Attractability of trajectories from an exponentially stable invariant torus

1984 ◽  
Vol 35 (3) ◽  
pp. 237-241
Author(s):  
S. O. Baekova
Keyword(s):  
Invariant Torus ◽  
Exponentially Stable ◽  
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).

scholarly journals Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Modelin the Presence of Static Obstacles

2021 ◽  
Vol 9 (6) ◽  
pp. 652
Author(s):  
Haitong Xu ◽  
Miguel A. Hinostroza ◽  
C. Guedes Guedes Soares
Keyword(s):  
Control System ◽  
Vector Field ◽  
Path Following ◽  
Field Method ◽  
Integrated System ◽  
Ship Model ◽  
Surface Ship ◽  
Path Following Control ◽  
Exponentially Stable ◽  
Vector Field Method

A modified path-following control system using the vector field method for an underactuated autonomous surface ship model is proposed in the presence of static obstacles. With this integrated system, autonomous ships are capable of following the predefined path, while avoiding the obstacles automatically. It is different from the methods in most published papers, which usually study path-following and obstacle collision avoidance, separately. This paper considers the coupled path following and collision avoidance task as a whole. Meanwhile, the paper also shows the heading control design method in the presence of static obstacles. To obtain a strong stability property, a nonlinear autopilot is designed based on the manoeuvring tests of the free-running ship model. The equilibrium point of the controller is globally exponentially stable. For the guidance system, a novel vector field method was proposed, and the proof shows the coupled guidance and control system is uniform semi-global exponentially stable (USGES). To prevent the obstacles near the predefined path, the proposed guidance law is augmented by integrating the repelling field of obstacles so that it can control the ship travel toward the predefined path through the obstacles safely. The repelling field function is given considering the obstacle shape and collision risk using the velocity obstacle (VO) algorithm. The simulations and ship model test were performed to validate the integrated system of autonomous ships.

Robust Stability Analysis of Delay-Dependence for a Class of Switched Uncertain Systems with Time Delay

2013 ◽  
Vol 415 ◽  
pp. 139-142
Author(s):  
Chi Jo Wang ◽  
Juing Shian Chiou
Keyword(s):  
Time Delay ◽  
Stability Margin ◽  
Uncertain System ◽  
Switching Law ◽  
Delay Dependence ◽  
Robust Stability Analysis ◽  
Exponentially Stable ◽  
Delay Dependent ◽  
Interval Systems ◽  
Delay Dependent Stability

Some new criteria of delay-dependent stability for the switched time-delay uncertain system are deduced by employing time-switched method and the comparison theorem in this paper. The total activation time ratio of the switching law can be determined to guarantee the switched time-delay uncertain system is exponentially stable with stability margin . Finally, this method can be extended to switched interval systems with time-delay. Some examples are exploited to illustrate the proposed schemes..

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