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Vidal-Henriquez E, Zykov V, Bodenschatz E, Gholami A. Convective instability and boundary driven oscillations in a reaction-diffusion-advection model. CHAOS (WOODBURY, N.Y.) 2017; 27:103110. [PMID: 29092427 DOI: 10.1063/1.4986153] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In a reaction-diffusion-advection system, with a convectively unstable regime, a perturbation creates a wave train that is advected downstream and eventually leaves the system. We show that the convective instability coexists with a local absolute instability when a fixed boundary condition upstream is imposed. This boundary induced instability acts as a continuous wave source, creating a local periodic excitation near the boundary, which initiates waves travelling both up and downstream. To confirm this, we performed analytical analysis and numerical simulations of a modified Martiel-Goldbeter reaction-diffusion model with the addition of an advection term. We provide a quantitative description of the wave packet appearing in the convectively unstable regime, which we found to be in excellent agreement with the numerical simulations. We characterize this new instability and show that in the limit of high advection speed, it is suppressed. This type of instability can be expected for reaction-diffusion systems that present both a convective instability and an excitable regime. In particular, it can be relevant to understand the signaling mechanism of the social amoeba Dictyostelium discoideum that may experience fluid flows in its natural habitat.
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Affiliation(s)
| | - Vladimir Zykov
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
| | - Eberhard Bodenschatz
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
| | - Azam Gholami
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
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Feng J, Green JR, Johnson SA, Wang J. CO2production in the bromate-1,4-cyclohexanedione oscillatory reaction. J PHYS ORG CHEM 2010. [DOI: 10.1002/poc.1795] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Yanagita T, Suetani H, Aihara K. Bifurcation analysis of solitary and synchronized pulses and formation of reentrant waves in laterally coupled excitable fibers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:056208. [PMID: 19113201 DOI: 10.1103/physreve.78.056208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2008] [Indexed: 05/27/2023]
Abstract
We study the dynamics of a reaction-diffusion system comprising two mutually coupled excitable fibers. We consider a case in which the dynamical properties of the two fibers are nonidentical due to the parameter mismatch between them. By using the spatially one-dimensional FitzHugh-Nagumo equations as a model of a single excitable fiber, synchronized pulses are found to be stable in some parameter regime. Furthermore, there exists a critical coupling strength beyond which the synchronized pulses are stable for any amount of parameter mismatch. We show the bifurcation structures of the synchronized and solitary pulses and identify a codimension-2 cusp singularity as the source of the destabilization of synchronized pulses. When stable solitary pulses in both fibers disappear via a saddle-node bifurcation on increasing the coupling strength, a reentrant wave is formed. The parameter region, where a stable reentrant wave is observed in direct numerical simulation, is consistent with that obtained by bifurcation analysis.
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Affiliation(s)
- Tatsuo Yanagita
- Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0020, Japan.
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Manz N, Ginn BT, Steinbock O. Propagation failure dynamics of wave trains in excitable systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:066218. [PMID: 16906957 DOI: 10.1103/physreve.73.066218] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2006] [Indexed: 05/11/2023]
Abstract
We report experimental and numerical results on temporal patterns of propagation failures in reaction-diffusion systems. Experiments employ the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagation failures occur in the frontier region of the wave train and can profoundly affect its expansion speed. The specific rhythms observed vary from simple periodic to highly complex and possibly chaotic sequences. All but the period-1 sequences are found in the transition region between "merging" and "tracking" dynamics, which correspond to wave behavior caused by two qualitatively different types of anomalous dispersion relations.
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Affiliation(s)
- Niklas Manz
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, USA
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Manz N, Steinbock O. Dynamics of excitation pulses with attractive interaction: kinematic analysis and chemical wave experiments. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:066213. [PMID: 15697490 DOI: 10.1103/physreve.70.066213] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2004] [Indexed: 05/24/2023]
Abstract
We present a theoretical analysis of stacking and destacking wave trains in excitable reaction-diffusion systems with anomalous velocity-wavelength dependence. For linearized dispersion relations, kinematic analysis yields an analytical function that rigorously describes front trajectories. The corresponding accelerations have exactly one extremum that slowly decays with increasing pulse number. For subsequent pulses these maxima occur with a lag time equal to the inverse slope of the linearized dispersion curve. These findings are reproduced in experiments with chemical waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction but should be also applicable to step bunching on crystal surfaces and certain traffic phenomena.
