## 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.

**Additional Topics (2 hrs):** Cascades, Lotka-Volterra models, Van der Pol system.

**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.