Map of fixed points and Lyapunov functions for one class of discrete dynamical systems

1994 ◽  
Vol 56 (5) ◽  
pp. 1125-1131 ◽  
Author(s):  
R. N. Ganikhodzaev
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Lyapunov Functions ◽  
Discrete Dynamical Systems

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.

Analysis of Snapback Repellers Using Methods of Symbolic Computation

2019 ◽  
Vol 29 (04) ◽  
pp. 1950054 ◽  
Author(s):  
Bo Huang ◽  
Wei Niu
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Unit Circle ◽  
Symbolic Computation ◽  
Complex Plane ◽  
Discrete Dynamical Systems ◽  
Algorithmic Approach ◽  
Algebraic Criterion ◽  
Algebraic Problem

This paper presents an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multidimensional discrete dynamical systems based on a modified Marotto’s theorem developed by Li and Chen (called “Marotto–Li–Chen theorem”). Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and propose an algorithmic approach to solve this algebraic problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.

NUMERICAL COMPUTATIONS OF CONNECTING ORBITS IN DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1281-1293 ◽  
Author(s):  
FENGSHAN BAI ◽  
GABRIEL J. LORD ◽  
ALASTAIR SPENCE
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete System ◽  
Numerical Technique ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Connecting Orbits

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in nonautonomous and autonomous systems of ordinary differential equations. First, the existence and computation of connecting orbits between fixed points in discrete dynamical systems is discussed; then it is shown that the problem of finding connections between equilibria and periodic solutions in continuous systems may be reduced to finding connections between fixed points in a discrete system. Implementation of the method is considered: the choice of a linear solver is discussed and phase conditions are suggested for the discrete system. The paper concludes with some numerical examples: connections for equilibria and periodic orbits are computed for discrete systems and for nonautonomous and autonomous systems, including systems arising from the discretization of a partial differential equation.

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.

ON THE COMPLEXITY OF COUNTING FIXED POINTS AND GARDENS OF EDEN IN SEQUENTIAL DYNAMICAL SYSTEMS ON PLANAR BIPARTITE GRAPHS

2006 ◽  
Vol 17 (05) ◽  
pp. 1179-1203 ◽  
Author(s):  
PREDRAG T. TOŠIĆ
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Large Scale ◽  
Bipartite Graphs ◽  
Discrete Dynamical Systems ◽  
Boolean Formula ◽  
Multi Agent Systems ◽  
Combinatorial Structures ◽  
Garden Of Eden ◽  
Update Rules

We study counting various types of configurations in certain classes of graph automata viewed as discrete dynamical systems. The graph automata models of our interest are Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively). These models have been proposed as the mathematical foundation for a theory of large-scale simulations of complex multi-agent systems. Our emphasis in this paper is on the computational complexity of counting the fixed point and the garden of Eden configurations in Boolean SDSs and SyDSs. We show that counting these configurations is, in general, computationally intractable. We also show that this intractability still holds when both the underlying graphs and the node update rules of these SDSs and SyDSs are severely restricted. In particular, we prove that the problems of exactly counting fixed points, gardens of Eden and two other types of S(y)DS configurations are all #P-complete, even if the SDSs and SyDSs are defined over planar bipartite graphs, and each of their nodes updates its state according to a monotone update rule given as a Boolean formula. We thus add these discrete dynamical systems to the list of those problem domains where counting combinatorial structures of interest is intractable even when the related decision problems are known to be efficiently solvable.

scholarly journals Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Héctor Barge ◽  
José M. R. Sanjurjo
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Fixed Points ◽  
General Result ◽  
Discrete Dynamical Systems ◽  
Hopf Bifurcations ◽  
Substantial Role ◽  
Higher Dimensional ◽  
Low Dimensional

<p style='text-indent:20px;'>In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.</p>

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