Topics in dynamics: First passage probabilities and chaotic properties of the physical wind-tree model

Dynamical systems;Chaos

Bolding, Mark M. ; Matzinger, Heinrich Bunimovich, Leonid Bonetto, Federico Goldsztein, Guillermo Mathematics ; Matzinger, Heinrich

University:Georgia Institute of Technology

Department:Mathematics

关键词: Dynamical systems;    Chaos;   

Others  :  https://smartech.gatech.edu/bitstream/1853/59931/1/BOLDING-DISSERTATION-2018.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We prove that the evolution of the "most chaotic" dynamical systems consists of three different stages. Consider a finite Markov partition (coarse graining) X of the phase space of a system. In the first short time interval there is a hierarchy with respect to the values of the first passage probabilities for the elements of X and therefore predictions can be made about which element an orbit will most likely hit. In the third infinitely long time interval there is an opposite hierarchy of the elements of X and therefore analogous finite time predictions can be made. Our results demonstrate that finite time predictions for the evolution of strongly chaotic dynamical systems are possible. We then show that mathematical billiards generated by the motion of a point particle do not adequately describe the dynamics of real physical particles within the same domain (billiard table), and introduce a new model for diffusion intended to revitalize the Ehrenfests’ Wind-Tree model.