Dynamical systems theory is concerned with how systems change in time. These systems can be physical, chemical, biological, or of any nature.

 

When the rules that describe the change of quantities in time result in nonlinear relationships, it becomes impossible to solve the equations.

Then, we cannot describe the evolution of the dynamical system in a traditional sense. This is the situation where chaos may arise.

Instead, we adopt the approach of Poincaré and study the geometrical characteristics of solutions of the dynamical system.

The geometrical approach provides new and remarkable insights into previously hopeless problems.  Presently, the geometrical approach to dynamical systems theory is providing new results in a wide range of problems in science, engineering, economics, and the social sciences.