Definition Of Dynamical System Research Articles

What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics.

Geometric realization of simplicial complexes makes them a unique representation of complex systems. The existence of local continuous spaces at the simplices level with global discrete connectivity between simplices makes the analysis of dynamical systems on simplicial complexes a challenging problem. In this work, we provide some examples of complex systems in which this representation would be a more appropriate model of real-world phenomena. Here, we generalize the concept of metaplexes to embrace that of geometric simplicial complexes, which also includes the definition of dynamical systems on them. A metaplex is formed by regions of a continuous space of any dimension interconnected by sinks and sources that works controlled by discrete (graph) operators. The definition of simplicial metaplexes given here allows the description of the diffusion dynamics of this system in a way that solves the existing problems with previous models. We make a detailed analysis of the generalities and possible extensions of this model beyond simplicial complexes, e.g., from polytopal and cell complexes to manifold complexes, and apply it to a real-world simplicial complex representing the visual cortex of a macaque.

Analytical Solutions for Advanced Functional Differential Equations with Discontinuous Forcing Terms and Studying Their Dynamical Properties

This paper aims to find new analytical closed-forms to the solutions of the nonhomogeneous functional differential equations of the nth order with finite and constants delays and various initial delay conditions in terms of elementary functions using Laplace transform method. As well as, the definition of dynamical systems for ordinary differential equations is used to introduce the definition of dynamical systems for delay differential equations which contain multiple delays with a discussion of their dynamical properties: The exponential stability and strong stability

Dynamical Systems in Categories

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

¿Sistemas dinámicos en ciencias sociales?

Dynamical systems are a specific kind of complex systems, namely: self-reference systems, autopoietic systems, disipative systems, adaptative complex systems, etc. In the context of mathematical modelling in social sciences, this article introduces a basic definition of dynamical systems and its main concepts. A background about key applications in social sciences is shown, and a specific case of the artisanal fisheries in which dynamical systems are certainly useful. Finally, some of the current research challenges of dynamical systems in social sciences are discussed.

¿Sistemas dinámicos en ciencias sociales?

Dynamical systems are a specific kind of complex systems, namely: self-reference systems, autopoietic systems, disipative systems, adaptative complex systems, etc. In the context of mathematical modelling in social sciences, this article introduces a basic definition of dynamical systems and its main concepts. A background about key applications in social sciences is shown, and a specific case of the artisanal fisheries in which dynamical systems are certainly useful. Finally, some of the current research challenges of dynamical systems in social sciences are discussed.

How to deal with potentially huge dimensional state space: The meta-dynamics approach—application to a model of the co-evolution of bacterio-phage populations

In many problems coming from the “complexity sciences”, the presence of high-dimensional state spaces and non-linear equations renders traditional mathematical objects useless. To deal with these issues, several approaches have recently been studied, most of them presenting the particularity of splitting the dynamical events in two categories; an upper level in which the events describe how the structure of the system changes (the meta-dynamics) and a lower level with the events describing the evolution of the elements in the structure (the dynamics). The state space is then not defined in extenso but unfolds or contracts during the evolution of the system. In term of simulation, this view allows us to deal with only a small part of the state space at each step avoiding the time and memory limitations. In this paper, we develop the implementation of an example of co-evolving population using a recently published unified mathematical formalism of the concept of meta-dynamics, based on an extension of Kalman's definition of dynamical systems. This general framework describes how to combine the two levels of dynamics, in order to allow the upper level (the meta-dynamics) to be able to modify the lower one (the dynamics). We explain here in detail how to use this formal approach in terms of simulation and algorithms. It allows us to present simulation results combining both changes of the structure of the system and changes at a lower population level.

REMARKS ON THE CONCEPT OF DYNAMICAL SYSTEMS

In this paper, the difference between the concepts of the states of a time system introduced in [1] and of the global state sets of general systems introduced in [3] is discussed. A generalization of the concepts of the global state sets and the global systems-response functions of general systems in given. The constructions of the global state sets and the corresponding global systems-response functions of a system are studied. A list of different definitions of dynamical systems is given, from which a few problems are asked, and a new definition of dynamical systems is given, which is a concept based on the mathematical models of general systems and time systems posed in [2, 3].