Exact coherent structures in spatiotemporal chaos: From qualitative description to quantitative predictions

Symmetries;Nonlinear dynamics;Chaos;Spatiotemporal chaos;Periodic orbit theory;Group theory

Budanur, Nazmi Burak ; Cvitanović, Predrag Physics Schatz, Michael Grigoriev, Roman Gangbo, Wilfrid Uzer, Ahmet Turgay ; Cvitanović, Predrag

University:Georgia Institute of Technology

Department:Physics

关键词: Symmetries;    Nonlinear dynamics;    Chaos;    Spatiotemporal chaos;    Periodic orbit theory;    Group theory;   

Others  :  https://smartech.gatech.edu/bitstream/1853/54445/1/BUDANUR-DISSERTATION-2015.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

The term spatiotemporal chaos refers to physical phenomena that exhibit irregular oscillations in both space and time. Examples of such phenomena range from cardiac dynamics to fluid turbulence, where the motion is described by nonlinear partial differential equations. It is well known from the studies of low dimensional chaotic systems that the state space, the space of solutions to the governing dynamical equations, is shaped by the invariant sets such as equilibria, periodic orbits, and invariant tori. State space of partial differential equations is infinite dimensional, nevertheless, recent computational advancements allow us to find their invariant solutions (exact coherent structures) numerically. In this thesis, we try to elucidate the chaotic dynamics of nonlinear partial differential equations by studying their exact coherent structures and invariant manifolds. Specifically, we investigate the Kuramoto-Sivashinsky equation, which describes the velocity of a flame front, and the Navier-Stokes equation for an incompressible fluid in a circular pipe. We argue with examples that this approach can lead to a theory of turbulence with predictive power.