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 The Stability Criteria with Compound Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Yazhuo Zhang ◽  
Baodong Zheng
Keyword(s):  
Dynamical Systems ◽  
Spectral Property ◽  
Discrete Dynamical Systems ◽  
Asymptotical Stability ◽  
Hopf Bifurcations ◽  
Stability Criteria ◽  
Bifurcation Problem ◽  
Compound Matrices ◽  
The Stability ◽  
Schur Stability

The bifurcation problem is one of the most important subjects in dynamical systems. Motivated by M. Li et al. who used compound matrices to judge the stability of matrices and the existence of Hopf bifurcations in continuous dynamical systems, we obtained some effective methods to judge the Schur stability of matrices on the base of the spectral property of compound matrices, which can be used to judge the asymptotical stability and the existence of Hopf bifurcations of discrete 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 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.

Exponential Stability Regions Estimation of Nonlinear Dynamical Systems

2020 ◽  
Vol 21 (3) ◽  
pp. 131-135
Author(s):  
V. V. Grigoriev ◽  
S. V. Bystrov ◽  
O. K. Mansurova ◽  
I. M. Pershin ◽  
A. B. Bushuev ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Exponential Stability ◽  
Nonlinear Dynamical Systems ◽  
Average Rate ◽  
Living Organism ◽  
Control Algorithms ◽  
Computational Techniques ◽  
Nonlinear Dynamical ◽  
Stability And Instability ◽  
Analysis Of Stability

The main purpose of the research is the extending of the concept of qualitative exponential stability and instability for a wider class of dynamical systems and plants with estimating of regions of exponential stability as well as developing of analytic and calculating technologies for analyzing the quality of processes and projecting of control devices for control systems. And if the property of asymptotic stability indicates the convergence or divergence of the processes in time, the exponential stability provides information about the speed of convergence or divergence processes, thereby characterizing the rapidness of the system. Meeting the conditions of quality exponential stability evaluates the average rate of convergence or divergence of processes, as well as ongoing processes of deviations of the time-average behavior, the last gives information about the behavior of transients (oscillation, overshoot). Development of analytical and computational techniques for the analysis of stability and instability of comparison systems and, as a result of mul tiply-connected systems, as well as the processes quality is almost an essential task for the study multi-agent control algorithms and biologically inspired control algorithms in nonlinear systems, because the same systems should be related to each other and has a predetermined degree of exponential stability for the required control. Proposed results can be using in the developing flying and terrestrial robots based on biological control algorithm of living organism such as insects or bees.

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