The stability of fixed points of discrete dynamical systems in the space conv ℝ n

2016 ◽  
Vol 50 (2) ◽  
pp. 163-165 ◽  
Author(s):  
V. I. Slyn’ko
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Discrete Dynamical Systems ◽  
The Stability

scholarly journals Local Stability in 3D Discrete Dynamical Systems: Application to a Ricker Competition Model

2017 ◽  
Vol 2017 ◽  
pp. 1-16
Author(s):  
Rafael Luís ◽  
Elias Rodrigues
Keyword(s):  
Population Dynamics ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Local Stability ◽  
Three Dimensional ◽  
Discrete Dynamical Systems ◽  
Competition Model ◽  
Centre Manifold ◽  
The Stability ◽  
Centre Manifold Theorem

A survey on the conditions of local stability of fixed points of three-dimensional discrete dynamical systems or difference equations is provided. In particular, the techniques for studying the stability of nonhyperbolic fixed points via the centre manifold theorem are presented. A nonlinear model in population dynamics is studied, namely, the Ricker competition model of three species. In addition, a conjecture about the global stability of the nontrivial fixed points of the Ricker competition model is presented.

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.

INFINITE DIMENSIONAL KRAWCZYK OPERATOR FOR FINDING PERIODIC ORBITS OF DISCRETE DYNAMICAL SYSTEMS

2007 ◽  
Vol 17 (12) ◽  
pp. 4261-4272 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete Dynamical Systems ◽  
Krawczyk Operator ◽  
Infinite Dimensional ◽  
Dispersal Model ◽  
Period 2 ◽  
Existence Of Zeros

In this work, we introduce the Krawczyk operator for infinite dimensional maps. We prove two properties of this operator related to the existence of zeros of the map. We also show how the Krawczyk operator can be used to prove the existence of periodic orbits of infinite dimensional discrete dynamical systems and for finding all periodic orbits with a given period enclosed in a specified region. As an example, we consider the Kot–Schaffer growth-dispersal model, for which we find all fixed points and period-2 orbits enclosed in the region containing the attractor observed numerically.

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.

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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