Differential Equations and Dynamical Systems
Existence and uniqueness theorems. Linear systems; stability theory, perturbation theory. Poincare-Bendixson theory; boundary value problems.
Differential Equations and Dynamical Systems, Perko, Springer-Verlag, 7th Edition.
The purpose of this course is to give a rigorous mathematical development of many classical topics in the study of ordinary differential equations and, more generally, dynamical systems. Throughout, we take a geometric point of view seeking qualitative information on solutions of nonlinear dynamical systems even when explicit solution formulas do not exist. The theories and applications encountered in this course will create a strong foundation for studying nonlinear systems (ODE and PDE) and nonlinear science in general. Topics to be included are:
- Basic Theory: existence, uniqueness and regularity of solutions, continuous dependence on initial data and parameters, and extendability of solutions.
- Linear Dynamics: the fundamental theory for linear systems (global existence theorems, solution by operator exponentiation), the structure of solutions for autonomous linear systems, analysis in the constant coefficient (autonomous) case, Jordan normal forms and associated stability conditions, the periodic-coefficient case and Floquet theory.
- Introduction to Stability Theory: definitions, examples, and counterexamples of various notions of stability, the method of Lyapunov functions, and stability via linearization.
- Nonlinear Systems -- Local Theory: flows and linear approximation, critical points and hyperbolicity, stable and unstable manifold theorems, the Hartman-Grobman theorem, the center manifold theorem and Carr's theorem, normal form theory, and analysis of gradient and Hamiltonian systems.
- Nonlinear Systems -- Global Theory: important invariant sets (periodic, homoclinic, and heteroclinic orbits, etc.) and their associated invariant manifolds, the Poincare map and the orbital stability of periodic orbits, Bendixson-Dulac criteria for the nonexistence of periodic solutions, and the Poincare-Bendixson theorem for planar systems.
- Bifurcation Theory: bifurcations of equilibria, homoclinic and heteroclinic bifurcations (Melnikov functions), center manifolds and the Lyapunov-Schmidt reduction, global bifurcations, and the Smale Horse-shoe.
(Liu 2016 )