Ordinary Differential Equations

Overview

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ABOUT THE COURSE: Differential Equations are one of the central topics which results when studying models arising out of physical systems. Most often than not the nonlinear nature of the equation forces us to study the qualitative and geometric theory of equations and dynamical systems. In this course we start by first studying about the various aspect of Linear Systems and then analyze local and global behavior of nonlinear systems using techniques from Linear theory.INTENDED AUDIENCE: Undergraduate students in Mathematics, Physics and Engineering, Graduate students in Mathematics.PREREQUISITES: Linear Algebra and Several Variable Calculus.

Syllabus

Week 1: Review of Linear Algebra and Several Variable Calculus.
Week 2: Picard-Lindeloff Theorem for Existence and Uniqueness of Solution, Continuous Dependence on Initial Data.
Week 3: Linear System: Fundamental Existence Theorem and Uniqueness theorem, Solution Subspace, Linear Independence and Wronskian.
Week 4: Matrix Exponential: Convergence, Properties. Matrix Exponential for Diagonlizable Matrix, Fundamental Matrix, Generalized Eigenvalues, Matrix Exponential for non-Diagonlizable Matrix.
Week 5: Phase Plane Analysis: Phase Portrait, Fixed point and Linearization, Reversible System.
Week 6: Comparison Principle, Maximum Principle and Oscillation theory.
Week 7: Periodic System, Floquet theory, Hill’s equation.
Week 8: Limit Cycle: Closed Orbit, Poincare Map, Poincare-Bendixson theorem.
Week 9: Sturm-Liouville Theory for second order equations. Properties of the eigenfunctions, Spectrum of Linear Operators.
Week 10: Linearization and Stable Manifold Theorem.
Week 11: Stability in Lyapunov Sense, Lyapunov Direct Method.
Week 12: Limit Cycle and Poincare Bendixson Theorem.