| The mathematics of fractals and fractional calculus: New applications in science Anastassios Bountis1; 1UNIVERSITY OF PATRAS, Patras, Greece; PAPER: 335/Mathematics/Regular (Oral) SCHEDULED: 14:50/Wed. 30 Nov. 2022/Arcadia 3 ABSTRACT: The theory of Nonlinear Dynamics refers to the study of time - evolving processes in physical systems, whose evolution, although deterministic, often leads to behaviors that are unpredictable for long times. When time is a continuous variable, these processes are described by systems of nonlinear differential equations. However, it is often mathematically convenient to view time as a discrete variable, and describe the dynamics via nonlinear difference equations, which are computationally much easier to analyze [1,2]. In this lecture, I will begin by reviewing the geometry of fractals, originally developed to study objects that are geometrically complex, and often form structures called strange attractors to which the solutions of differential or difference equations converge as time grows indefinitely. Next, I will describe results of physical importance regarding energy transmission in Hamiltonian lattices, which constitute excellent models of dynamical processes occurring in Solid State Physics. In such processes, it is highly desirable to consider, beyond the short range effects of nearest neighbors, long range phenomena in which particles are influenced by distant neighbors. In this framework, it is often mathematically relevant to replace ordinary derivatives by their fractional form, in which ordinary differentiation is replaced by an operator Dα where α is not a positive integer [3,4]. For example, regarding anomalous diffusive processes in complex media, one often employs fractional-order differential equations to account for nonlocal diffusion effects. References: [1] T. Bountis, ''Fundamental Concepts of Chaos: Part I'', Open Systems and Information Dynamics, 3 (1), 23-95 (1995). [2] T. Bountis, ''Fundamental Concepts in Classical Chaos, Part II: Fractals and Chaotic Dynamics'', Open Systems and Information Dynamics, 4,281-322 (1997). [3] J. E. Macias-Diaz, A. Bountis, H. Christodoulidi, “Energy Transmission in Hamiltonian Systems with Globally Interacting Particles and On-Site Potentials”, Mathematics in Engineering, 1(2): 343–358 (April 2019). [4] E. Macias Diaz and A. Bountis, “Nonlinear Supratransmission in Quartic Hamiltonian Lattices, with Globally Interacting Particles and On--Site Potentials”, Journal of Computational Nonlinear Dynamics, 16(2) 021001 (2021). |
