PH.D. COMPREHENSIVE EXAMINATION
IN DYNAMICAL SYSTEMS (2006) SYLLABUS

Topics:
Existence-uniqueness theory and elementary properties of flows:
The generation of a flow via autonomous differential equations (regular flows), continuity in initial conditions, Implicit Function Theorem, Contraction Mapping Principle, topologically conjugate flows, Poincaré maps for periodic orbits and periodically forced systems. Differentiability of Poincaré maps and the variational equation.
Two dimensional systems:
Phase portraits, Hamiltonian systems, invariant sets, the fixed point index, Poincaré Bendixson theory, relaxation oscillations, the Van der Pol Oscillator and Liénard's equation.
Geometric Techniques:
Stability and attraction, invariant sets, properties of ω-limit sets of sets, Lasalle's invariance principle, Wazewski's principle, invariant manifold theory, Hartman-Grobman theorem, systems on the cylinder, stability of periodic orbits and the Poincaré map, the method of averaging for non-autonomous systems, Melnikov's method.
Linear systems:
Jordan canonical form, differentiability of flows, the variational system, Abel's formula, Liouville's Theorem for Hamiltonian systems, regular Sturm-Liouville theory, differential inequalities, Floquet theory.
Invariant manifold theory:
the stable, unstable, center-stable, center-unstable and center manifolds. Calculations on the center manifold, invariant manifolds for hyperbolic periodic orbits.
Elementary bifurcations:
bifurcations of rest points for flows and fixed points for maps, relation to the center manifold, the Hopf bifurcation theorem.
Continuation of Periodic Orbits:
Autonomous and nonautonomous perturbations, Melnikov functions.
Complicated behavior:
The horseshoe map, the Two-shift, symbolic dynamics, Periodic points, dense orbits, ω-limit sets, homoclinic and heteroclinic points, period three points for interval maps, the Smale-Birkoff homoclinic theorem, Melnikov's method.

Texts:
1. Carmen Chicone, Ordinary Differential Equations With Applications.
2. Hale and Koçac, Dynamics and Bifurcations.
3. K. Alligood, T. Sauer, and J. Yorke, Chaos: An Introduction to Dynamical Systems.
4. J. Hale, Ordinary Differential Equations.
5. V. Arnold, Ordinary Differential Equations.
6. Hirsch and Smale, Differential Equations, Dynamical Systems, and Linear Algebra.
7. Guckenhimer and Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields.

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