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Ramírez-Ávila GM, Kapitaniak T, Gonze D. Dynamical analysis of a periodically forced chaotic chemical oscillator. CHAOS (WOODBURY, N.Y.) 2024; 34:073154. [PMID: 39047162 DOI: 10.1063/5.0213913] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2024] [Accepted: 07/11/2024] [Indexed: 07/27/2024]
Abstract
We present a comprehensive dynamical analysis of a chaotic chemical model referred to as the autocatalator, when subject to a periodic administration of one substrate. Our investigation encompasses the dynamical characterization of both unforced and forced systems utilizing isospikes and largest Lyapunov exponents-based parameter planes, bifurcation diagrams, and analysis of complex oscillations. Additionally, we present a phase diagram showing the effect of the period and amplitude of the forcing signal on the system's behavior. Furthermore, we show how the landscapes of parameter planes are altered in response to forcing application. This analysis contributes to a deeper understanding of the intricate dynamics induced by the periodic forcing of a chaotic system.
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Affiliation(s)
- Gonzalo Marcelo Ramírez-Ávila
- Namur Institute for Complex Systems (naXys), Université de Namur, Rue de Bruxelles 61, B-5000 Namur, Belgium
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
- Instituto de Investigaciones Físicas, and Planetario Max Schreier, Universidad Mayor de San Andrés, Campus Universitario, C. 27 s/n Cota-Cota, 0000 La Paz, Bolivia
- Unit of Theoretical Chronobiology, Université Libre de Bruxelles, CP231, Boulevard du Triomphe, 1050 Brussels, Belgium
| | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Didier Gonze
- Unit of Theoretical Chronobiology, Université Libre de Bruxelles, CP231, Boulevard du Triomphe, 1050 Brussels, Belgium
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Folke Olsen L. Complex dynamics in an unexplored simple model of the peroxidase-oxidase reaction. CHAOS (WOODBURY, N.Y.) 2023; 33:023102. [PMID: 36859227 DOI: 10.1063/5.0129095] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Accepted: 01/09/2023] [Indexed: 06/18/2023]
Abstract
A previously overlooked version of the so-called Olsen model of the peroxidase-oxidase reaction has been studied numerically using 2D isospike stability and maximum Lyapunov exponent diagrams and reveals a rich variety of dynamic behaviors not observed before. The model has a complex bifurcation structure involving mixed-mode and bursting oscillations as well as quasiperiodic and chaotic dynamics. In addition, multiple periodic and non-periodic attractors coexist for the same parameters. For some parameter values, the model also reveals formation of mosaic patterns of complex dynamic states. The complex dynamic behaviors exhibited by this model are compared to those of another version of the same model, which has been studied in more detail. The two models show similarities, but also notable differences between them, e.g., the organization of mixed-mode oscillations in parameter space and the relative abundance of quasiperiodic and chaotic oscillations. In both models, domains with chaotic dynamics contain apparently disorganized subdomains of periodic attractors with dinoflagellate-like structures, while the domains with mainly quasiperiodic behavior contain subdomains with periodic attractors organized as regular filamentous structures. These periodic attractors seem to be organized according to Stern-Brocot arithmetics. Finally, it appears that toroidal (quasiperiodic) attractors develop into first wrinkled and then fractal tori before they break down to chaotic attractors.
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Affiliation(s)
- Lars Folke Olsen
- PhyLife, Institute of Biochemistry and Molecular Biology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
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Gallas JAC, Hauser MJB, Olsen LF. Complexity of a peroxidase-oxidase reaction model. Phys Chem Chem Phys 2021; 23:1943-1955. [PMID: 33463636 DOI: 10.1039/d0cp06153k] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study a state-of-the-art model of the peroxidase-oxidase reaction. Using the model, we report systematic numerical experiments exploring the impact of changing the enzyme concentration on the dynamics of the reaction. Specifically, we report high-resolution phase diagrams predicting and describing how the reaction unfolds over a quite extended range of enzyme concentrations. Surprisingly, such diagrams reveal that the enzyme concentration has a huge impact on the reaction evolution. The highly intricate dynamical behaviours predicted here are difficult to establish theoretically due to the total absence of an adequate framework to solve nonlinearly coupled differential equations. But such behaviours may be validated experimentally.
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Affiliation(s)
- Jason A C Gallas
- Instituto de Altos Estudos da Paraiba, Rua Silvino Lopes 419-2502, 58039-190 João Pessoa, Brazil
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Olsen LF, Lunding A. Chaos in the peroxidase-oxidase oscillator. CHAOS (WOODBURY, N.Y.) 2021; 31:013119. [PMID: 33754781 DOI: 10.1063/5.0022251] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2020] [Accepted: 12/18/2020] [Indexed: 06/12/2023]
Abstract
The peroxidase-oxidase (PO) reaction involves the oxidation of reduced nicotinamide adenine dinucleotide by molecular oxygen. When both reactants are supplied continuously to a reaction mixture containing the enzyme and a phenolic compound, the reaction will exhibit oscillatory behavior. In fact, the reaction exhibits a zoo of dynamical behaviors ranging from simple periodic oscillations to period-doubled and mixed mode oscillations to quasiperiodicity and chaos. The routes to chaos involve period-doubling, period-adding, and torus bifurcations. The dynamic behaviors in the experimental system can be simulated by detailed semiquantitative models. Previous models of the reaction have omitted the phenolic compound from the reaction scheme. In the current paper, we present new experimental results with the oscillating PO reaction that add to our understanding of its rich dynamics, and we describe a new variant of a previous model, which includes the chemistry of the phenol in the reaction mechanism. This new model can simulate most of the experimental behaviors of the experimental system including the new observations presented here. For example, the model reproduces the two main routes to chaos observed in experiments: (i) a period-doubling scenario, which takes place at low pH, and a period-adding scenario involving mixed mode oscillations (MMOs), which occurs at high pH. Our simulations suggest alternative explanations for the pH-sensitivity of the dynamics. We show that the MMO domains are separated by narrow parameter regions of chaotic behavior or quasiperiodicity. These regions start as tongues of secondary quasiperiodicity and develop into strange attractors through torus breakdown.
