LECTURE 5 : Attracting & Repelling Fixed Points | Banach Fixed Point Theorem Proof

In this Dynamical Systems Lecture 5, we explore attracting and repelling fixed points, local vs global attraction, contraction mappings, and a detailed proof of the Banach Fixed Point Theorem. We start by recalling fixed points and orbits, then classify fixed points using clear examples (f(x) = x/2 and f(x) = x²). Learn the difference between local and global attraction, how repelling points work via inverses, and the full step-by-step proof of Banach’s theorem using Cauchy sequences and complete metric spaces. Perfect for students of dynamical systems, real analysis, and metric spaces. Chapters: 0:00 Introduction & Recap of Previous Lectures 0:39 Attracting and Repelling Fixed Points Overview 2:42 Recalling Fixed Points 7:34 Example: f(x) = x/2 – Attracting Fixed Point 12:18 Attracting Fixed Point Definition 15:00 Local vs Global Attraction 25:28 Local Attraction Example: f(x) = x² 34:08 Repelling Fixed Points 42:22 Inverse Approach to Repelling Points 49:25 Key Definitions (Cauchy Sequence, Complete Space, Contraction) 52:38 Banach Fixed Point Theorem Statement 1:00:02 Proof – Constructing the Sequence 1:07:28 Estimating Distances Between Consecutive Terms 1:22:00 Proving the Sequence is Cauchy 1:43:24 Convergence to a Fixed Point 1:58:14 Uniqueness of the Fixed Point 2:05:26 Error Estimate 2:10:18 Summary & Practice Questions Like, subscribe, and share if this helped! Drop your questions in the comments.