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Affiliation(s)
- Niklas Manz
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, FL 32306-4390, USA
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Manz N, Hamik CT, Steinbock O. Tracking waves and vortex nucleation in excitable systems with anomalous dispersion. PHYSICAL REVIEW LETTERS 2004; 92:248301. [PMID: 15245136 DOI: 10.1103/physrevlett.92.248301] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2004] [Indexed: 05/24/2023]
Abstract
We report experimental results obtained from a chemical reaction-diffusion system in which wave propagation is limited to a finite band of wavelengths and in which no solitary pulses exist. Wave patterns increase their size through repeated annihilation events of the frontier pulse that allow the succeeding pulses to advance farther. A related type of wave dynamics involves a stable but slow frontier pulse that annihilates subsequent waves in front-to-back collisions. These so-called merging dynamics give rise to an unexpected form of spiral wave nucleation. All of these phenomena are reproduced by a simple, three-species reaction-diffusion model that reveals the importance of the underlying anomalous dispersion relation.
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Affiliation(s)
- N Manz
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, USA
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Manz N, Steinbock O. Tracking Waves and Spiral Drift in Reaction−Diffusion Systems with Finite Bandwidth Dispersion Relations. J Phys Chem A 2004. [DOI: 10.1021/jp048456g] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Niklas Manz
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390
| | - Oliver Steinbock
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390
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Szalai I, Kurin-Csörgei K, Epstein IR, Orbán M. Dynamics and Mechanism of Bromate Oscillators with 1,4-Cyclohexanedione. J Phys Chem A 2003. [DOI: 10.1021/jp0360523] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- István Szalai
- Department of Inorganic and Analytical Chemistry, L. Eötvös University, P.O. Box 32, H-1518 Budapest 112, Hungary, and Department of Chemistry, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110
| | - Krisztina Kurin-Csörgei
- Department of Inorganic and Analytical Chemistry, L. Eötvös University, P.O. Box 32, H-1518 Budapest 112, Hungary, and Department of Chemistry, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110
| | - Irving R. Epstein
- Department of Inorganic and Analytical Chemistry, L. Eötvös University, P.O. Box 32, H-1518 Budapest 112, Hungary, and Department of Chemistry, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110
| | - Miklós Orbán
- Department of Inorganic and Analytical Chemistry, L. Eötvös University, P.O. Box 32, H-1518 Budapest 112, Hungary, and Department of Chemistry, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110
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Ko HK, Lee KJ, Lee J. Parametrically forced surface wave with a nonmonotonic dispersion relation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:026218. [PMID: 12636787 DOI: 10.1103/physreve.67.026218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2002] [Indexed: 05/24/2023]
Abstract
Surface wave patterns that arise in a mechanically driven ferrofluid system under constant magnetic field are investigated (1) to find out what kind of spatial patterns emerge when the system acquires a nonmonotonic dispersion relation and (2) to compare its surface wave patterns with those produced in the magnetically driven system studied earlier. As the strength of the applied magnetic field increases, the initial subharmonic square lattice formed by the Faraday instability first transforms to rolls, then becomes a rhomboid lattice. The rolls and the rhomboid lattice are found to coexist for a finite range of parameter space forming patterns with mixed domains. Possible underlying mechanisms for the observed rhomboid lattice is discussed. None of the diverse superlattices observed in the magnetically driven ferrofluid system appears in the mechanically driven system studied here.
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Affiliation(s)
- Hee-kyoung Ko
- National Creative Research Initiative Center for Neuro-dynamics and Department of Physics, Korea University, Seoul 136-701, Korea
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Vanag VK, Epstein IR. Comparative Analysis of Packet and Trigger Waves Originating from a Finite Wavelength Instability. J Phys Chem A 2002. [DOI: 10.1021/jp026081y] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Vladimir K. Vanag
- Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110
| | - Irving R. Epstein
- Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110
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Steinbock O. Excitable front geometry in reaction-diffusion systems with anomalous dispersion. PHYSICAL REVIEW LETTERS 2002; 88:228302. [PMID: 12059458 DOI: 10.1103/physrevlett.88.228302] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2001] [Indexed: 05/23/2023]
Abstract
Two-dimensional excitable systems with anomalous dispersion provide a discrete set of interpulse distances for the stable propagation of planar wave trains. Numerical simulations show that the trailing front of a pulse pair can undergo transitions between these stable distances. In response to localized perturbations, the trailing front converges towards one of numerous, sigmoidal shapes. Their transition segments move at constant speeds and can collide and fuse with each other. A complementing kinematic analysis of the front dynamics yields a reaction-diffusion-like equation.
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Affiliation(s)
- Oliver Steinbock
- Florida State University, Department of Chemistry, Tallahassee, Florida 32306-4390, USA
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