## Course Information:

MAT 207, Fall Quarter, 2014
**Lectures:**
MWF 10:00–10:50 a.m., Physics 140

**Text:**
*Dynamics and Bifurcations*, J. Hale and H. Kocak, Springer-Verlag, 1991.

## Professor:

John HunterOffice: MSB 3230

Office hours: F 2:30–3:30 p.m., Th 3:00–4:00 p.m.

## TA:

Yuanyuan XuOffice: MSB 3206

Office hours: MW 1:00–2:00 p.m.

If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.(Henri Poincaré, 1903)

## Announcement

Solutions to the final are here. Course grades have been submitted to the registrar.

## Important Dates

- First class: Fri, Oct 3
- Last day to add: Fri, Oct 17
- Last day to drop: Wed, Oct 29
- Last class: Fri, Dec 12
- Academic holidays: Tue, Nov 11; Thu-Fri Nov 27-28

## Exams

There will be one midterm and an in-class final

- Midterm: Fri, Nov 7 (in class)
- Final: Fri, December 19, 10:30 a.m. – 12:30 p.m. (Exam Code V)

## Midterm

Solutions to the midterm are here.

## Final exam

Solutions to the final are here. Here is a rough list of topics we've covered. This material is covered in the course notes, homework solutions, and in the indicated chapters of the text.

General theory of ODEs. Reduction to first-order autonomous systems. Existence, uniqueness, and continuous dependence. Phase spaces and flows. Hamiltonian systems and gradient flows. [Course notes]

Linear first-order systems of ODEs and the matrix exponential. Equilibria. Linearization. Stability. Bifurcations. [Course notes]

General theory of iterated maps. Fixed points. Linearization. Stability. Poincare maps. [Course notes]

Scalar first-order ODEs. Phase lines and qualitative behavior of solutions. Bifurcations of equilibria (saddle-node, transcritical, pitchfork). [Ch. 1-2]

Scalar maps. Qualitative behavior of iterates. Fixed points, stability, and bifurcations. Period doubling bifurcation. [Ch. 3]

2x2 systems of ODEs. Phase planes. Topological equivalence of flows. Classification and stability of equilibria (saddle, node, spiral, center; source, sink; hyperbolic, non-hyperbolic). Stable/unstable manifolds. Homoclinic and heteroclinic orbits. Traveling wave solutions of PDEs. Bifurcations of equilibria. [Ch. 7-10]

Conservative and gradient 2 x 2 systems. [Ch. 14]

Closed orbits and limit cycles. Poincare-Bendixson theorem. Hopf bifurcation. [Ch. 11-12]

## Homework

Problems sets will be assigned each week on this web page, and collected in class on Monday.

## Grade

The course grade will be based on (weights in parentheses):

- Homework (30%)
- Midterms (30%)
- Final (40%)

## Some Papers

E. N. Lorenz, Deterministic nonperiodic flow (1963)

R. M. May, Simple mathematical models with very complicated dynamics (1976)

D. Ruelle and F. Takens,
On the nature of turbulence (1971)

## Homework

**Problem set 1** (due Mon, Oct 13)

**Solutions: Problem set 1**

**Problem set 2** (due Mon, Oct 20)

**Solutions: Problem set 2**

**Problem set 3** (due Mon, Oct 27)

**Solutions: Problem set 3**

**Problem set 4** (due Mon, Nov 3)

**Solutions: Problem set 4**

**Problem set 5** (due Mon, Nov 17)

**Solutions: Problem set 5**

**Problem set 6** (due Mon, Nov 24)

**Solutions: Problem set 6**

**Problem set 7** (due Fri, Dec 12)