Poincaré map approach to limit cycles of a simplified ultradiscrete Sel'kov model

Ohmori, Shousuke; Yamazaki, Yoshihiro

doi:10.14495/jsiaml.14.127

Cited by 4 publications

(1 citation statement)

References 17 publications

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“…In this letter, we focus on types and stability of fixed points in two-dimensional dynamical systems. So far, we have numerically investigated the Sel'kov model as a specific example [10][11][12]. However, the previous studies have not been treated as generally and analytically as in the above one-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Ohmori,

Yamazaki

2023

JSIAM Letters

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Relationship for dynamical properties in the vicinity of fixed points between two-dimensional continuous and its positivity-preserving discretized dynamical systems is studied. Based on linear stability analysis, we reveal the conditions under which the dynamical structures of the original continuous dynamical systems are retained in their discretized dynamical systems, and the types of fixed points are identified if they change due to discretization. We also discuss stability of the fixed points in the discrete dynamical systems. The obtained general results are applied to Sel'kov model and Lengyel-Epstein model.

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Section: Introductionmentioning

confidence: 99%

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Ohmori,

Yamazaki

2023

JSIAM Letters

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Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: One dimensional cases

Ohmori

Yamazaki

2023

Journal of Mathematical Physics

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The stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation parameter in the discrete dynamical systems, and the emergence condition of an additional bifurcation, flip bifurcation, is identified. The correspondence between the discrete dynamical systems with positivity and the ultradiscrete ones derived from them is discussed. It is found that the derived ultradiscrete max-plus dynamical systems can retain the bifurcations of the original continuous ones via tropical discretization and ultradiscretization.

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Emergence of ultradiscrete states due to phase lock caused by saddle-node bifurcation in discrete limit cycles

Yamazaki

Ohmori

2023

Progress of Theoretical and Experimental Physics

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Dynamical properties of limit cycles for a tropically discretized negative feedback model are numerically investigated. This model has a controlling parameter τ, which corresponds to time interval for the time evolution of phase in the limit cycles. By considering τ as a bifurcation parameter, we find that ultradiscrete state emerges due to phase lock caused by saddle-node bifurcation. Furthermore, focusing on limit cycles for the max-plus negative feedback model, it is found that the unstable limit cycle in the max-plus model corresponds to the unstable fixed points emerging by the saddle-node bifurcation in the tropically discretized model.

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Poincaré map approach to limit cycles of a simplified ultradiscrete Sel'kov model

Cited by 4 publications

References 17 publications

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: One dimensional cases

Emergence of ultradiscrete states due to phase lock caused by saddle-node bifurcation in discrete limit cycles

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