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Erban R, Kang HW. Chemical Systems with Limit Cycles. Bull Math Biol 2023; 85:76. [PMID: 37402077 DOI: 10.1007/s11538-023-01170-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2023] [Accepted: 05/19/2023] [Indexed: 07/05/2023]
Abstract
The dynamics of a chemical reaction network (CRN) is often modeled under the assumption of mass action kinetics by a system of ordinary differential equations (ODEs) with polynomial right-hand sides that describe the time evolution of concentrations of chemical species involved. Given an arbitrarily large integer [Formula: see text], we show that there exists a CRN such that its ODE model has at least K stable limit cycles. Such a CRN can be constructed with reactions of at most second-order provided that the number of chemical species grows linearly with K. Bounds on the minimal number of chemical species and the minimal number of chemical reactions are presented for CRNs with K stable limit cycles and at most second order or seventh-order kinetics. We also show that CRNs with only two chemical species can have K stable limit cycles, when the order of chemical reactions grows linearly with K.
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Affiliation(s)
- Radek Erban
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
| | - Hye-Won Kang
- Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, 21250, Maryland, USA
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Zhang F, Wang J. The onset of dissipative chaos driven by nonequilibrium conditions. J Chem Phys 2022; 156:024103. [PMID: 35032982 DOI: 10.1063/5.0072294] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Dissipative chaos appears widely in various nonequilibrium systems; however, it is not clear how dissipative chaos originates from nonequilibrium. We discuss a framework based on the potential-flux approach to study chaos from the perspective of nonequilibrium dynamics. In this framework, chaotic systems possess a wide basin on the potential landscape, in which the rotational flux dominates the system dynamics, and chaos occurs with the appearance of this basin. In contrast, the probability flux is particularly associated with the detailed balance-breaking in nonequilibrium systems. This implies that the appearance of dissipative chaos is driven by nonequilibrium conditions.
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Affiliation(s)
- Feng Zhang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, China
| | - Jin Wang
- Department of Chemistry and of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, New York 11794-3400, USA
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Zhang F, Xu L, Wang J. The dynamic and thermodynamic origin of dissipative chaos: chemical Lorenz system. Phys Chem Chem Phys 2020; 22:27896-27902. [PMID: 33284916 DOI: 10.1039/d0cp03580g] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Chaos appears widely in various chemical and physical systems and is often accompanied by nonequilibrium due to its dissipative nature. However, it is still not clear how dissipative chaos is influenced by nonequilibrium conditions. Here, we study chaos from the perspective of nonequilibrium dynamics by considering a chemical Lorenz system. We found that its nonequilibrium nature can be quantified from the steady-state probability flux in the state space. The dynamic origin for the onset and offset of dissipative chaos was from the sudden appearance and disappearance of such nonequilibrium fluxes. Meanwhile, the dissipation associated with the flux as quantified by the entropy production rate provides the thermodynamic origin of dissipative chaos. Sharp changes in the degree of nonequilibrium also provide alternative quantitative indicators for the onset and offset of dissipative chaos.
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Affiliation(s)
- Feng Zhang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, China
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Gaspard P. Stochastic approach to entropy production in chemical chaos. CHAOS (WOODBURY, N.Y.) 2020; 30:113103. [PMID: 33261359 DOI: 10.1063/5.0025350] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2020] [Accepted: 10/14/2020] [Indexed: 06/12/2023]
Abstract
Methods are presented to evaluate the entropy production rate in stochastic reactive systems. These methods are shown to be consistent with known results from nonequilibrium chemical thermodynamics. Moreover, it is proved that the time average of the entropy production rate can be decomposed into the contributions of the cycles obtained from the stoichiometric matrix in both stochastic processes and deterministic systems. These methods are applied to a complex reaction network constructed on the basis of Rössler's reinjection principle and featuring chemical chaos.
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Affiliation(s)
- Pierre Gaspard
- Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles (U.L.B.), Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium
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Magyar A. Generalized mass action realizations of temperature dependent chemical reaction networks. REACTION KINETICS MECHANISMS AND CATALYSIS 2019. [DOI: 10.1007/s11144-018-1500-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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Ghosh S, Ray DS. Parametric spatiotemporal oscillation in reaction-diffusion systems. Phys Rev E 2016; 93:032209. [PMID: 27078346 DOI: 10.1103/physreve.93.032209] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2015] [Indexed: 06/05/2023]
Abstract
We consider a reaction-diffusion system in a homogeneous stable steady state. On perturbation by a time-dependent sinusoidal forcing of a suitable scaling parameter the system exhibits parametric spatiotemporal instability beyond a critical threshold frequency. We have formulated a general scheme to calculate the threshold condition for oscillation and the range of unstable spatial modes lying within a V-shaped region reminiscent of Arnold's tongue. Full numerical simulations show that depending on the specificity of nonlinearity of the models, the instability may result in time-periodic stationary patterns in the form of standing clusters or spatially localized breathing patterns with characteristic wavelengths. Our theoretical analysis of the parametric oscillation in reaction-diffusion system is corroborated by full numerical simulation of two well-known chemical dynamical models: chlorite-iodine-malonic acid and Briggs-Rauscher reactions.
