Application for Morse-Smale systems All my life I do topological classification of Morse-Smale systems (flows and cascades). Today, fundamental science is not in fashion, to receive grants require an application. Can you please tell where to look.
 A: I know this is an older question, but since it may interest some, here are some papers about Morse-Smale theory applications:


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*This paper describes applications of Morse-Smale theory to sensor networks. It has great pictures, and describes how Morse-Smale theory allows one to avoid slow points in computer computations, among other benefits.

*This paper describes applications of Morse-Smale theory to video segmentation. In part, it says, 



In computer vision, detail has been often identi- fied with scale, and so-called scale-space approaches have achieved simplification by blurring. However, blurring obliterates small detail together with the boundaries of large regions. A more discriminating treatment of scale would instead eliminate small detail while keeping large parts crisply delineated.
  In this paper, we propose a representation of video that captures structure and affords flexible control of detail, without sacrificing crispness. Specifically, we represent structure through the so-called Morse- Smale complex of a scalar function f(x,y,t) associated with the video data, and we obtain a hierarchy of increasingly simple complexes through the process of topological simplification.



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*This thesis also describes how Morse-Smale theory can be used to simplify data storage.


All of these papers have references to other papers doing similar things. It seems like Morse-Smale theory has very practical applications to real-world issues!
The main ideas in these papers seem to be that having transverse stable and unstable manifolds gives a nice cell structure or helps us find saddle points more easily.
