Local and global dynamics in 1-D Hamiltonian lattices: From physics to engineering

Anastasios (Tassos) Bountis

Professor Emeritus, Professor EmeritusUniversity of Patras, Patras, Greece, and Center for Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia tassosbountis@gmail.com

Abstract: Local and global stability properties of 1-D Hamiltonian lattices of N interacting particles have been studied extensively for more than 60 years, in view of their important implications for statistical mechanics and solid-state physics [1]. Most studies so far have focused on analytic interparticle interactions, ranging from nearest neighbor to full range, often in the presence of on-site potentials [2]. Moreover, under periodic driving at one end of the lattice, the phenomenon of energy supratransmission in such lattices has been observed and thoroughly documented [2,3]. In the present lecture, I will first describe an approach from local to global dynamics and statistics in these systems based on the study of some of their simple periodic solutions (nonlinear normal modes) as the total energy is increased. Next, I will apply this approach to study analogous phenomena in 1-D Hamiltonian lattices that arise in various mechanical engineering applications, such as graphene elasticity, Hollomon’s law of “work hardening” and hysteretic damping. These involve nearest-neighbor interactions that are: (a) either purely non-analytic, (b) harmonic plus non-analytic or (c) analytic with non-analytic hysteretic damping effects [4 – 6]. In some cases, I will apply periodic driving at one end of the lattice and demonstrate the occurrence of energy supratransmission for the first time in these systems.

References:

  1. T. Bountis and H. Skokos, “Complex Hamiltonian Dynamics”, Synergetics series of Springer Verlag, April 2012.
  2. A. Bountis, “Complex Dynamics and Statistics of 1-D Hamiltonian Lattices:Long Range Interactions and Supratransmission”, Proc. of 6th Ph.D. School on “Mathematical Modeling of Complex Systems”, Nonlinear Phenomena in Complex Systems, 2 (3) (2020).
  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, 2019.
  4. A. Bountis, K. Kaloudis, Th. Oikonomou, B. Many Manda, Ch. Skokos, “Stability Properties of 1-D Hamiltonian Lattices with Non-Analytic Potentials”, International Journal of Bifurcation and Chaos, Volume No. 30, Issue No. 15, Article No. 2030047 (2020).
  5. A. Bountis, K. Kaloudis and Ch. Spitas, “Periodically Forced Nonlinear Oscillators With Hysteretic Damping”, Journal of Computational Nonlinear Dynamics,15 (12) 121006 (2020).
  6. Bountis T., Kaloudis K., Shena J., Skokos Ch., and Spitas Ch., “Energy Transport in 1-Dimensional Oscillator Arrays with Hysteretic Damping”,Eur. Phys. J. Spec. Top. 231: 225-236 (2022).