# PH3160 Non-Linear Phenomena and Chaos.

### More Detailed Syllabus.

(~3 sessions)
Intro. linear and nonlinear operators, recognizing nonlinear eqns.
Models - role of differential eqns and difference eqns.
Some history. Examples of complexity and nonlinearity in physics.
Driven dissipative systems - role of drivers, role of dissipation.
What's new in the mathematics and physics?
Dynamical Systems; States and Rules. Three categories.

(~4 sessions)
Flows with finite numbers of ODE's, autonomous and non-autonomous types.
1-D flows, fixed points. Examples from physics.
Linear stability analysis, characteristic times. Potential.
Bifurcations: Saddle-Node, Transcritical eg. laser threshold.
Pitchfork bifurcations, Euler strut. Hysteresis, Bistability egs.
Flows on Circle.

(~7 sessions)
2-D Flows, Phase plane, Linearizing 2-D Nonlinear systems.
2-D stability. Flow directions.
Nullclines, Limit cycle. Lienard eqns. Van der Pol and Duffing
2D bifurcations. Hopf bifurcations.
Driven 2nd order ODE's, Basins of attraction, Poincare maps.
How is chaos identified? Lyapunov exponents.

(~7 sessions)
1-D maps, cobwebs.
Logistic map, origin of period-2 and period-3 and ghosts.
Bernoulli shift map.
Lyapunov for maps. Fractals. Dimensions. Div test for dissipation.
Kaplan-Yorke conjecture.
Examples of stretching + folding, 2-D maps: Henon map.
Hamiltonian chaos. Standard map. (Henon-Heiles)
Turbulence. Routes to chaos. Ergodicity and K-entropy.
Briefly: (Chaology. Use of chaos for encryption.) Control of chaos.

### Course Work and Self-Study Assignments:

Six pieces of compulsory Course Work will be set during the course.
Marks for these count towards the final examination.

The self-study assignments are designed to assist your understanding ...
some of the required Course Work
may also ask for some of the results you obtain from working through these.

### Requirements for Final Examination:

You will be expected to be able to define the terminology, describe the
various concepts and procedures covered in lectures
and to be able to solve a variety of
problems (both large and small) at the same level and of the same type as in
the Examples and Exercises dealt with in class and as Course Work.
See previous years exam paper for an example of level and coverage required.

*Mike Wilson *

Jan 7 1998