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Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere.

Jaume Llibre, Claudio Pessoa (2006)

Extracta Mathematicae

Let X be a homogeneous polynomial vector field of degree 2 on S2 having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of S2, and at most there are two invariant circles. We characterize the global phase portrait of these vector fields. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles.

Homogeneous systems of higher-order ordinary differential equations

Mike Crampin (2010)

Communications in Mathematics

The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation...

Homogeneous Systems with a Quiescent Phase

K. P. Hadeler (2008)

Mathematical Modelling of Natural Phenomena

Recently the effect of a quiescent phase (or dormant/resting phase in applications) on the dynamics of a system of differential equations has been investigated, in particular with respect to stability properties of stationary points. It has been shown that there is a general phenomenon of stabilization against oscillations which can be cast in rigorous form. Here we investigate, for homogeneous systems, the effect of a quiescent phase, and more generally, a phase with slower dynamics. We show that...

Homogenization of codimension 1 actions of n near a compact orbit

Marcos Craizer (1994)

Annales de l'institut Fourier

Let Φ be a C n -action on an orientable ( n + 1 ) -dimensional manifold. Assume Φ has an isolated compact orbit T and let W be a small tubular neighborhood of it. By a C change of variables, we can write W = n / n × I and T = 𝕋 n × [ 0 ] , where I is some interval containing 0.In this work, we show that by a C 0 change of variables, C outside T , we can make Φ | W invariant by transformations of the type ( x , z ) ( x + a , z ) , a n , where x n / n and z I . As a corollary one cas describe completely the dynamics of Φ in W .

Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel

M. Neamţu, L. Buliga, F. R. Horhat, D. Opriş (2010)

Mathematical Modelling of Natural Phenomena

The aim of this paper is to study the steady states of the mathematical models with delay kernels which describe pathogen-immune dynamics of infectious diseases. In the study of mathematical models of infectious diseases it is important to predict whether the infection disappears or the pathogens persist. The delay kernel is described by the memory function that reflects the influence of the past density of pathogen in the blood and it is given by a nonnegative bounded and normated function k defined...

Hopf bifurcations in a three-species food chain system with multiple delays

Xiaoliang Xie, Wen Zhang (2017)

Open Mathematics

This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations....

Hopf-like bifurcations in planar piecewise linear systems.

Emilio Freire, Enrique Ponce, Francisco Torres (1997)

Publicacions Matemàtiques

Continuous planar piecewise linear systems with two linear zones are considered. Due to their low differentiability specific techniques of analysis must be developed. Several bifurcations giving rise to limit cycles are pointed out.

Hoptf bifurcation from infinity for planar control systems.

Jaume Llibre, Enrique Ponce (1997)

Publicacions Matemàtiques

Symmetric piecewise linear bi-dimensional systems are very common in control engineering. They constitute a class of non-differentiable vector fields for which classical Hopf bifurcation theorems are not applicable. For such systems, sufficient and necessary conditions for bifurcation of a limit cycle from the periodic orbit at infinity are given.

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