Discrete Dynamical Systems Research Papers (original) (raw)

Several endomorphisms of a plane have been constructed by coupling two logistic maps. Here we study the dynamics occurring in one of them, a twisted version due to J. Dorband, which (like the other models) is rich in global bifurcations. By use of critical curves, absorbing and invariant areas are determined, inside which global bifurcations of the attracting sets (fixed points, closed invariant curves, cycles or chaotic attractors) take place. The basins of attraction of the absorbing areas are determined together with their bifurcations.

- by and +1
- •
- Mathematics, Chaos Theory, Dynamical Systems and Bifurcation Theory, Discrete Dynamical Systems

A fractal structure is a countable family of coverings which displays accurate information about the irregularities that a set presents when being explored with enough level of detail. It is worth noting that fractal structures become especially appropriate to provide new definitions of fractal dimension, which constitutes a valuable measure to test for chaos in dynamical systems. In this paper, we explore several approaches to calculate the fractal dimension of a subset with respect to a fractal structure. These models generalize the classical box dimension in the context of Euclidean subspaces from a discrete viewpoint. To illustrate the flexibility of the new models, we calculate the fractal dimension of a family of self-affine sets associated with certain discrete dynamical systems.

- by Manuel Fernández-Martínez and +1
- •
- Fractal Structure, Fractal Dimension, Box-Counting Dimension, Discrete Dynamical Systems

El objetivo del presente artículo es describir un método mediante el cual sea posible la obtención de todas las soluciones de un sistema lineal para el caso de dos variables, mediante la utilización de una única función. Dicha función resultará ser una función de variable real, que toma valores en el campo complejo. En el procedimiento a seguir nos resultará de gran ayuda la serie de Fibonacci, cuya presencia en procesos naturales aparece en numerosas referencias ligado a procesos como la Phyllotaxis, la Simetría Pentagonal, los Quasicristales e incluso en procesos Biomoleculares como la estructura de las Proteínas y del ADN. Finalmente veremos que una generalización de la serie de Fibonacci será la función que nos permita describir cualquier sistema lineal de dos variables.

- by Angel Deleito García
- •
- Discrete Dynamical Systems

In this paper a new family of hyperbolic functions is presented. The behavior of the new hyperbolic functions is studied in specific cases. These functions present a marked asymmetry between left-handed or counterclockwise function (CCW) and right-handed or clockwise function (CW). Also they present a great variety of periods in the entire parametric range, exhibiting chaos in certain regions and order in others.

- by Angel Deleito García
- •
- Discrete Dynamical Systems
- by Sarker RANA
- •
- Computational Biology, Discrete Dynamical Systems
- by Enrique Peacock-Lopez
- •
- Engineering, Mathematics, Seasonality, Mathematical Sciences
- by Wenwei Zeng
- •
- Chemistry, Turbulent Flows, Discrete Dynamical Systems

Difference equations model the evolution of many processes of interest in physics, biology, economics, etc., in a discrete or iterative manner instead of a continuous parameter (e.g., time). In this regard, the renowned Fibonacci sequence constitutes an interesting example of iterative sequence that can be modeled in terms of such equations, more specifically a second-order homogeneous linear difference equation. This work intends to go a step beyond, introducing a generalized Fibonacci function based, on the one hand, on the close resemblance between such an equation and the one resulting from recasting a general two-variable linear system of difference equations as an also single second-order homogeneous linear difference equation. On the other hand, the discrete index of the usual Fibonacci sequence is replaced by a continuous (evolution) variable or parameter (although the Fibonacci initial conditions are kept). This leads to a rich variety of dynamical behaviors in the complex plane depending on the value of the parameters involved in the associated characteristic equation. The dynamics exhibited by the corresponding generalized Fibonacci functions are investigated and analyzed here, finding how apparently simple relations may describe relatively complex behaviors on the complex plane even in the case of regular or periodic solutions. Even though, it is seen that the curves in the complex plane displayed by the generalized Fibonacci functions during their evolution enable a better understanding of the behavior exhibited by the starting discrete model, such as the regimes of stability and instability, or the appearance of single and multiple fixed points.

- by Angel Deleito García
- •
- Discrete Dynamical Systems

- by Ralph Abraham
- •
- Mechanical Engineering, Mathematics, Applied Mathematics, Chaos Theory

Nonlinear difference equations, such as the logistic map, have been used to study chaos and also to model population dynamics. Here we propose a model that extends the " lose + lose = win " behavior found in Parrondo's Paradox, where switching between chaotic parameters in the logistic map yields periodic behavior (" chaos + chaos = order "). The model uses twelve parameters each reflecting the conditions of one of the twelve months. In this paper we study the effects of smooth-transitioning parameters and the robust system that emerges.

- by Enrique Peacock-Lopez and +1
- •
- Discrete Dynamical Systems
- by Emily Silva
- •
- Engineering, Mathematics, Seasonality, Mathematical Sciences
- by David POUVREAU
- •
- Mathematics, Algebraic Geometry, Topology, Linear Algebra

The thesis is concerned with the notion of hierarchical modularity in complex systems, its algorithmic detection and its use in explaining structure and dynamical behavior of such systems by means of hierarchical modular models. It proposes a new notion, antimodularity, in order to capture the possible occurrence of difficulties in scientific explanation due to the excessive computational complexity of algorithms used to find modular structure in big scientific datasets, this way highlighting a probable impending major shift of paradigm in certain special sciences.

- by Luca Rivelli
- •
- Philosophy of Science, Philosophy of Biology, Systems Biology, Complex Systems

This paper develops a dynamic model of North-South trade in which environment plays an important role. Our model is based on Chichilnisky North-South model for the macroeconomic interaction between two sectors of the world economy. The latter was introduced in a static context. We introduce dynamics in the original North-South model by allowing endogenous accumulation of capital. As a second extension of [1], we introduce here a variable which represents the system of property rights on the environmental asset which is used as an input to production. This could represent, for example, the property rights on forests from which wood is extracted to be used as an input to the production of traded goods or the property rights on water which is similarly used, perhaps for agricultural goods for export. The paper explains mathematically and through simulations the dynamics of a two-region world. There are two produced goods and two inputs to production. We show that as we vary the property rights of the environment the dynamics of the system changes. The less well defined are the property rights, the more chaotic are the model's dynamics.

In this paper, we study a classical discrete dynamical system, the Chip Firing Game, used as a model in physics, economics and computer science. We use order theory and show that the set of reachable states (i.e. the configuration space) of such a system started in any configuration is a lattice, which implies strong structural properties. The lattice structure of

- by Ha My Phan
- •
- Convergence, Lattice, Structural Properties, Dynamic System

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter λ and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real mth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of canonical polynomial maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.

- by Fermin Franco-Medrano
- •
- Dynamical Systems, Mathematical Biology, Chemotherapy, Chaos Theory