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Asymptotics and Perturbation Methods

Asymptotics and Perturbation Methods (Spring 2021). Instructor: Prof. Steven Strogatz, Department of Mathematics, Cornell University. Asymptotic methods and perturbation theory are clever techniques for finding approximate analytical solutions to complicated problems, by exploiting the presence of a large or small parameter. This course is an introduction to such methods and their applications in various branches of science and engineering. The prerequisites are a knowledge of basic calculus and differential equations at an undergraduate level. The course emphasizes concrete examples, intuition, and applications to science and engineering, rather than theorems, proofs, and mathematical rigor. The treatment is friendly yet careful. Topics include asymptotic expansion of integrals via Laplace's method, stationary phase, steepest descent, and saddle points. Perturbation methods for differential equations include dominant balance, boundary layer theory, multiple scales, and WKB theory. Most of the examples in the course deal with integrals or ordinary differential equations, but if time permits, we might also discuss some applications involving partial differential equations and difference equations.

Lecture 21 - Delayed Bifurcation


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Lecture 01 - Asymptotic Expansions
Lecture 02 - Properties of Asymptotic Expansions
Lecture 03 - Integration by Parts
Lecture 04 - Laplace's Method
Lecture 05 - Stationary Phase
Lecture 06 - Steepest Descent
Lecture 07 - Saddle Points
Lecture 08 - Integral Representations and an Introduction to Dominant Balance
Lecture 09 - Dominant Balance
Lecture 10 - Perturbation Methods for Algebraic Equations
Lecture 11 - Regular Perturbation Methods for ODEs
Lecture 12 - Introduction to Boundary Layer Theory
Lecture 13 - Higher Order Matching in Boundary Layer Theory
Lecture 14 - Location and Thickness of Boundary Layers
Lecture 15 - Corner Layers
Lecture 16 - A Tricky Nonlinear Boundary Value Problem
Lecture 17 - An Application to Systems Biology: the Michaelis-Menten Model
Lecture 18 - Introduction to WKB Theory
Lecture 19 - Turning Points and Airy Functions
Lecture 20 - WKB for Eigenvalue Problems
Lecture 21 - Delayed Bifurcation
Lecture 22 - Introduction to the Method of Multiple Scales
Lecture 23 - Two-Timing
Lecture 24 - Aging Spring and Adiabatic Invariants
Lecture 25 - Difference Equations and Multiple Scales
Lecture 26 - PDEs and Boundary Layers
Lecture 27 - Renormalization and Envelopes