In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by ...using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.
We aim to provide a theoretical framework to explain the discrete transitions of mood connecting ideas from network theory and dynamical systems theory. It was recently shown how networks (graphs) ...can be used to represent psychopathologies, where symptoms of, say, depression, affect each other and certain configurations determine whether someone could transition into a depression. To analyse changes over time and characterise possible future behaviour is in general rather difficult for large graphs. We describe the dynamics of graphs using one-dimensional discrete time dynamical systems theory obtained from a mean field approximation to stochastic cellular automata (SCA). Often the mean field approximation is used on a regular graph (a grid or torus) where each node has the same number of edges and the same probability of becoming active. We provide quantitative results on the accuracy of using the mean field approximation for the grid and random and small-world graph to describe the dynamics of the SCA. Bifurcation diagrams for the mean field of the different graphs indicate possible phase transitions for certain parameter settings of the mean field. Simulations confirm for different graph sizes (number of nodes) that the mean field approximation is accurate.
•Mean field framework to analyse complex and dynamic graphs.•Extensions of the mean field to random and small-world graphs.•High accuracy of the mean field approximation to stochastic process.•Mean parameter of majority function determines stability or bistability.•Possibility to use this framework to explain psychopathology.
In this paper we analyze the dynamics of a nonlinear Cournot-type duopoly game with differentiated goods for two bounded rational players with different objective functions. Specifically, the first ...player is a semi-public company and cares about a percentage of the social welfare and the second player is a private company which cares only about its own profit maximization. The game is modelled with a system of two difference equations We examine the effect of the parameters on the equilibria of the model and we analyse their stability conditions. Complex dynamic features including period doubling bifurcations of the unique Nash equilibrium are also investigated. Numerical simulations are carried out to show the complex behaviour. The chaotic features are justified numerically via computing Lyapunov numbers, sensitive dependence on initial conditions, bifurcation diagrams and strange attractors.
A new bio-inspired method for optimizing the objective function on a parallelepiped set of admissible solutions is proposed. It uses a model of the behavior of tomtits during the search for food. ...This algorithm combines some techniques for finding the extremum of the objective function, such as the memory matrix and the Levy flight from the cuckoo algorithm. The trajectories of tomtits are described by the jump-diffusion processes. The algorithm is applied to the classic and nonseparable optimal control problems for deterministic discrete dynamical systems. This type of control problem can often be solved using the discrete maximum principle or more general necessary optimality conditions, and the Bellman’s equation, but sometimes it is extremely difficult or even impossible. For this reason, there is a need to create new methods to solve these problems. The new metaheuristic algorithm makes it possible to obtain solutions of acceptable quality in an acceptable time. The efficiency and analysis of this method are demonstrated by solving a number of optimal deterministic discrete open-loop control problems: nonlinear nonseparable problems (Luus–Tassone and Li–Haimes) and separable problems for linear control dynamical systems.
We introduce the Ellis semigroup of a nonautonomous discrete dynamical system (X, f
1,∞
) when X is a metric compact space. The underlying set of this semigroup is the pointwise closure of
in the ...space X
X
. By using the convergence of a sequence of points with respect to an ultrafilter it is possible to give a precise description of the semigroup and its operation. This notion extends the classical Ellis semigroup of a discrete dynamical system. We show several properties that connect this semigroup and the topological properties of the nonautonomous discrete dynamical system.
Jordan canonical form (JCF) is one of the most important, and useful, concepts in linear algebra. Mathematics, physics, biology, science and engineering undergraduates often find the first ...application of real JCF in the discipline of differential equations (continuous models) to solving systems of differential equations. In this work, we apply real JCF to understand, through analytical and numerical methods, the two-dimensional linear discrete dynamical systems. After that, we present a schematic model of red blood cell production. An advantage of discrete models over continuous models is that they readily yield numerical exploration, whether by calculator or computer (in our case we will use Mathematica software). This fact provides us with a valuable pedagogical resource because we can find a simple model to show the links between mathematics and other areas of knowledge without the need for sophisticated mathematical concepts beyond linear algebra. Following Paul Halmos's suggestion, 'The only way to learn mathematics is to do mathematics', some classroom exercises are given to better understand the subject, and a project for further student investigations is suggested.
The function ω ƒ on simple n-ods Vidal-Escobar, Ivon; Garcia-Ferreira, Salvador
Applied general topology,
10/2019, Letnik:
20, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Given a discrete dynamical system (X, ƒ), we consider the function ωƒ-limit set from X to 2x asωƒ(x) = {y ∈ X : there exists a sequence of positive integers
n1 < n2 < … such that limk→∞ ƒnk (x) = ...y},for each x ∈ X. In the article 1, A. M. Bruckner and J. Ceder established several conditions which are equivalent to the continuity of the function ωƒ where ƒ: 0,1 → 0,1 is continuous surjection. It is natural to ask whether or not some results of 1 can be extended to finite graphs. In this direction, we study the function ωƒ when the phase space is a n-od simple T. We prove that if ωƒ is a continuous map, then Fix(ƒ2) and Fix(ƒ3) are connected sets. We will provide examples to show that the inverse implication fails when the phase space is a simple triod. However, we will prove that:Theorem A 2. If ƒ: T → T is a continuous function where T is a simple triod then ωƒ is a continuous set valued function iff the family {ƒ0, ƒ1, ƒ2,} is equicontinuous.As a consequence of our results concerning the ωƒ function on the simple triod, we obtain the following characterization of the unit interval.Theorem A 1. Let G be a finite graph. Then G is an arc iff for each continuous function ƒ: G → G the following conditions are equivalent:
(1) The function ωƒ is continuous.
(2) The set of all fixed points of ƒ2 is nonempty and connected.
In this study, we consider the dynamics of a fuzzy system extended by a crisp compact system, which has a mixing regular periodic decomposition. First, we describe the complete dynamics of Zadeh's ...extension of a crisp compact system, which has a mixing regular periodic decomposition, by splitting the space of fuzzy numbers on the compact space. Next, using the results obtained, we prove that the topological entropies of the crisp system and its Zadeh's extension to the space of fuzzy numbers are the same.
Let
X
be a compact metric countable space, let
f
:
X
→
X
be a homeomorphism and let
E
(
X
,
f
) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (1) ...(
X
,
f
) is equicontinuous, (2) (
X
,
f
) is distal and (3) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (
X
,
f
) is distal if, and only if,
E
(
X
,
f
) is a group.
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