The objective of dynamical systems is to study the evolution equations to try to obtain information on their long-term behavior, knowing the topological and metric characteristics of the space where they are defined. The study of dynamical systems dates back to the end of the 19th century, when Henri Poincaré published a work on Celestial Mechanics (Les Méthodes Nouvelles de la Mécanique Céleste). In this work, Poincaré proposed to use knowledge from other areas such as topology, geometry, algebra and analysis. With this he managed to describe, qualitatively (without trying to solve the equations explicitly), the evolution of the system. This proposal is what marks the beginning of Dynamic Systems.
Currently, in addition to evolution equations, dynamical systems include the study of transformation iterations, difference equations, and partial differential evolution equations.
It is noteworthy that the results and methods of dynamical systems have helped to explain complex phenomena in chemistry, physics, biology, economics, among others.
One such method uses both ergodic theory and analysis. An example of this is the Lyapunov exponents, which give an idea of the long-term behavior of solutions that arise from close initial conditions. This knowledge allows us to know if the system is stable or not. Positive Lyapunov exponents indicate a system that is very sensitive to initial conditions. On the other hand, it is also necessary to study the structural stability of the system, since this gives us an indication that small errors in the determination of the system will not affect the long-term global behavior.
The areas of interest of the group of dynamic systems in the IMCA are:
- Stochastic stability;
- Lyapunov exponents;
- Singular hyperbolicity;
- Invariant sets and stability indicators.
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