Small Amplitude Limit Cycles and Local Bifurcation of Critical Periods for a Quartic Kolmogorov System

He, Dongping; Huang, Wentao; Wang, Qinlong

doi:10.1007/s12346-020-00401-5

Cited by 9 publications

(4 citation statements)

References 24 publications

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“…According to the algebraic equivalence shown in (6), we can easily get the first eight focal values of the origin for system (13) or its conjugate real system (12) with D 001 = −1:…”

Section: Hopf Bifurcation At the Equilibrium Ementioning

confidence: 99%

“…For the planar Kolmogorov system, it is well known that system (1) does not have limit cycles if f 1 and f 2 are linear, namely it is the classical Lotka-Volterra-Gause model. When f 1 and f 2 are not linear, many results have been obtained in [3][4][5][6]. For the three-dimensional Kolmogorov system, if f 1 , f 2 and f 3 are linear, then system (1) is a quadratic Lotka-Volterra system.…”

mentioning

confidence: 99%

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Hopf and zero-Hopf bifurcations for a class of cubic Kolmogorov systems in R3

Wang,

Lu,

Wang

et al. 2023

Preprint

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In this paper, Hopf and zero-Hopf bifurcations are investigated for a class of three-dimensional cubic Kolmogorov systems with one positive equilibrium. Firstly, by computing the singular point quantities and figuring out center conditions, we determined that the highest order of the positive equilibrium is eight as a fine focus, which yields that there exist at most seven small amplitude limit cycles restricted to one center manifold and Hopf cyclicity 8 at the positive equilibrium. Secondly, by using the normal form algorithm, we discuss the existence of stable periodic solution via zero-Hopf bifurcation around the positive equilibrium. At the same time, the relevance between zero-Hopf bifurcation and Hopf bifurcation is analyzed via its special case, which is rarely considered. Finally, some related illustrations are given by means of numerical simulation.

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“…According to the algebraic equivalence shown in (6), we can easily get the first eight focal values of the origin for system (13) or its conjugate real system (12) with D 001 = −1:…”

Section: Hopf Bifurcation At the Equilibrium Ementioning

confidence: 99%

mentioning

confidence: 99%

Hopf and zero-Hopf bifurcations for a class of cubic Kolmogorov systems in R3

Wang,

Lu,

Wang

et al. 2023

Preprint

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“…Small amplitude limit cycles and the local bifurcations of critical pe-riods for a quartic Kolmogorov system at the positive equilibrium are investigated, and the maximum number of small amplitude limit cycles bifurcating is obtained to be seven [10].…”

Section: Introductionmentioning

confidence: 99%

Qualitative Analysis of the Minimal Higgins Model of Glycolysis

Ferčec¹,

Mencinger²,

Petek³

et al. 2023

MATCH

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Glycolysis, one of the leading metabolic pathways, involves many different periodic oscillations emerging at positive steady states of the biochemical models describing this essential process. One of the models employing the molecular diffusion of intermediates is the Higgins biochemical model to explain sustained oscillations. In this paper, we investigate the center-focus problem for the minimal Higgins model for general values of the model parameters with the help of computational algebra. We demonstrate that the model always has a stable focus point by finding a general form of the first Lyapunov number. Then, varying two of the model parameters, we obtain the first three coefficients of the period function for the stable focus point of the model and prove that the singular point is actually a bi-weak monodromic equilibrium point of type [1, 2]. Additionally, we prove that there are two (small) intervals for a chosen parameter a > 0 for which one critical period bifurcates from this singular point after small perturbations

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“…Lloyd et al [12] more than 20 years ago considered a class of cubic Kolmogorov systems and showed that six limit cycles can exist in the vicinity of a positive equilibrium point. For other relevant results, one can refer to the previous studies [13][14][15] and references therein. However, it can be seen from the above references that a majority of research on the number of limit cycles for Kolmogorov systems considered the case of a two-dimensional vector field only, whereas the higher-dimensional (n ≥ 3) case has not received as much attention.…”

mentioning

confidence: 99%