On Stability of N-Dimensional Dynamical Systems

2009 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Distance Function ◽  
Relative Distance ◽  
Stability Of Equilibrium ◽  
The Stability

This paper presents a theory of the stability of equilibrium in dynamical systems. The measuring function is introduced through a relative distance function. The kth -order, G – functions at the equi-measuring function surface and the increment of the equi-measuring function are introduced. Based on the kth -order, G – functions, a theory for the stability of dynamical system is presented, including the definitions and theorems.

scholarly journals STABILITY OF NON-LINEAR DYNAMICAL SYSTEM

2021 ◽  
Vol 9 (07) ◽  
pp. 275-283
Author(s):  
Anas Salim Youns ◽  
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Linear System ◽  
Three Dimension ◽  
Linear Dynamical System ◽  
Linear Dynamical Systems ◽  
Linearization Technique ◽  
The Public ◽  
Non Linear ◽  
The Stability

The mainobjective of this research is to study the stability of thenon-lineardynamical system by using the linearization technique of three dimension systems toobtain an approximate linear system and find its stability. We apply this technique to reaches to the stability of the public non linear dynamical systems of dimension. Finally, some proposed examples (example (1) and example (2)) are given to explain this technique and used the corollary.

On the Dynamical Systems Stability Control and Applications

2014 ◽  
Vol 555 ◽  
pp. 361-368
Author(s):  
Marcel Migdalovici ◽  
Daniela Baran ◽  
Gabriela Vlădeanu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear System ◽  
Stability Control ◽  
Necessary Condition ◽  
Linear Dynamical Systems ◽  
System A ◽  
First Approximation ◽  
Free Parameters ◽  
The Stability

The stability control analyzed by us, in this show, is based on our results in the domain of dynamical systems that depend of parameters. Any dynamical system can be considered as dynamical system that depends of parameters, without numerical particularization of them. All concrete dynamical systems, meted in the specialized literature, underline the property of separation between the stable and unstable zones, in sense of Liapunov, for two free parameters. This property can be also seen for one or more free parameters. Some mathematical conditions of separation between stable and unstable zones for linear dynamical systems are identified by us. For nonlinear systems, the conditions of separation may be identified using the linear system of first approximation attached to nonlinear system. A necessary condition of separation between stable and unstable zones, identified by us, is the sufficient order of differentiability or conditions of continuity for the functions that define the dynamical system. The property of stability zones separation can be used in defining the strategy of stability assurance and optimizing of the parameters, in the manner developed in the paper. The cases of dynamical systems that assure the separations of the stable and unstable zones, in your evolution, and permit the stability control, are analyzed in the paper.

Stability of Dynamical Systems With Multiple Nonlinearities

1987 ◽  
Vol 109 (4) ◽  
pp. 410-413 ◽  
Author(s):  
Norio Miyagi ◽  
Hayao Miyagi
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Stability Region ◽  
Direct Method ◽  
Essential Feature ◽  
Point Of View ◽  
Stability Of Dynamical Systems ◽  
The Stability

This note applies the direct method of Lyapunov to stability analysis of a dynamical system with multiple nonlinearities. The essential feature of the Lyapunov function used in this note is a non-Lure´ type Lyapunov function which surpasses the Lure´-type Lyapunov function from the point of view of the stability region guaranteed. A modified version of the multivariable Popov criterion is used to construct non-Lure´ type Lyapunov function, which allow for the dynamical sytems with multiple nonlinearities.

Estimation of Lyapunov Exponents Using a Semi-Discrete Formulation

1993 ◽  
Vol 46 (11S) ◽  
pp. S229-S233
Author(s):  
Josef S. To¨ro¨k
Keyword(s):  
Dynamical System ◽  
Steady State ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Lie Series ◽  
State Behavior ◽  
The Stability ◽  
Exponential Rates ◽  
Discrete Formulation ◽  
Steady State Behavior

Lyaponov exponents are a generalization of the eigenvalues of a dynamical system at an equilibrium point. They are used to determine the stability of any type of steady-state behavior, including chaotic solutions. More specifically, Lyapunov exponents measure the exponential rates of divergence or convergence associated with nearby trajectories. This paper presents an efficient method of estimating the Lyapunov spectrum of continuous dynamical systems. Based on the Lie series expansion of the flow, the technique can be readily implemented to estimate the Lyapunov exponents of dynamical systems governed by ordinary differential equations.

scholarly journals Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems

2019 ◽  
Vol 24 (1) ◽  
pp. 13 ◽  
Author(s):  
Francisco Solis
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Dynamical Behavior ◽  
Discrete Dynamical Systems ◽  
Exponential Polynomial ◽  
Linear Dynamical System ◽  
The Stability ◽  
Polynomial Family ◽  
Complex Dynamical Behavior

In this paper, we introduce and analyze a family of exponential polynomial discrete dynamical systems that can be considered as functional perturbations of a linear dynamical system. The stability analysis of equilibria of this family is performed by considering three different parametric scenarios, from which we show the intricate and complex dynamical behavior of their orbits.

Dynamical systems: stability and simulability

2007 ◽  
Vol 17 (2) ◽  
pp. 247-259 ◽  
Author(s):  
MATHIEU HOYRUP
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Continuous System ◽  
Finite Precision ◽  
Mathematical Structures ◽  
Continuous Dynamical System ◽  
Precision Machine ◽  
The Stability ◽  
Reachability Property

Computers are used extensively to simulate continuous dynamical systems. However, different conceptual and mathematical structures underlie discrete machines and continuous dynamics, so the question arises as to the ability of the computer to simulate or, more generally, to check the properties of a continuous system.We discuss and compare two notions of stability for a continuous dynamical system,viz.shadowing and robustness, and relate them to both the practical and theoretical computability of the system. We first discuss what we can learn from the stability of a system, using a finite-precision machine. We then show, following the work in Collins (2005), that shadowing fails but robustness succeeds in ensuring the checkability of a reachability property.

General dynamical systems and conditional stability

1967 ◽  
Vol 63 (1) ◽  
pp. 199-207 ◽  
Author(s):  
A. A. Kayande ◽  
V. Lakshmikantham
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Lyapunov Functional ◽  
Conditional Stability ◽  
Stability Properties ◽  
Lyapunov's Method ◽  
Lyapunov’S Method ◽  
The Stability ◽  
General Dynamical System

The notion of a general dynamical system was introduced by Barbashin(1). In this paper we consider a general dynamical system in a metric space following Zubov(6) where Lyapunov's method has been extended to investigate the stability properties using a single Lyapunov functional.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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