Lyapunov coefficients for degenerate Hopf bifurcations and an application in a model of competing populations

Ferreira, Jocirei D.; Nieva, A; Yepez, Wilmer Molina

doi:10.1016/j.jmaa.2017.05.040

Cited by 3 publications

(8 citation statements)

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“…where L means direct sum of subspaces. It can be proved 32,45 that a given vector y belongs to T s if and only if p, y h i¼ 0, that is, y T s ⟺ p, y h i¼ 0: ð6:18Þ…”

Section: Basic Equations Eigenvalues and Eigenvectorsmentioning

confidence: 99%

“…Only the most important and basic results are given here. More details on these matters can be found in Kuznetsov, 32 Hassard et al, 33,42 Ferreira et al, 45 Sotomayor et al, 47,48 and Yu and Cao. 53…”

Section: ð6:38eþmentioning

confidence: 99%

“…The mathematical theory of the Poincaré-Andronov-Hopf bifurcation is discussed in numerous scientific books [32][33][34][35][36][37][38][39][40] and articles. [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58] Poincaré-Andronov-Hopf bifurcation theory, along with the projection method, the theory of central manifold, and the theory of normal forms, makes it possible to establish the existence of a stable limit cycle, respectively, stable sinusoidal oscillations in nonlinear systems of differential equations. It is especially important that this theory allows limit cycles to be analyzed not only in two-dimensional spaces (on the plane) but also in spaces with dimensions greater than two.…”

Section: Introductionmentioning

confidence: 99%

“…47,48 The first Lyapunov coefficient at the critical parameter value (when it is nonzero) determines the type of Hopf bifurcation that occurs in the considered system. 32,33,36,42,45 The Poincaré-Andronov-Hopf bifurcation theory is used to analyze dynamical systems in various areas of science and technology. A phase-shift oscillator considered in this paper consists of an inverting amplifier with a positive feedback network.…”

Section: Introductionmentioning

confidence: 99%

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The Poincaré–Andronov–Hopf bifurcation theory and its application to nonlinear analysis of RC phase‐shift oscillator

Georgiev,

Uzunov,

Todorov

et al. 2023

Circuit Theory & Apps

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SummaryIn the paper, a nonlinear analysis of RC phase‐shift oscillator with operational amplifier is done. Using Kirchhoff's laws, a nonlinear system of three differential equations that describes the behavior of the oscillator is obtained. This system is analyzed using the Poincaré–Andronov–Hopf bifurcation theory. The basic principles of the Poincaré–Andronov–Hopf bifurcation theory are presented in advance, including the Center Manifold Theory and the Theory of Normal Forms. It was found that, under certain conditions, the oscillator system has a limit cycle, which means that the oscillator under consideration generates periodic oscillations. The period, amplitude, and stability of the generated periodic oscillations are determined theoretically. The obtained analytical results are confirmed by a numerical analysis of the nonlinear system of differential equations performed by MATLAB.

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“…where L means direct sum of subspaces. It can be proved 32,45 that a given vector y belongs to T s if and only if p, y h i¼ 0, that is, y T s ⟺ p, y h i¼ 0: ð6:18Þ…”

Section: Basic Equations Eigenvalues and Eigenvectorsmentioning

confidence: 99%

Section: ð6:38eþmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Poincaré–Andronov–Hopf bifurcation theory and its application to nonlinear analysis of RC phase‐shift oscillator

Georgiev,

Uzunov,

Todorov

et al. 2023

Circuit Theory & Apps

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show abstract

“…If we introduce another parameter and with the change of that parameter, Hopf bifurcation points forms a curve in the two-parameter plane. Thus, at the intersection of the supercritical Hopf bifurcation and subcritical Hopf bifurcation curves, there will be a point with zero first Lyapunov coefficient that makes possible Bautin bifurcation happen which is of codimension two [10][11][12]14].…”

Section: Introductionmentioning

confidence: 99%

Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model

Guo¹,

Niu²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

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Bautin bifurcation in delayed reaction-diffusion systems with application to the Segel-Jackson model

Guo¹,

Niu²

2018

Preprint

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Lyapunov coefficients for degenerate Hopf bifurcations and an application in a model of competing populations

Cited by 3 publications

References 10 publications

The Poincaré–Andronov–Hopf bifurcation theory and its application to nonlinear analysis of RC phase‐shift oscillator

The Poincaré–Andronov–Hopf bifurcation theory and its application to nonlinear analysis of RC phase‐shift oscillator

Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model

Bautin bifurcation in delayed reaction-diffusion systems with application to the Segel-Jackson model

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