On the Classification of Dynamical Systems by Chris Zeeman in 1988
One is emboldened to raise the question of classifying all ordinary differential equations. Of course, such a task is impossible because it is too complicated, so perhaps we should first try and classify the ‘generic’ ones. Since any higher order equation can be written as a first order equation on a higher dimensional manifold the problem reduces to the classification of vector fields on a manifold. So let V denote the space of smooth vector fields on a manifold X. What should a classification programme consist of? We suggest four steps:
1. Choose an equivalence relation on V, and define a vector field to be stable if it has a neighbourhood of equivalents in V.
2. Prove that the stables are dense in V.
3. Classify the stable classes.
4. Classify the unstable classes of codimension 1,2,…, etc.