Plenary Lecture
Prof. Mark. Edelman
Department of Physics, Yeshiva University, New York, NY, USA
Title: General properties (periodic points, stability, and bifurcations) of discrete fractional dynamics
Abstract: Fractional calculus is a natural extension of regular calculus, and it has a rich history associated with the names of many great mathematicians. The recent increase in interest in fractional calculus is due to its multiple applications in various areas of science and engineering. But general properties of fractional dynamics are poorly investigated. It is easier to start the investigation of general properties by using fractional maps which we define as the discreet Volterra equations of a convolution type with the diverging kernels and absolutely converging first differences of the kernels. We derive the asymptotic stability conditions and the equations that define asymptotically periodic and bifurcation points in generalized fractional maps. We use those equations to draw bifurcation diagrams of the fractional logistic map and to show that the fractional Feigenbaum numbers may exist and have the same values as the regular Feigenbaum numbers. New results allow extension of the general theory to dissipative systems like the Hénon map.
Bio: Dr. Mark Edelman received his PhD in Astrophysics from the Odessa State University, former USSR. For a long time he worked as a Research Scientist and a Senior Research Scientist at the Rostov State University and Rostov Pedagogical Institute, Rostov-on-Don, USSR. From 1993 to 2010 he worked at the Courant Institute of Mathematical Sciences, New York University, New York City, USA, as a research scientist. His broad scientific interests include cosmic gas dynamics, corrugation instability of shock waves, nonlinear dynamics, chaos theory, fractional calculus. His present scientific activity is concentrated on the fractional dynamical systems and systems with power-law memory. He is a member of the editorial boards of “Fractional Calculus and Applied Analysis”, “Fractal and Fractional”, “Communications in Nonlinear Science and Numerical Simulations” and “Journal of Vibration Testing and System Dynamics”. His publications include edited book, ten book chapters, and numerous journal articles. He was a Plenary/Invited Speaker at numerous international conferences. Dr. Edelman is the winner of 2016 Zaslavsky Award.
Since 2009 Dr. Edelman has been teaching various physics courses and supervising student’s research at Stern College for Women, Yeshiva University, simultaneously continuing his research at the Courant Institute.