Lecture |
Sections |
Topics |
1 |
. |
General course overview. |
2 |
1.0 - 1.3 |
Definition of dynamical
systems. Discussion of importance and difficulty of nonlinear systems. Examples
of applications giving rise to nonlinear models. |
3 |
2.0 - 2.3 |
Elementary one-dimensional
flows. Flows on the line, fixed points, and stability. Application to population
dynamics. Discussion of how geometric "dynamical systems" approach
is different from approach in Math 33. |
4 |
2.4
- 2.6 |
"Advanced"
one-dimensional flows. Linear stability analysis (with numerous examples),
existence and uniqueness, impossibility of oscillations. |
5 |
2.6 - 2.7 |
Potentials. Introduction
to the idea of numerical solutions of nonlinear equations, including discussion
of basic methods, software tools (Matlab, Maple, Mathematica, DSTool, xppaut,
etc.). Advertisement for Math 151A/B. |
6 |
3.0 - 3.1 |
Introduction to
bifurcations, saddle-node bifurcation. Physical relevance of bifurcations,
introduction to bifurcation diagrams, notion of normal forms. For saddle-node
bifurcation, incorporate treatment in Crawford. |
7 |
3.2 - 3.3 |
Transcritical bifurcation.
Incorporate treatment in Crawford. Extended example on laser threshold. |
8 |
3.4 - 3.5 |
Pitchfork bifurcation.
Incorporate treatment in Crawford. Extended example on overdamped bead on
rotating hoop. |
9 |
3.5 |
Dimensional analysis.
Basic technique. Relate to overdamped bead example. |
10 |
3.6 - 3.7 |
Imperfect bifurcations.
Basic theory and bifurcation diagrams. Insect outbreak model, time permitting. |
11 |
4.0 - 4.3 |
Flows on the circle.
Definition, beating, nonuniform oscillators, ghosts and bottlenecks. |
12 |
4.4 - 4.6 |
Oscillator examples.
Instructor should choose one or two of the examples (overdamped pendulum,
fireflies, superconducting Josephson junctions) to cover in depth. |
13 |
5.0 - 5.1 |
Introduction to
two-dimensional linear systems. Motivating examples, mathematical set-up,
definitions, different types of stability. Phase portraits, stable and unstable
eigenspaces. |
14 |
5.2 |
Classification of
linear systems. Eigenvalues, eigenvectors. Characteristic equation, trace
and determinant. Different types of fixed points. (Suggestion: cover example
material in Section 5.3 and related problems on homework.) |
15 |
. |
Midterm |
16 |
6.0 - 6.2 |
Introduction to
two-dimensional nonlinear systems. Phase portraits and null-clines. Existence,
uniqueness, and strong topological consequences for two-dimensions. |
17 |
6.3 |
Equiliria and stability.
Fixed points and linearization. Effect of nonlinear terms. Hyperbolicity
and the Hartman-Grobman theorem. |
18 |
6.5 - 6.6 |
Special nonlinear
systems. Conservative and reversible systems. Heteroclinic and homoclinic
orbits. |
19 |
6.7 |
Extended application
of nonlinear phase plane analysis to classic pendulum problem without restricting
to small-angle regime. (Alternatively: another application of the instructor's
choice.) |
20 |
6.8 |
Index theory. Discussion
of local versus global methods. Definition and useful properties of the
index, with examples. |
21 |
7.0 - 7.1 |
Introduction to
limit cycles. Definition. Polar coordinates. Van der Pol oscillator and
other examples. |
22 |
7.2 |
Ruling out limit
cycles. Gradient systems, Liapunov functions, and Dulac's criterion, with
examples. |
23 |
7.3 |
Proving existence
of closed orbits. Poincare-Bendixson theorem, trapping regions. Examples.
Impossibility of chaos in the phase plane. |
24 |
8.0 - 8.1 |
Bifurcations in
two (and more) dimensions. Revisitation of saddle-node, transcritical, and
pitchfork bifurcations, with examples. |
25 |
8.2 - 8.3 |
Hopf bifurcation.
Definition. Supercritical, subcritical, and degenerate types. Application
to oscillating chemical reactions if time permits. |
26 |
8.4 |
Global bifurcations
of cycles. Saddle-node, infinite-period, and homoclinic bifurcations. Scaling
laws for amplitude and period of limit cycle. |
27 |
. |
leeway |
28 |
. |
Review |
