Singular Perturbation Analysis and Gene Regulatory Networks with Delay

Shlykova, Irina; Ponosov, Arcady

doi:10.1063/1.3241446

AIP Conference Proceedings 2009

DOI: 10.1063/1.3241446

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Singular Perturbation Analysis and Gene Regulatory Networks with Delay

Irina Shlykova

Arcady Ponosov

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2015

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“…The results presented by the authors, in particular Lemma 5.1, make use of tools which are similar to the one we consider although they do not provide a complete classification of the hidden dynamics that we present here. An extension to gene regulatory systems with delays is studied in [19], where the authors use a technique to remove delays which determines a system of singularly perturbed differential equations.…”

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Classification of Hidden Dynamics in Discontinuous Dynamical Systems

Guglielmi¹,

Hairer²

2015

SIAM J. Appl. Dyn. Syst.

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Ordinary differential equations with discontinuous right-hand side, where the discontinuity of the vector field arises on smooth surfaces of the phase space, are the topic of this work. The main emphasis is the study of solutions close to the intersection of two discontinuity surfaces. There, the so-called hidden dynamics describes the smooth transition from ingoing to outgoing solution directions, which occurs instantaneously in the jump discontinuity of the vector field. This article presents a complete classification of such transitions (assuming the vector fields surrounding the intersection are transversal to it). Since the hidden dynamics is realized by standard space regularizations, much insight is obtained for them. One can predict, in the case of multiple solutions of the discontinuous problem, which solution (classical or sliding mode) will be approximated after entering the intersection of two discontinuity surfaces. A novel modification of space regularizations is presented that permits to avoid (unphysical) high oscillations and makes a numerical treatment more efficient.

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confidence: 99%

Classification of Hidden Dynamics in Discontinuous Dynamical Systems

Guglielmi¹,

Hairer²

2015

SIAM J. Appl. Dyn. Syst.

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Singular Perturbation Analysis and Gene Regulatory Networks with Delay

Cited by 1 publication

References 2 publications

Classification of Hidden Dynamics in Discontinuous Dynamical Systems

Classification of Hidden Dynamics in Discontinuous Dynamical Systems

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