarram S, Luss D. Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors. J Phys Chem B 2004. [DOI: 10.1021/jp049803p] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Bharat Marwaha
- Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792
| | - Sandhya Sundarram
- Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792
| | - Dan Luss
- Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792
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Bîrzu A, Plenge F, Jaeger NI, Hudson JL, Krischer K. Complex Spatiotemporal Antiphase Oscillations during Electrodissolution of a Metal Disk Electrode: Model Calculations. J Phys Chem B 2003. [DOI: 10.1021/jp022457w] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Adrian Bîrzu
- Department of Physics, Technical University of München, James-Franck-Strasse 1, 85748 Garching, Germany, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany, Institute for Applied and Physical Chemistry, University of Bremen, FB 2-Biology/Chemistry, Postfach 330 440, 28334 Bremen, Germany, and Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
| | - Florian Plenge
- Department of Physics, Technical University of München, James-Franck-Strasse 1, 85748 Garching, Germany, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany, Institute for Applied and Physical Chemistry, University of Bremen, FB 2-Biology/Chemistry, Postfach 330 440, 28334 Bremen, Germany, and Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
| | - Nils I. Jaeger
- Department of Physics, Technical University of München, James-Franck-Strasse 1, 85748 Garching, Germany, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany, Institute for Applied and Physical Chemistry, University of Bremen, FB 2-Biology/Chemistry, Postfach 330 440, 28334 Bremen, Germany, and Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
| | - John L. Hudson
- Department of Physics, Technical University of München, James-Franck-Strasse 1, 85748 Garching, Germany, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany, Institute for Applied and Physical Chemistry, University of Bremen, FB 2-Biology/Chemistry, Postfach 330 440, 28334 Bremen, Germany, and Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
| | - Katharina Krischer
- Department of Physics, Technical University of München, James-Franck-Strasse 1, 85748 Garching, Germany, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany, Institute for Applied and Physical Chemistry, University of Bremen, FB 2-Biology/Chemistry, Postfach 330 440, 28334 Bremen, Germany, and Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
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Beck O, Amann A, Schöll E, Socolar JES, Just W. Comparison of time-delayed feedback schemes for spatiotemporal control of chaos in a reaction-diffusion system with global coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:016213. [PMID: 12241467 DOI: 10.1103/physreve.66.016213] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2001] [Indexed: 05/23/2023]
Abstract
Time-delayed feedback control for stabilizing time periodic spatial patterns is investigated in a generic reaction-diffusion system with global coupling. We focus on the case of low-dimensional chaos where unstable patterns admit only a single unstable mode. Spatial degrees of freedom are taken into account to define different control schemes. The efficiency of these schemes is discussed, where control forces are motivated by physical requirements as well as by the possibility of obtaining analytically exact results. We find that control schemes that contain the full feedback of the inhibitor variable may finally destroy the control performance. Thus schemes that omit the inhibitor might be more efficient. Our numerical findings are explained in terms of Floquet spectra and compared with analytical solutions of particular coupling schemes.
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Affiliation(s)
- O Beck
- Institut für Theoretische Physik, TU Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany
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