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Rohe K, Cisternas J. Propagation failure in discrete reaction-diffusion system based on the butterfly bifurcation. CHAOS (WOODBURY, N.Y.) 2022; 32:053124. [PMID: 35649997 DOI: 10.1063/5.0086239] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2022] [Accepted: 04/29/2022] [Indexed: 06/15/2023]
Abstract
Reaction-diffusion systems are used in biology, chemistry, and physics to model the interaction of spatially distributed species. Particularly of interest is the spatial replacement of one equilibrium state by another, depicted as traveling waves or fronts. Their profiles and traveling velocity depend on the nonlinearities in the reaction term and on spatial diffusion. If the reaction occurs at regularly spaced points, the velocities also depend on lattice structures and the orientation of the traveling front. Interestingly, there is a wide region of parameters where the speeds become zero and the fronts do not propagate. In this paper, we focus on systems with three stable coexisting equilibrium states that are described by the butterfly bifurcation and study to what extent the three possible 1D traveling fronts suffer from propagation failure. We demonstrate that discreteness of space affects the three fronts differently. Regions of propagation failure add a new layer of complexity to the butterfly diagram. The analysis is extended to planar fronts traveling through different orientations in regular 2D lattices. Both propagation failure and the existence of preferred orientations play a role in the transient and long-time evolution of 2D patterns.
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Affiliation(s)
- K Rohe
- Surface Science Group, Institute for Integrated Natural Sciences, University of Koblenz-Landau, Universitätsstraße 1, 56070 Koblenz, Germany
| | - J Cisternas
- Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Chile, Monseñor Alvaro del Portillo 12455, Santiago 7620001, Chile
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Rohe K, Cisternas J, Wehner S. Competing ternary surface reaction CO + O 2 + H 2 on Ir(111). Proc Math Phys Eng Sci 2020; 476:20190712. [PMID: 32398931 DOI: 10.1098/rspa.2019.0712] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2019] [Accepted: 03/19/2020] [Indexed: 11/12/2022] Open
Abstract
The CO oxidation on platinum-group metals under ultra-high-vacuum conditions is one of the most studied surface reactions. However, the presence of disturbing species and competing reactions are often neglected. One of the most interesting additional gases to be treated is hydrogen, due to its importance in technical applications and its inevitability under vacuum conditions. Adding hydrogen to the reaction of CO and O2 leads to more adsorbed species and competing reaction steps towards water formation. In this study, a model for approaching the competing surface reactions CO+O 2 + H2 is presented and discussed. Using the framework of bifurcation theory, we show how the steady states of the extended system correspond to a swallowtail catastrophe set with a tristable regime within the swallowtail. We explore numerically the possibility of reaching all stable states and illustrate the experimental challenges such a system could pose. Lastly, an approximative first-principle approach to diffusion illustrates how up to three stable states balance each other while forming heterogeneous patterns.
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Affiliation(s)
- Kevin Rohe
- Surface Science Group, Institute for Integrated Natural Sciences, University of Koblenz-Landau, Universitätsstraße 1, 56070 Koblenz, Germany
| | - Jaime Cisternas
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Alvaro del Portillo 12455, Las Condes, Santiago, Chile
| | - Stefan Wehner
- Surface Science Group, Institute for Integrated Natural Sciences, University of Koblenz-Landau, Universitätsstraße 1, 56070 Koblenz, Germany
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Cartes C, Descalzi O, Brand HR. Simultaneous influence of additive and multiplicative noise on stationary dissipative solitons. Phys Rev E 2019; 100:012214. [PMID: 31499916 DOI: 10.1103/physreve.100.012214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2019] [Indexed: 06/10/2023]
Abstract
We investigate the simultaneous influence of spatially homogeneous multiplicative noise as well as of spatially δ-correlated additive noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. Depending on the ratio between the strength of additive and multiplicative noise we find a number of distinctly different types of behavior including explosions, collapse, filling in, and spatio-temporal disorder as well as intermittent behavior of all types listed. Techniques used to analyze the results include snapshots, x-t plots and plots of the spatially and temporally averaged amplitude as a function of the strength of multiplicative noise while keeping the strength of additive noise fixed. Typically 50 realizations are used for averaging to obtain the corresponding data points in these diagrams. For the widths of these distribution as a function of additive noise we obtain a linear decrease in the limit of fairly large, but fixed values of the multiplicative noise. To summarize our findings concisely we show three-dimensional plots of the mean pattern amplitude and the generalized susceptibility as a function of the strengths of additive and multiplicative noise. We critically compare the results of our investigations with those obtained in the two limiting cases of purely additive and of purely multiplicative noise.
