Henri Poincare's inventions in dynamical systems and topology

The purpose of this article is to trace the invention of images and concepts that became part of Poincaré’s dynamical systems theory and the Analysis Situs. We will argue that these different topics are intertwined whereas for topology Riemann surfaces and automorphic functions play an additional part. The introduction explains the term invention in the context of Poincaré’s philosophical ideas. Poincaré was educated in the school of Chasles and Darboux that emphasized the combination of analysis and geometry to perform mathematics fruitfully. This will be illustrated in the second section where we list his new concepts and inventions in dynamical systems, followed by the descriptions of theory available before Poincaré started his explorations and the theory he developed. The third section studies in the same way the development of Poincaré’s topological thinking that took place in the same period of time as his research in dynamical systems theory.