Bifurcation of a stable invariant torus from an equilibrium

1990 ◽  
Vol 48 (1) ◽  
pp. 632-635 ◽  
Author(s):  
Yu. N. Bibikov
Keyword(s):  
Invariant Torus ◽  
Stable Invariant ◽  
Stable Invariant Torus

scholarly journals On preservation of an exponentially stable invariant torus

2015 ◽  
Vol 63 (1) ◽  
pp. 215-222 ◽  
Author(s):  
Mykola Perestyuk ◽  
Petro Feketa
Keyword(s):  
Simple Structure ◽  
Linear Extensions ◽  
Qualitative Behaviour ◽  
Small Perturbations ◽  
One Dimensional ◽  
Exponentially Stable ◽  
Invariant Toroidal Manifold ◽  
Dimensional Dynamical System ◽  
Stable Invariant ◽  
Stable Invariant Torus

Abstract New conditions of the preservation of an exponentially stable invariant toroidal manifold of linear extension of one-dimensional dynamical system on torus under small perturbations in ω-limit set are established. This approach is applied to the investigation of the qualitative behaviour of solutions of linear extensions of dynamical systems with simple structure of limit sets.

Invariant torus and its destruction for an oscillator with dry friction

2021 ◽  
Author(s):  
Xiaoming Zhang ◽  
Chao Zeng ◽  
Denghui Li ◽  
Jianhua Xie ◽  
Celso Grebogi
Keyword(s):  
Dry Friction ◽  
Invariant Torus

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

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