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Affiliation(s)
- Carlos Cartes
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
| | - Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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Cisternas J, Wehner S. Detailed analysis of transitions in the CO oxidation on palladium(111) under noisy conditions. J Chem Phys 2018; 149:044706. [PMID: 30068191 DOI: 10.1063/1.5040704] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022] Open
Abstract
It has been shown that CO oxidation on Pd(111) under ultrahigh vacuum conditions can suffer rare transitions between two stable states triggered by weak intrinsic perturbations. Here we study the effects of adding controlled noise by varying the concentrations of O2 and CO that feed the vacuum chamber, while the total flux stays constant. In addition to the regime of rare transitions between states of different CO2 reaction rates induced by intrinsic fluctuations, we found three distinct effects of external noise depending on its strength: small noise suppresses transitions and stabilizes the upper rate state; medium noise induces bursting; and large noise gives rise to reversible transitions in both directions. To explain some of the features present in the dynamics, we propose an extended stochastic model that includes a global coupling through the gas phase to account for the removal of CO gas caused by the adsorption of the Pd surface. The numerical simulations based in the model show a qualitative agreement with the noise-induced transitions found in experiments, but suggest that more complex spatial phenomena are present in the observed fluctuations.
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Affiliation(s)
- Jaime Cisternas
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Alvaro del Portillo 12455, Las Condes, Santiago, Chile
| | - Stefan Wehner
- Surface Science Group, Institute for Integrated Natural Sciences, University of Koblenz-Landau, Universitätsstraße 1, 56070 Koblenz, Germany
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Descalzi O, Cartes C, Brand HR. Multiplicative noise can lead to the collapse of dissipative solitons. Phys Rev E 2016; 94:012219. [PMID: 27575135 DOI: 10.1103/physreve.94.012219] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2016] [Indexed: 06/06/2023]
Abstract
We investigate the influence of spatially homogeneous multiplicative noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. We find that for sufficiently large multiplicative noise the formation of stationary and temporally periodic dissipative solitons is suppressed. This result is characterized by a linear relation between the bifurcation parameter and the noise amplitude required for suppression. For the regime associated with exploding dissipative solitons we find a reduction in the number of explosions for larger noise strength as well as a conversion to other types of dissipative solitons or to filling-in and eventually a collapse to the zero solution.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Carlos Cartes
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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Descalzi O, Cartes C, Brand HR. Noisy localized structures induced by large noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:020901. [PMID: 25768449 DOI: 10.1103/physreve.91.020901] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2014] [Indexed: 06/04/2023]
Abstract
We investigate the influence of large noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. The interaction of localization and noise can lead to filling in or noisy localized structures for fixed noise strength. To focus on the interaction between noise and localization we cover a region in parameter space, in particular, subcriticality, for which stationary stable deterministic pulses do not exist. Possible experimental tests of the work presented for autocatalytic chemical reactions and bioinspired systems are outlined.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Carlos Cartes
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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Cisternas J, Wehner S, Descalzi O. CO oxidation on Ir(111) surfaces under large non-Gaussian noise. J Chem Phys 2012; 137:064105. [PMID: 22897253 DOI: 10.1063/1.4742191] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jaime Cisternas
- Complex Systems Group, College of Engineering and Applied Sciences, Universidad de los Andes, Santiago, Chile.
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