scholarly journals Countably Expansiveness for Continuous Dynamical Systems

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1228
Author(s):  
Manseob Lee ◽  
Jumi Oh
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Stability Theory ◽  
Smooth Manifold ◽  
The Stability ◽  
Expansive Flows ◽  
Dense Set

Expansiveness is very closely related to the stability theory of the dynamical systems. It is natural to consider various types of expansiveness such as countably-expansive, measure expansive, N-expansive, and so on. In this article, we introduce the new concept of countably expansiveness for continuous dynamical systems on a compact connected smooth manifold M by using the dense set D of M, which is different from the weak expansive flows. We establish some examples having the countably expansive property, and we prove that if a vector field X of M is C 1 stably countably expansive then it is quasi-Anosov.

Periodic Motions in an Autonomous System With a Discontinuous Vector Field

2020 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Autonomous System ◽  
Parameter Study ◽  
Spring Stiffness ◽  
Algebraic Equations ◽  
Periodic Motions ◽  
Mapping Structures ◽  
The Stability

Abstract In this paper, periodic motions in an autonomous system with a discontinuous vector field are discussed. The periodic motions are obtained by constructing a set of algebraic equations based on motion mapping structures. The stability of periodic motions is investigated through eigenvalue analysis. The grazing bifurcations are presented by varying the spring stiffness. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Numerical simulations are conducted for motion illustrations. The parameter study helps one understand autonomous discontinuous dynamical systems.

scholarly journals The stability theorems for discrete dynamical systems on two-dimensional manifolds

1983 ◽  
Vol 90 ◽  
pp. 1-55 ◽  
Author(s):  
Atsuro Sannami
Keyword(s):  
Dynamical Systems ◽  
Smooth Manifold ◽  
Riemannian Metric ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stability Theorems ◽  
The Stability

One of the basic problems in the theory of dynamical systems is the characterization of stable systems.Let M be a closed (i.e. compact without boundary) connected smooth manifold with a smooth Riemannian metric and Diffr (M) (r ≥ 1) denote the space of Cr diffeomorphisms on M with the uniform Cr topology.

scholarly journals Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Codruţa Stoica
Keyword(s):  
Dynamical Systems ◽  
Exponential Stability ◽  
Stability Theory ◽  
Exponential Dichotomy ◽  
Evolution Equations ◽  
Discrete Dynamical Systems ◽  
Asymptotic Approach ◽  
Stability And Instability ◽  
Unified Study ◽  
The Stability

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.

scholarly journals Matrix measure and application to stability of matrices and interval dynamical systems

2003 ◽  
Vol 2003 (2) ◽  
pp. 75-85 ◽  
Author(s):  
Ziad Zahreddine
Keyword(s):  
Dynamical Systems ◽  
Convex Hull ◽  
General Condition ◽  
Stability Theory ◽  
Stability Criteria ◽  
Matrix Measure ◽  
The Real ◽  
The Matrix ◽  
The Stability

Using some properties of the matrix measure, we obtain a general condition for the stability of a convex hull of matrices that will be applied to study the stability of interval dynamical systems. Some classical results from stability theory are reproduced and extended. We present a relationship between the matrix measure and the real parts of the eigenvalues that make it possible to obtain stability criteria.

Stability Theory via Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Stability Theory ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Comparison System ◽  
Nonlinear Dynamical ◽  
Vector Lyapunov Functions ◽  
Lasalle Theorem ◽  
System States ◽  
The Stability

This chapter describes a fundamental stability theory for nonlinear dynamical systems using vector Lyapunov functions. It first introduces the notation and definitions before developing stability theorems via vector Lyapunov functions for continuous-time and discrete-time nonlinear dynamical systems. It then extends the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. It also presents a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii–LaSalle theorem. In the analysis of large-scale nonlinear interconnected dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem.

scholarly journals ATTRACTOR–REPELLER PAIR, MORSE DECOMPOSITION AND LYAPUNOV FUNCTION FOR RANDOM DYNAMICAL SYSTEMS

2008 ◽  
Vol 08 (04) ◽  
pp. 625-641 ◽  
Author(s):  
ZHENXIN LIU ◽  
SHUGUAN JI ◽  
MENGLONG SU
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Metric Space ◽  
Stability Theory ◽  
Lyapunov Functions ◽  
Random Dynamical Systems ◽  
Morse Decomposition ◽  
Compact Metric Space ◽  
Morse Decompositions ◽  
The Stability

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor–repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [7,17], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor–repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.

CHAOS SYNCHRONIZATION OF CHEN SYSTEMS WITH TIME-VARYING DELAYS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250183 ◽  
Author(s):  
JIANENG TANG ◽  
CAIRONG ZOU ◽  
SHAOPING WANG ◽  
LI ZHAO ◽  
PINGXIANG LIU
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Stability Theory ◽  
Chaos Synchronization ◽  
Delay Systems ◽  
Time Varying ◽  
Time Delay Systems ◽  
Synchronization Problem ◽  
The Stability

In this paper, the synchronization problem of Chen systems with time-varying delays is discussed based on the stability theory of time-delay systems. Through the analysis of the error dynamical systems, the time-delay correlative synchronization controller is designed to achieve chaos synchronization. And finally, numerical simulations are provided to verify the effectiveness and feasibility of the developed method.

Stability Theory of Dynamical Systems

1970 ◽  
Author(s):  
Nam Parshad Bhatia ◽  
George Philip Szegö
Keyword(s):  
Dynamical Systems ◽  
Stability Theory
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