2.5 Qualitative and quantitative approach

There are basically two contrary approaches to the study and understanding of dynamical systems. In the first case, we concretise a particular problem as a dynamical system and gather as much information as possible about its behaviour. The logical consequence is a complicated set of equations, particularly since the equations must be formulated as realistically as possible in order to incorporate all the participating effects. In the second case, we are interested in the characteristics of dynamical systems in general and not in entering into detail. Here, too, we must differentiate between two cases:

i. A mathematical approximation in the classical sense which evolves and eluci­dates this new qualitative analysis of differential equations and develops it on the basis of assumptions and strict argumentation. Qualitative approaches are based on geometrical or topological methods which the great mathematicianHenri Poincaré (1854-1912) first applied to the investigation of the stability of our solar system. Topology, the study of qualitative geometry, has become an indispensible mathematical tool for describing the behaviour of dynamical systems in all their complexibility.

ii. An approximation or simulation in the sense of experimental mathematics, an approach made possible by the computer. In this case, the aim is to arrive at generally valid statements on dynamical systems on the basis of simple non-linear dynamical systems, the behaviour of which is studied numerically. The choice of representative examples in dynamical systems has to be guided by intuition and experience in order to discover characteristic patterns of response and thus obtain informative results for a broader class of dynamical systems.3 Mathematical introduction to dynamical systems