Quasiperiodicity and Chaos Chaos and Quasiperiodicity

Title of dissertation: QUASIPERIODICITY AND CHAOS Suddhasattwa Das, Doctor of Philosophy, 2015 Dissertation directed by: Professor James A. Yorke Department of Mathematics In this work, we investigate a property called “multi-chaos” is which a chaotic set has densely many hyperbolic periodic points of unstable dimension k embedded in it, for at least 2 different values of k. We construct a family of maps on the torus having this property. They serve as a paradigm for multi-chaos occurring in higher dimensional systems. One of the factors that leads to this strong form of chaos is the occurrence of a quasiperiodic orbit transverse to an expanding subbundle of the tangent bundle. Hence, a key step towards identifying multi-chaos numerically is finding quasiperiodic orbits in high dimensional systems. To analyze quasiperiodic orbits, we develop a method of weighted ergodic averages and prove that these averages have super-polynomial convergence to the Birkhoff average. We also show how this accelerated convergence of the ergodic averages over quasiperiodic trajectories enable us to compute the rotation number, Fourier series and Lyapunov exponents of quasiperiodic orbits with a high degree of precision (≈ 10−30). CHAOS AND QUASIPERIODICITY