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The talk introduces innovations in the areas of dynamical systems, chaos, and differential equations. It first shows that the classical theory of dynamical systems, pioneered by H.Poincare and developed by G. Birkhoff, can be further advanced if one extends the line of equilibria, periodic, almost periodic, recurrent, and Poisson stable motions with unpredictable points, as has been done in Marat Akhmet’s work. The dynamics is accompanied by chaotic phenomenon, labelled as Poincare chaos. This novel type of chaos is produced in quasi minimal sets and considered first of all with Poisson stable ingredients. The history of dynamical systems and differential equations (non-autonomous) is connected to mutual enrichment. This case demonstrates that the unpredictable point in special functional spaces can be an unpredictable function and, consequently, that a new type of chaos-generating oscillations are to be found in differential equations.
Marat Akhmet is a professor at the Department of Mathematics, Middle East Technical University, Turkey, and is a specialist in dynamical models and differential equations. He has published six books and more than a hundred scientific papers. In the last several years, he has been investigating dynamics of neural networks: periodic and almost periodic motions, stability, chaos and its control.