Fabíolo Moraes Amaral

STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

TYPE-ZERO SADDLE-NODE BIFURCATIONS AND STABILITY REGION ESTIMATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS

A complete characterization of the stability boundary of a class of nonlinear dynamical systems that admit energy functions is developed in this paper. This characterization generalizes the existing results by allowing the type-zero saddle-node nonhyperbolic equilibrium points on the stability boundary. Conceptual algorithms to obtain optimal estimates of the stability region (basin of attraction) in the form of level sets of a given family of energy functions are derived. The behavior of the stability region and the corresponding estimates are investigated for parameter variation in the neighborhood of a type-zero saddle-node bifurcation value.

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

ABSTRACT A complete characterization of the stability boundary of an asymptotically stable equilibrium point in the presence of type-k saddle-node non-hyperbolic equilibrium points, with k ≥ 0, on the stability boundary is developed in this paper. Under the transversality condition, it is shown that the stability boundary is composed of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-0 saddle-node equilibrium points on the stability boundary and the stable centre and centre manifolds of the type-r saddle-node equilibrium points with r ≥ 1 on the stability boundary. This characterization is the first step to understanding the behaviour of stability regions and stability boundaries in the occurrence of saddle-node bifurcations on the stability boundary.

Supercritical Hopf equilibrium points on the boundary of the stability region

A complete characterization of the boundary of the stability region of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of non-hyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium points. Under condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

Characterization of saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical system

A dynamical characterization of the stability boundary for a fairly large class of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddle-node equilibrium points on the stability boundary. The stability boundary of an asymptotically stable equilibrium point is shown to consist of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-zero saddle-node equilibrium points on the stability boundary and the stable center and center manifolds of the type-k saddle-node equilibrium points with k≥ 1 on the stability boundary.

Bifurcações do Tipo Hopf da Região de Estabilidade de Sistemas Dinâmicos Autônomos Não Lineares

Exploraremos o comportamento da regiao de estabilidade de sistemas dinâmicos sujeitos a variacoes de parâmetros na vizinhanca de um ponto de equilibrio [...]

Bifurcations and Stability Regions of Nonlinear Dynamical Systems

Stability regions of nonlinear dynamical systems may suffer drastic changes as a consequence of parameter variation. These changes are triggered by local or global bifurcations of the vector field. In this chapter, these changes are studied for two types of local bifurcations on the stability boundary: saddle-node bifurcations and Hopf bifurcations on the stability boundary. Local and global characterizations of the stability boundary (the topological boundary of stability region) will be developed at the bifurcation points and the behavior (changes) of stability boundaries and stability regions at these bifurcations will be studied.

Robustness of stability regions of nonlinear circuits and systems under parameter variation

The behavior of stability regions of nonlinear dynamical systems subjected to parameter variation is studied in this paper. Sufficient conditions to guarantee the persistence of the stability boundary characterization under parameter variation are presented. When these conditions are violated, the stability region may undergo a bifurcation and may suffer drastic changes. In this paper, the behavior of stability region and stability boundary when the system undergoes a type-zero saddle-node bifurcation on the stability boundary is investigated. A complete characterization of these changes in the neighborhood of a type-zero saddle-node bifurcation point on the stability boundary is developed. These results are applied to the analysis of stability region of a simple Hopfield artificial neural network.