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 Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 138
Author(s):  
Zhixin Zhang ◽  
Yufeng Zhang ◽  
Jia-Bao Liu ◽  
Jiang Wei
Keyword(s):  
Neural Networks ◽  
Fractional Order ◽  
Asymptotical Stability ◽  
Time Varying ◽  
Global Asymptotical Stability ◽  
Stability Criteria ◽  
Lmi Approach ◽  
Time Varying Delay ◽  
The Stability ◽  
Varying Delay

In this paper, the global asymptotical stability of Riemann-Liouville fractional-order neural networks with time-varying delays is studied. By combining the Lyapunov functional function and LMI approach, some sufficient criteria that guarantee the global asymptotical stability of such fractional-order neural networks with both discrete time-varying delay and distributed time-varying delay are derived. The stability criteria is suitable for application and easy to be verified by software. Lastly, some numerical examples are presented to check the validity of the obtained results.

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.

BIFURCATIONS OF LIMIT CYCLES CREATED BY A MULTIPLE NILPOTENT CRITICAL POINT OF PLANAR DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (02) ◽  
pp. 497-504 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Dynamical Systems ◽  
Critical Point ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Hopf Bifurcations ◽  
Bifurcation Phenomena ◽  
Cubic System ◽  
The Stability ◽  
Nilpotent Critical Point ◽  
Planar Dynamical Systems

Bifurcations of limit cycles created from a multiple critical point of planar dynamical systems are studied. It is different from the usual Hopf bifurcations of limit cycles created from an elementary critical point. This bifurcation phenomena depends on the stability of the multiple critical point and the multiple number of the critical point. As an example, a cubic system which can created four small amplitude limit cycles from the origin (a multiple critical point) is given.

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.

The global dynamics of a discrete nonlinear hierarchical population system

2019 ◽  
Vol 12 (02) ◽  
pp. 1950022
Author(s):  
Ze-Rong He ◽  
Huai Chen ◽  
Shu-Ping Wang
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Population Model ◽  
Numerical Experiments ◽  
Discrete Dynamical Systems ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Population System ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the global dynamics of a hierarchical population model, in which the fertility of an individual depends on the total number of higher-ranking members. We investigate the stability of equilibria, nonexistence of periodic orbits and the persistence of the population by means of eigenvalues, Lyapunov function, and several results in discrete dynamical systems. Our work demonstrates that the reproductive number governs the evolution of the population. Besides the theoretical results, some numerical experiments are also presented.

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