Computation of the Stability Condition for the Hopf Bifurcation of Diffeomorphisms on $\mathbb{R}^2 $

Wan, Yieh-Hei

doi:10.1137/0134013

Cited by 89 publications

(45 citation statements)

References 4 publications

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“…The coefficient c 1 (a) can be computed directly using the formula below derived by Wan in the context of the Hopf bifurcation theory [27]. In [20] it is shown that when one uses area-preserving coordinate changes, this formula by Wan yields the twist coefficient τ 1 that is used to verify the non-degeneracy condition necessary to apply the KAM theorem.…”

Section: Im(λ) Re(λ)mentioning

confidence: 99%

Applications of KAM theory to population dynamics

Gidea

Meiss

Ugarcovici

et al. 2011

Journal of Biological Dynamics

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Computer simulations have shown that several classes of population models, including the May host-parasitoid model and the Ginzburg-Taneyhill 'maternal-quality' single species population model, exhibit extremely complicated orbit structures. These structures include islands-around-islands, ad infinitum, with the smaller islands containing stable periodic points of higher period. We identify the mechanism that generates this complexity and we discuss some biological implications.

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Section: Im(λ) Re(λ)mentioning

confidence: 99%

Applications of KAM theory to population dynamics

Gidea

Meiss

Ugarcovici

et al. 2011

Journal of Biological Dynamics

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“…First, the period ofthe cycle for values of r near the bifurcation point is always greater than 100 generations, and the cycle period increases as r is moved away from the point ofbifurcation. The period at the bifurcation point can be simply calculated from the imaginary part ofthe eigenvalues at the bifurcation point (7). The eigenvalues at the bifurcation point are ofthe form e" , and the period is 2Xr/@.…”

Section: Resultsmentioning

confidence: 99%

Stable cycling in discrete-time genetic models.

Hastings¹

1981

Proc. Natl. Acad. Sci. U.S.A.

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Examples of stable cycling are discussed for twolocus, two-allele, deterministic, discrete-time models with constant fitnesses. The cases that cycle were found by using numerical techniques to search for stable Hopf bifurcations. One consequence ofthe results is that apparent cases ofdirectional selection may be due to stable cycling.The causes ofcycling in populations, which include predator-prey oscillations, the role of time delays, and amplification of environmental disturbances have long been a topic of research in population biology (1, 2). Selection within a single population would be an alternate, and possibly quite general, explanation ofcycling. These cycles could provide an explanation ofcurious behavior observed in one-locus genetic experiments. In this paper, one example of stable cycling in a discrete-time, constant-fitness, two-locus, two-allele model will be discussed, and general features of all the examples I have found thus far will also be included. In ref. 3 cycling that results from genotype-environment interactions is discussed. The only other reported instance of stable cycling due to genetic causes alone is the independent work ofAkin (4), who recently proved the existence of stable cycles arising from a Hopfbifurcation in the continuous time, two-locus, two-allele model. This paper complements and extends the work of Akin by dealing with a discrete-time model and examining the behavior of the cycling populations, rather than proving the existence of cycles. Akin employed an asymptotic, analytic approach, whereas I have used a numerical approach. BACKGROUND AND METHODSThe model used in this study is the usual deterministic two-allele discrete-generation model (5) with alleles A and a at the A locus and B and b at the B locus. The frequencies (and "names") of the four chromosomal types AB, Ab, aB, ab are x1, X2, X3, X4,

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“…The coefficient c 1 can be computed directly using the formula below derived by Wan in the context of Hopf bifurcation theory [21]. In [22], it is shown that, when one uses area-preserving coordinate changes, Wan's formula yields the twist coefficient τ 1 that is used to verify the non-degeneracy condition necessary to apply the KAM theorem.…”

Section: Theorem 3 Assume That τ(ζζ)mentioning

confidence: 99%

Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

2016

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By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system:where the parameter β > 0, and initial conditions x 0 and y 0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse's lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits.

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Computation of the Stability Condition for the Hopf Bifurcation of Diffeomorphisms on $\mathbb{R}^2 $

Cited by 89 publications

References 4 publications

Applications of KAM theory to population dynamics

Applications of KAM theory to population dynamics

Stable cycling in discrete-time genetic models.

Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

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