How do I know if a differential equation leads to chaos or catastrophe? When I use catastrophe here, I mean a system exhibiting a finite number of bifurcations and by chaos, I mean a system exhibiting a (very) large number of bifurcations. I do know that catastrophe theory is based on Thom's theorem and chaos theory on qualitative analysis but I can't get over the fact that they are 2 different theories. They seem so similar in terms of bifurcations. So, which theory do I use before-hand to know if a differential equation leads to chaos or catastrophe and furthermore, can you please explain the exact difference?

I am not familiar with catastrophe theory but would like to comment on the 'chaos' part. Apriori, it is only in very simple systems that you can know if there will be chaos. Over the last 100 years, there have been many tools developed to analytically and computationally find out if a given parameterized systems is chaotic. E.g.:

1). Melnikov's method

2). Thurston-Nielsen classification of diffeos on surfaces

3). Detection of horseshoes.