Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point

2012 ◽  
pp. 85-116
Author(s):  
Mark I. Freidlin ◽  
Alexander D. Wentzell
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Point

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

2013 ◽  
Vol 23 (12) ◽  
pp. 1350196 ◽  
Author(s):  
JOSAPHAT R. R. GOUVEIA ◽  
FABÍOLO MORAES AMARAL ◽  
LUÍS F. C. ALBERTO
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Stability Boundary ◽  
Complete Characterization ◽  
Center Manifolds ◽  
The Stability ◽  
Hyperbolic Equilibrium ◽  
Autonomous Dynamical Systems

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.

scholarly journals Complex fuzzy dynamical systems and Stability of the equilibrium Point

2016 ◽  
Vol 2016 (3) ◽  
pp. 223-233 ◽  
Author(s):  
S. Melliani ◽  
A. El Allaoui ◽  
L. S. Chadli
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Point ◽  
Fuzzy Dynamical Systems

ADAPTABILITY AND SENSITIVITY OF COMPLEX SYSTEMS

2013 ◽  
Vol 23 (09) ◽  
pp. 1350163 ◽  
Author(s):  
ZHI-CHENG YE ◽  
QING-DUAN FAN ◽  
QIN-BIN HE ◽  
ZENG-RONG LIU
Keyword(s):  
Dynamical Systems ◽  
Complex Systems ◽  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Stable Equilibrium Point ◽  
Complex Dynamical Systems ◽  
Scientific Communities ◽  
Mathematical Definitions

Recently, the study on the dynamical behavior of complex dynamical systems has become a focal subject in the field of complexity. In particular, the system's adaptability and sensitivity have attracted increasing attention from various scientific communities. In this paper, we focus on some properties of complexity to gain a better understanding of it. Two descriptive mathematical definitions of attractors' adaptability and sensitivity are introduced from the viewpoint of dynamical systems. Then, these new descriptions are applied to analyze the adaptability and sensitivity of stable equilibrium points. In addition, a method is introduced for improving both the adaptability and sensitivity of a system with a stable equilibrium point.

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