Multiple steady states, complex oscillations, and the devil’s staircase in the peroxidase–oxidase reaction

Larter, Raima; Bush, Christopher L.; Lonis, Timothy R.; Aguda, Baltazar D.

doi:10.1063/1.453550

Cited by 40 publications

(12 citation statements)

References 23 publications

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“…They seem to occur most frequently in higherdimensional chemical reactions such as the BZ reaction [159,157,158,106], the Briggs-Rauscher reaction [122], and the PO reaction [154,151]; see also section 7. We discussed one possibility of obtaining L > 1 LAOs as part of an MMO: that there are several large oscillations in the fast subsystem before an MMO trajectory reenters the SAO region.…”

Section: Mmos: Bringing Together Theory and Datamentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

498

635

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Abstract. Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale "mechanism." Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Section: Mmos: Bringing Together Theory and Datamentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

498

635

View full text Add to dashboard Cite

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“…The choice of subscripts in (56) will become clear in Section 6. We are most interested in the two cases when (56) is either given by (49) for the jump case, or by (52) in the canard case.…”

Section: Large Loopsmentioning

confidence: 99%

“…An important starting point are the numerical simulations by Olsen [60] showing that (2) can exhibit various types of oscillations depending on the choice of parameters. Bifurcations of equilibria and sequences of MMOs are investigated in [52]. Considering the chemical reaction mechanisms, it was already realized by Aguda, Larter and Clarke [2] -based on chemical considerations -that the Olsen model can probably be best understood by decomposition into smaller subsystems.…”

Section: Introduction and Reviewmentioning

confidence: 99%

Multiscale Geometry of the Olsen Model and Non-classical Relaxation Oscillations

Kuehn

Szmolyan

2015

J Nonlinear Sci

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We study the Olsen model for the peroxidase-oxidase reaction. The dynamics is analyzed using a geometric decomposition based upon multiple time scales. The Olsen model is four-dimensional, not in a standard form required by geometric singular perturbation theory and contains multiple small parameters. These three obstacles are the main challenges we resolve by our analysis. Scaling and the blow-up method are used to identify several subsystems. The results presented here provide a rigorous analysis for two oscillatory modes. In particular, we prove the existence of non-classical relaxation oscillations in two cases. The analysis is based upon desingularization of lines of transcritical and submanifolds of fold singularities in combination with an integrable relaxation phase. In this context our analysis also explains an assumption that has been utilized, based purely on numerical reasoning, in a previous bifurcation analysis by Desroches, Krauskopf and Osinga [Discret. Contin. Dyn. Syst. S, 2(4), p.807-827, 2009]. Furthermore, the geometric decomposition we develop forms the basis to prove the existence of mixed-mode and chaotic oscillations in the Olsen model, which will be discussed in more detail in future work.

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“…To achieve such personalized and predictive treatment, we need to be able to describe the behavior of observable biochemical systems accurately and quantitatively. and are currently investigated and developed [1][2][3][4]. However, despite the volume of these biochemical systems models, they have not been effectively used in the modeling of real biological and biomedical problems of interest.…”

Section: Introductionmentioning

confidence: 99%

Quantitative comparison of numerical solvers for models of oscillatory biochemical systems

Chang

Tsai

Kao

et al. 2007

Second International Multi-Symposiums on Computer and Computational Sciences (IMSCCS 2007)

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The goal of this work is to find optimal numerical solvers to study models of biological systems of interest. We report 2 simple rules to select the optimal numerical solver(s) for solving stiff, complex oscillatory systems. Numerical solvers currently used to solve mathematical models in systems biology can be ill-conditioned due to stiffness. As a case study, we choose the classic Belousov-Zhabotinskii (BZ) reaction, described by the Oregonator model. We determine the optimal numerical solver(s) to handle stiff initial-value problems that lead to limit cycle behavior, by systematically comparing qualitative and quantitative performance measures i.e. convergence, accuracy and computational cost of numerical solvers that are widely used by the engineering and modeling community. This is a cornerstone for our long-term research objective to systematically study a variety of molecular-level models for biomedical systems for human disease diagnosis and therapeutic treatment in order to understand and predict disease mechanism and progression.

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Multiple steady states, complex oscillations, and the devil’s staircase in the peroxidase–oxidase reaction

Cited by 40 publications

References 23 publications

Mixed-Mode Oscillations with Multiple Time Scales

Mixed-Mode Oscillations with Multiple Time Scales

Multiscale Geometry of the Olsen Model and Non-classical Relaxation Oscillations

Quantitative comparison of numerical solvers for models of oscillatory biochemical systems

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