Softwares # Dynamic Systems In Mathematics, Conceived A Dynamic System.

Dynamic systems allow obtaining or predicting values of a certain magnitude to respond to problems or needs in the real world (physical, economic, medical.

This page, attached to the Colombian section of the Society of Industrial and Applied Mathematics (COSIMA), seeks to link and create a bond between those somehow related to Colombia’s dynamic systems. You are a springboard to publicize the work of researchers in the area, the different organized events, and the trajectories of groups and individuals.

In historical terms, we can think of dynamical systems as a branch of mathematics founded by Henry Poincare (1854 – 1912) 1, who, in his studies of periodic orbits for three-body problems, provided a brilliant set of tools, which today we know as qualitative theory of differential equations.

as a field of vectors in phase space solution tangent curve at each point to the vectors of said field. His most significant interest, which was the problems of celestial mechanics, was the description of the phase portrait, that is, of the entire set of solutions and the stability of the answers, which consisted of qualitative analysis of the results.

It can be explained from a simple pendulum’s movement to the planetary motion. In addition to the comprehensive phenomenological coverage, those first tools developed by have multiplied today, so much so that to name a few, we would have to see with a magnifying glass a particular problem. And also It is enough to mention that dynamical systems feed on ordinary and partial differential equations, geometry, topology, analysis, computation, probability, etc.

According to systems theory, all entities or objects are systems or components of another higher-level system. Dynamic systems are those in which their behavior changes or evolves, and these modifications or transformations of their state can be analyzed or modeled using mathematical models.

The methodology that allows us to study and describe how the environment and the iteration between the components and parts that make up the system affect the behavior or state of the system over time are called “system dynamics.”

Mathematically, And also we define a dynamic system as one whose behavior or state depends on a variable magnitude (starting now variable) over time or that changes or evolves (or a set of these).

In mathematics, dynamic systems allow us to analyze and describe the change or evolution experienced by a variable over time. And also This change or evolution variable will be by an equation (or system of equations) that will predict its value at any moment. For simplicity in describing this type of system, we consider systems modeled by a single equation in this post.

Depending on the time scale considered to study the evolution of the value of the variable, we will have two types of dynamic systems:

- Discrete: the time values are determined or specific among a set of possible values. For example, t = {1, 2, 3} , where t can only take the values 1, 2, or 3.
- Continuous: Time values are any value in a specific range. For example, t is any real value in the interval [0, 10]. That is, t can take any value between 0 and 10 (1, 2.1, 3.768).

Based on the equation that predicts the behavior of the variable as a function of time, we will have:

- Linear systems are those in which the expression of the equation or the relationship between its parameters is linear. For example, x(t+1) = 3x(t) .
- Nonlinear systems are those in which the relationship between the parameters of the equation is not linear. For example, x(t+1)=3(x(t))2 .

- Difference equations
- Ordinary Differential Equations (ODE)
- integral equations
- Differential equations with delays
- Partial Differential Equations

As discussed in the previous point. Dynamic systems have multiple applications for studying and modeling natural systems to solve numerous problems or respond to various needs.

And also calculating the profits or profitability obtained with a specific economic investment over time, predicting the population of. And also one particular region after a certain number of years. Or the temporal evolution.

The mathematical theory behind dynamic systems is address in regular study programs. And also as in the Master’s Degree in Mathematical Engineering and Computing at UNIR. It is an essential topic for the study and modeling of systems. And also Authentic in research of all kinds (physics, medicine, economics.

**Also Read: What Is The Scientific Method, Experimental Scientific, and More**

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