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Affiliation(s)
- Shyamolina Ghosh
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700032, India
| | - Deb Shankar Ray
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700032, India
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Li C, Wang E, Wang J. Potential flux landscapes determine the global stability of a Lorenz chaotic attractor under intrinsic fluctuations. J Chem Phys 2012; 136:194108. [PMID: 22612081 DOI: 10.1063/1.4716466] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023] Open
Abstract
We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.
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Affiliation(s)
- Chunhe Li
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, People's Republic of China
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Li Y, Qian H, Yi Y. Oscillations and multiscale dynamics in a closed chemical reaction system: Second law of thermodynamics and temporal complexity. J Chem Phys 2008; 129:154505. [DOI: 10.1063/1.2995855] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Wang M, Hou Z, Xin H. Internal noise-enhanced phase synchronization of coupled chemical chaotic oscillators. ACTA ACUST UNITED AC 2004. [DOI: 10.1088/0305-4470/38/1/010] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Xu WG, Li QS. A Chemical Chaotic System Derived from Chua's Circuit. JOURNAL OF CHEMICAL RESEARCH 2002. [DOI: 10.3184/030823402103172059] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Chua's circuit is converted into a mass action chemical system with Samardzija's nonlinear transformation method.
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Affiliation(s)
- Wei Guo Xu
- School of Chemical Engineering and Material Science, Beijing Institute of Technology, Beijing 100081, P.R. China
| | - Qian Shu Li
- School of Chemical Engineering and Material Science, Beijing Institute of Technology, Beijing 100081, P.R. China
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Affiliation(s)
- Hongli Wang
- School of Chemical Engineering and Material Science, Beijing Institute of Technology, Beijing 100081, P. R. China, and National Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun, Jilin 130023, P. R. China
| | - Qian-Shu Li
- School of Chemical Engineering and Material Science, Beijing Institute of Technology, Beijing 100081, P. R. China, and National Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun, Jilin 130023, P. R. China
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Abstract
In this article we elaborate on the structure of the generalized Lotka-Volterra (GLV) form for nonlinear differential equations. We discuss here the algebraic properties of the GLV family, such as the invariance under quasimonomial transformations and the underlying structure of classes of equivalence. Each class possesses a unique representative under the classical quadratic Lotka-Volterra form. We show how other standard modeling forms of biological interest, such as S-systems or mass-action systems, are naturally embedded into the GLV form, which thus provides a formal framework for their comparison and for the establishment of transformation rules. We also focus on the issue of recasting of general nonlinear systems into the GLV format. We present a procedure for doing so and point at possible sources of ambiguity that could make the resulting Lotka-Volterra system dependent on the path followed. We then provide some general theorems that define the operational and algorithmic framework in which this is not the case.
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Affiliation(s)
- B Hernández-Bermejo
- Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia, Madrid, Spain
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Matías MA, Güémez J. Chaos suppression in flows using proportional pulses in the system variables. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:198-209. [PMID: 9965061 DOI: 10.1103/physreve.54.198] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Güémez J, Matías MA. Intrinsic-noise-induced transitions in chaotic systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:3059-3068. [PMID: 9962985 DOI: 10.1103/physreve.51.3059] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Matías MA, Güémez J. On the effects of molecular fluctuations on models of chemical chaos. J Chem Phys 1995. [DOI: 10.1063/1.468892] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Strizhak PE. Classical and quantum properties of a simple autocatalytic mechanism ? The Lotka-Volterra system. THEOR EXP CHEM+ 1994. [DOI: 10.1007/bf00520259] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Györgyi L, Field RJ. Aperiodicity resulting from two‐cycle coupling in the Belousov–Zhabotinskii reaction. III. Analysis of a model of the effect of spatial inhomogeneities at the input ports of a continuous‐flow, stirred tank reactor. J Chem Phys 1989. [DOI: 10.1063/1.457432] [Citation(s) in RCA: 24] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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