Degenerate Hopf Bifurcation Formulas and Hilbert’s 16th Problem

Farr, W.W.; Li, Chengzhi; Labouriau, Isabel S.; Langford, William F.

doi:10.1137/0520002

Cited by 86 publications

(35 citation statements)

References 15 publications

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“…We used the method of [4] (see also [5]): after Liapunov-Schmidt reduction, the periodic solutions correspond to zeros of a function xa(x 2 , λ), with λ = v , and the derivatives of a(u, λ), u = x 2 can be computed from those of the original system using the formulas of [3]. The bifurcation is subcritical if a u = ∂a ∂u and a λ = ∂a ∂λ have the same sign, supercritical otherwise.…”

Section: Hopf Bifurcationmentioning

confidence: 99%

Loss of synchronization in partially coupled Hodgkin–Huxley equations

Labouriau

Pinto

2004

Bulletin of Mathematical Biology

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Abstract. We study the loss of synchronization of two partially coupled Hodgkin-Huxley equations, with symmetric coupling. This models the coupling ot two cells through an electrical synapse. For strong enough coupling it is known that all solutions of the equations approach a state where the two cells are perfectly synchronized.We describe the local bifurcations that arise when the coupling strength is reduced, using a mixture of analytical and numerical methods. We find that almost perfect synchrony is retained for very small positive values of the coupling strength. Although perfect synchrony is lost for negative values of the coupling constant, the system always retains some degree of synchronization until it becomes totally unstable. This happens in two ways: in many cases for almost all initial conditions the solutions still approach a perfectly synchronized state. Even when this is not true, the attracting solutions are still synchronized, with a half-period phase shift.

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Section: Hopf Bifurcationmentioning

confidence: 99%

Loss of synchronization in partially coupled Hodgkin–Huxley equations

Labouriau

Pinto

2004

Bulletin of Mathematical Biology

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“…a n;1 ! u a2 2 ( )u a3 3 ( ) u an;1 n;1 ( ) (4) where D k n is the following subset of indices (see 2] for more details): D k n = fa = ( a 1 a 2 a n;1 ) 2 N n;1 such that a 1 + + a n;1 = k a 1 + + j a j + + ( n ; 1)a n;1 = ng:…”

mentioning

confidence: 99%

An Analytic-Numerical Method for Computation of the Liapunov and Period Constants Derived from Their Algebraic Structure

Gasull¹,

Guillamón²,

Mañosa³

1999

SIAM J. Numer. Anal.

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Abstract. We consider the problem of computing the Liapunov and the period constants for a smooth di erential equation with a non degenerate critical point. First, we i n vestigate the structure of both constants when they are regarded as polynomials on the coe cients of the di erential equation. Secondly, w e t a k e advantadge of this structure to derive a method to obtain the explicit expression of the above-mentioned constants. Although this method is based on the use of the Runge-KuttaFehlberg methods of orders 7 and 8 and the use of Richardson's extrapolation, it provides the real expression for these constants.

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“…Determining the existence and the exact number of limit cycles is a difficult problem even for planar autonomous polynomial differential systems. This problem is closely related to the longstanding 16th problem of D. Hilbert [8,18,20,28,29]. The analysis of bifurcation and limit cycles is not only a challenging problem in the qualitative theory of dynamical systems, but also of practical value now for our understanding of the qualitative behaviors of biological systems.…”

Section: Introductionmentioning

confidence: 99%

“…We want to analyze the bifurcation and limit cycles for system (1.1) with n = 2. In the literature there are several methods for studying this problem [8], e.g., by using Poincaré-Birkhott normal forms, Liapunov constants, succession functions, averaging, and intrinsic harmonic balancing. To deal with the problem, we make a linear transformation…”

Section: Bifurcation Analysis For Two-dimensional Systemsmentioning

confidence: 99%

“…Prior to the algebraic analysis of bifurcation investigated by the authors [16], other relevant work has been done by El Kahoui and Weber [7] who applied quantifier elimination [6] to Hopf bifurcation analysis, by Lu and others in [12,13] where limit cycles for a class of biological systems are analyzed by using similar algebraic approaches, and by other researchers in the purely mathematical context of bifurcation and limit cycles (see, e.g., [8,18,19]). Hopf bifurcation analyzed in the mathematical context is usually for systems in the canonical form with the origin as their steady state.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

Wei

Wang

Algebraic Biology

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Abstract. In this paper, we show how to analyze bifurcation and limit cycles for biological systems by using an algebraic approach based on triangular decomposition, Gröbner bases, discriminant varieties, real solution classification, and quantifier elimination by partial CAD. The analysis of bifurcation and limit cycles for a concrete two-dimensional system, the self-assembling micelle system with chemical sinks, is presented in detail. It is proved that this system may have a focus of order 3, from which three limit cycles can be constructed by small perturbation. The applicability of our approach is further illustrated by the construction of limit cycles for a two-dimensional Kolmogorov prey-predator system and a three-dimensional Lotka-Volterra system.

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Degenerate Hopf Bifurcation Formulas and Hilbert’s 16th Problem

Cited by 86 publications

References 15 publications

Loss of synchronization in partially coupled Hodgkin–Huxley equations

Loss of synchronization in partially coupled Hodgkin–Huxley equations

An Analytic-Numerical Method for Computation of the Liapunov and Period Constants Derived from Their Algebraic Structure

Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

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