The use of the parameter space to uncover new structures in continuous-time dynamic systems modeled with sets of differential equations.

摘要

The parameter space structure is presented as a useful tool, to study dynamic systems represented by differential equations. Different structures are obtained when the values of a system's invariant (Lyapunov exponents, fractal dimension, etc) are associated to colors, and visualized in parameter space by means of a map. This color map technique allows quick access to quantitative information about the dynamics of the system. It also permits navigating through the parameter space while intentionally maintaining the system in a desired state, and avoiding regions where the system's behavior would be undesirable. Under this view, the rich structure of stability clusters the Rossler and Chua flows exhibit, is also reported for the first time. These clusters are composed of affine-similar repetitions of basic elementary cells that in this thesis are called swallows . The existence of swallows in flows is quite surprising since, up until recently, swallows have been only known to be associated with one- and two-dimensional discrete maps. Swallows tend to form dense groups where child swallows depart from a main cell along various directions following simple curves. The main cells are, at the same time, child swallows of other larger main cells. The study of these directions and structures could give valuable information about the system's dynamics.