Fractals & Chaos

Brief syllabus

Fractals, iterated function systems, stability of fixed points, period doubling, bifurcations, chaos, Lyapunov exponents, intermittency, quasiperiodicity, phase locking, basins of attraction, dissipative maps, strange attractors, area-preserving maps, Julia and Mandelbrot sets, phase plane analysis, structural stability, limit cycles, Poincaré sections, delay-differential equations, Lyapunov functions, time-series analysis.

Objectives

By the end of the course students should be able to:

• understand the concepts listed in the detailed syllabus

• find the fractal, embedding, and topological dimensions of objects

• find an iterated function system whose attractor is a given object

• solve linear and some nonlinear difference equations

• find and classify fixed points, 2-cycles, and bifurcations of maps

• use phase plane analysis to determine the nature of the solutions of sets of ODEs

• find and classify equilibria and bifurcations of sets of ODEs and delay-differential equations

• demonstrate the existence of a limit cycle and determine the period of a relaxation oscillator

Lectures

Week 1

Fractals: definition and examples of fractals and exact and statistical self-similarity; fractal dimensions and their measurement; topological and embedding dimensions; prefractals; Laplacian growth.

Week 2

Iterated function systems: collage theorem.

Week 3

Difference equations: solvable linear and nonlinear difference equations, fixed points and their stability, critical points, attractors, basins of attraction.

Week 4

Logistic map: period doubling, bifurcation diagrams, Feigenbaum constants, Lyapunov exponents, chaos, intermittency.

Week 5

Circle maps: quasiperiodicity, phase locking, Arnold tongues.

Week 6

Higher-dimensional maps: classification of fixed points, stable and unstable manifolds, dissipative maps.

Week 7

Area-preserving maps: KAM theorem; Poincaré-Birkhoff fixed-point theorem; application to celestial mechanics; standard map.

Week 8

Exam 1 (on weeks 1-7).

Week 9

Maps on the complex plane: Julia and Mandelbrot sets.

Week 10

Dynamics of continuous 2-d systems: autonomous and non-autonomous systems, phase plane analysis, stability analysis of fixed points, structural stability.

Week 11

Limit cycles: Poincaré-Bendixson theorem, predator-prey and other models.

Week 12

Types of bifurcation including pitchfork, saddle-node, Hopf, and Neimark bifurcations.

Week 13

Higher-dimensional continuous systems: Lyapunov exponents, Poincaré section, Lorenz equations, chaos, strange attractors.

Week 14

Introduction to delay-differential equations.

Week 15

Lyapunov functions; time-series analysis.

Week 16

Topics of current interest related to fractals or chaos.

Week 17

Exam 2 (on weeks 1-16).

Teaching methods

Lectures and detailed printed lecture notes will be provided.

Course assessment

Homework: 4%

Exam 1: 48%

Exam 2: 48%

Recommended books

• "Nonlinear Dynamics and Chaos" by Steven H. Strogatz
• "Techniques in Fractal Geometry" by Kenneth Falconer
• "Chaos and Time-Series Analysis" by Julien Clinton Sprott