|Class meetings:||Monday 2:30-5:30pm CB 120|
|Grade policy:||Assignment 50% + Final exam 50%|
|Office Hours:||Wednesday 3:30-4:30 pm or by appointment|
|Text Book||Differential Equations and Dynamical Systems. by Lawrence, Perko
Formatted lecture notes will be distributed. If you are interested in the notes, contact me.
Lectures: (Updated weekly)
- Chapter 1. Introduction
Typical examples from physics, biology and chemstry will be presented. Mass-spring oscillations, simple pendulum, Lienard systems, van der Pol oscillators, Duffing equation, predator-prey models with different type of response functions, compartmental models for infectious diseases. Equivalent forms between Lineard systems and predator-prey type of systems. Hilbert’s 16th problems.
- Chapter 2. Linear systems
- 2.1 Linear systems in plane. Equilibrium and classfication. Straight Line Solutions. T-D plane, phase plan and bifurcation of planar systems.
- 2.2. General theory of linear systems.
- 2.3. Linear systems with constant coefficients.
Exp(At). Jordan canonical form. Putzer’s Algorithm.
Math 6340 is a graduate course on ordinary differential equations and dynamical systems. Dynamical systems is a very active field of research that has a tremendous impact on our understanding of complicated, nonlinear phenomena in various systems of applied sciences. I will emphasize on both theoretical and applied aspects of differential equations and dynamical systems. We will cover most of the topics of the text book but the materials will be modified and reorganized.
- General properties of Differential Equations
- Linear Systems and Stability
General theory of linear systems, Periodic coefficients and Floquet Theory, Stability of linear and nonlinear systems.
- Nonlinear Systems: Local Theory
Linearization, Invariant manifolds, Hartman-Grobman Theorem, Normal form theory.
- Nonlinear Systems: Global Theory
Limit sets aned attractors, limit periodic sets and limit cycles; Poincare map, Poincare-Bendixson Theory.
Lotka-Volterra system, Predator-prey system, Lineard systems.
Introduction of Hilbert’s 16th problem.
- Nonlinear Systems: Bifurcation theory
Saddle-node bifurcation, Transcritical bifurcation, Pitchfork bifurcation, Hopf bifurcations, homoclinic bifurcations, Bogdanov-Takens bifurcation of codimension 2.
- Numerical tools (software) for normal forms and bifurcation diagrams.