# Syllabus Math 489

## Math 489. Dynamics & Differential Eqns Instructor Syllabus

Text: K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An introduction to dynamical systems, Springer 1996.

This syllabus starts with discrete dynamical systems in low-dimensions
and introduces basic concepts and tools (such as fixed points,
stability, Lyapunov exponents) on simple tractable problems: logistic
maps, the baker's map etc. (Some of the discrete systems can be also
viewed as discretizations of ODEs.) With such preparation it becomes
easier for students to understand ODE techniques such as Lyapunov
exponents for ODEs, chaos, chaotic attractors, and so on.

Accordingly, the course is divided roughly into two interconnected parts: discrete dynamical systems and ODEs.

### Lectures

One-dimensional maps, period 3 implies chaos
L1. 1.1-1.2 One-dimensional maps. Cobweb plots.
L2. 1.3 Stability of fixed points.
L3. 1.4-1.5 Periodic points. Logistic maps.
L4. 1.6-1.7 Logistic map. Periodic Table. Sensitive dependence.
L5. 1.8 Itineraries (ITN). Computer Experiments.
L6. 1.8 ITNs. Period 3 implies all other periods. Sharkovskii Theorem (no proof).

Two-dimensional maps
L7. 2.1-2.2 Introduction to 2-dimensional maps. Sinks, sources, saddles.
L8. 2.3-2.4 Linear maps. Coordinate changes.
L9. 2.5 Nonlinear maps.
L10. 2.6 Stable and unstable manifolds.
L11. Challenge 2. Linear map on the torus.

Chaos
L12. 3.1 Lyapunov exponents and Lyapunov numbers.
L13. 3.2 Chaotic orbits.
L14. 3.3 Conjugacy of nonlinear maps.
L15. 3.3 Chaos in the logistic map.
L16. 3.4 Transition graphs and fixed points.

Fractals
L17. 4.1 Cantor sets.
L18. 4.2-4.4 Probabilistic construction of fractals. Fractals in deterministic systems.
L19. 4.5 Fractal Dimension
L20. 4.6 Computing box counting dimension.

Chaos in two-dimensional maps
L21. 5.1 Lyapunov exponents for 2d maps.
L22. 5.5 Markov partitions.
L23. 5.6 Horseshoe map. 6.2 Chaotic attractors.

Differential Equations
L24. 7.1-7.2 Differential equations in one dimension.
L25. 7.3 Linear differential equations.
L26. 7.4 Nonlinear systems.
L27. 7.5 Motion in a potential field.
L28. 7.6 Lyapunov functions

Periodic orbits and limit sets in ODEs
L29. 8.1-8.2 Omega-limit sets.
L30. 8.3 Poincaré-Bendixson theorem.

Chaos in ODEs
L31. 9.1 Lorentz attractor.
L32. 9.2 Stability in the large.

Bifurcations
L33. 11.1 Saddle-node and period doubling bifurcations.
L34. 11.4 Bifurcation of one-dimensional maps.
L35. 11.5-6 Bifurcation of plane maps.
L36. 11.8 Hopf bifurcation.

Exams and leeway: 5 hours.

Comments: Students are expected to do simple numerical simulations on about half the homework assignments. The computer experiments were taken from the book.

Revised and approved by UAC 4/25/12.