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Affiliation(s)
- Lars F Olsen
- PhyLife, Institute of Biochemistry and Molecular Biology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
| | - Anita Lunding
- Institute of Biochemistry and Molecular Biology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
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Freire JG, Calderón-Cárdenas A, Varela H, Gallas JAC. Phase diagrams and dynamical evolution of the triple-pathway electro-oxidation of formic acid on platinum. Phys Chem Chem Phys 2020; 22:1078-1091. [DOI: 10.1039/c9cp04324a] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/30/2023]
Abstract
A detailed numerical study including stability phase diagrams for the dynamical evolution of the electro-oxidation of formic acid on platinum was reported. The study evidences the existence of intertwined stability phases and the absence of chaos.
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Affiliation(s)
- Joana G. Freire
- Instituto Dom Luiz (IDL)
- Faculdade de Ciências
- Universidade de Lisboa
- 1749-016 Lisboa
- Portugal
| | - Alfredo Calderón-Cárdenas
- Instituto de Química de São Carlos
- Universidade de São Paulo
- 13560-970 São Carlos
- Brazil
- GIFBA, Universidad de Nariño
| | - Hamilton Varela
- Instituto de Química de São Carlos
- Universidade de São Paulo
- 13560-970 São Carlos
- Brazil
- Max-Planck Institute for the Physics of Complex Systems
| | - Jason A. C. Gallas
- Max-Planck Institute for the Physics of Complex Systems
- 01187 Dresden
- Germany
- Instituto de Altos Estudos da Paraíba
- 58039-190 João Pessoa
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Freire JG, Gallas MR, Gallas JAC. Impact of predator dormancy on prey-predator dynamics. CHAOS (WOODBURY, N.Y.) 2018; 28:053118. [PMID: 29857665 DOI: 10.1063/1.5016434] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The impact of predator dormancy on the population dynamics of phytoplankton-zooplankton in freshwater ecosystems is investigated using a simple model including dormancy, a strategy to avoid extinction. In addition to recently reported chaos-mediated mixed-mode oscillations, as the carrying capacity grows, we find surprisingly wide phases of nonchaos-mediated mixed-mode oscillations to be present well before the onset of chaos in the system. Nonchaos-mediated cascades display spike-adding sequences, while chaos-mediated cascades show spike-doubling. A host of braided periodic phases with exotic shapes is found embedded in a region of control parameters dominated by chaotic oscillations. We describe the organization of these complicated phases and show how they are interconnected and how their complexity unfolds as control parameters change. The novel nonchaos-mediated phases are found to be large and stable, even for low carrying capacity.
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Affiliation(s)
- Joana G Freire
- Instituto de Altos Estudos da Paraíba, Rua Silvino Lopes 419-2502, 58039-190 João Pessoa, Brazil
| | - Marcia R Gallas
- Instituto de Altos Estudos da Paraíba, Rua Silvino Lopes 419-2502, 58039-190 João Pessoa, Brazil
| | - Jason A C Gallas
- Instituto de Altos Estudos da Paraíba, Rua Silvino Lopes 419-2502, 58039-190 João Pessoa, Brazil
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Beims MW, Gallas JAC. Manifold angles, the concept of self-similarity, and angle-enhanced bifurcation diagrams. Sci Rep 2016; 6:18859. [PMID: 26732416 PMCID: PMC4702165 DOI: 10.1038/srep18859] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2015] [Accepted: 11/27/2015] [Indexed: 11/09/2022] Open
Abstract
Chaos and regularity are routinely discriminated by using Lyapunov exponents distilled from the norm of orthogonalized Lyapunov vectors, propagated during the temporal evolution of the dynamics. Such exponents are mean-field-like averages that, for each degree of freedom, squeeze the whole temporal evolution complexity into just a single number. However, Lyapunov vectors also contain a step-by-step record of what exactly happens with the angles between stable and unstable manifolds during the whole evolution, a big-data information permanently erased by repeated orthogonalizations. Here, we study changes of angles between invariant subspaces as observed during temporal evolution of Hénon’s system. Such angles are calculated numerically and analytically and used to characterize self-similarity of a chaotic attractor. In addition, we show how standard tools of dynamical systems may be angle-enhanced by dressing them with informations not difficult to extract. Such angle-enhanced tools reveal unexpected and practical facts that are described in detail. For instance, we present a video showing an angle-enhanced bifurcation diagram that exposes from several perspectives the complex geometrical features underlying the attractors. We believe such findings to be generic for extended classes of systems.
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Affiliation(s)
- Marcus W Beims
- Departamento de Física, Universidade Federal do Paraná, 81531-990 Curitiba, Brazil.,Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Brazil.,Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil.,Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
| | - Jason A C Gallas
- Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Brazil.,Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil.,Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.,Institute for Multiscale Simulation, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91052 Erlangen, Germany
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Kiss IZ, Pojman JA. Introduction to Focus Issue: Oscillations and Dynamic Instabilities in Chemical Systems: Dedicated to Irving R. Epstein on occasion of his 70th birthday. CHAOS (WOODBURY, N.Y.) 2015; 25:064201. [PMID: 26117111 DOI: 10.1063/1.4922594] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Affiliation(s)
- István Z Kiss
- Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA
| | - John A Pojman
- Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, USA
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