Topics in Dynamical Systems
Topics to be covered include complex dynamical systems, perturbation theory, nonlinear analysis of time series, chaotic dynamical systems, and numerical methods as dynamical systems. This course may be repeated for credit.
Introduction to Spectral Theory, Hislop, Springer, 1996.
MATH 850 or consent of instructor.
The class will cover various aspects of the theory of spectral, linear and nonlinear stability of special solutions to partial differential equations. As a first ingredient, the spectral theory of linear operators will be developed, including point and essential spectrum for self-adjoint, compact and locally compact operators and their perturbations, properties of the resolvent and Riesz projections, Kato-Rellich and Weyl theorems. Several important examples will be studied in the second part of the class, including linear stability of pulses in dissipative systems, stability of waves in Hamiltonian equations and the relation between linear and nonlinear stability. This will be done by introducing a variety of contemporary tools such as the Fredholm properties and exponential dichotomies of the operators, the Evans function and several decomposition techniques.
(Stanislavova 2008 )