SYLLABUS
FOR
PHYS460:
NONLINEAR SYSTEMS AND CHAOS
Instructor: Paul Fahey , Room 279
Credits: three
Course Description:
This
course develops the equations that describe several important nonlinear systems
in mechanics and in electronics and then develops the solutions. Concepts such as limit cycles, chaotic
attractors, hysteresis, stability and phase space will be defined and used to
understand complex systems. Classically
important oscillators such as the Duffing oscillator, the van der Pol
oscillator and the Lorenz equations will be solved at several different levels
of approximation with several ODE solvers. Chaos, bifurcations, the routes to
chaos, chaotic maps and the correspondence between maps and Poincare sections
of physical systems will be studied. The
output spectra of nonlinear systems in general will be discussed and methods
for calculating the amplitudes of harmonics, sub-harmonics and intermodulation
distortion products will be developed.
It is anticipated that the application of all of the above to the
nonlinear mechanical response of the inner ear will cap this course. The software packages MAPLE and MATLAB will
be used throughout the course in the solution development and display. CHAOTIC DYNAMIC WORKBENCH and CHAOS
DEMONSTRATIONS will be used to develop some concepts and to experiment with
model nonlinear systems.
Texts: The required/primary text is from Nonlinear
Dynamics and Chaos by Steven H. Strogatz (Addison-Wesley, 1994). Supplementing the primary text are: Chaos
and Nonlinear Dynamics by Robert C. Hilborn (Oxford University Press,
1994); Chaotic Dynamics of Nonlinear Systems by S. Neil Rasband (John
Wiley and Sons, 1990); Nonlinear Oscillations by Peter Hagedorn
(Clarendon Press/Oxford, 1988); Nonlinear Physics by R.Z. Sagdeev, D.A.
Usikov and G.M. Zaslavsky (Harwood Academic Publishers, 1988); Nonlinear
Oscillations in Physical Systems by C. Hayashi (reprinted from 1964 by
Princeton University Press, 1985). Also,
there is some light reading that will be required: SYNC: The
Emerging Science of Spontaneous Order by Steven Strogatz
(Hyperon, 2003).
The
software documentation of MAPLE, MATLAB, CHAOS DEMONSTRATIONS and CHAOTIC
DYNAMICS WORKBENCH will also be used by way of reference. Other software will be available such as
CHAOS FOR THE CLASSROOM.
Prerequisite: Calculus and elementary
physics.
Grading:
Grades will be based upon both individual and team assignments/projects
(60%) assigned as the course proceeds.
The projects will be graded on both clarity of written expression (where
appropriate) and mathematical and scientific accuracy. About 2/3 of the way through the course students
will submit a 300 to 400 word research topic proposal. Three weeks before finals students will
submit a 3,500 to 4,500 word first draft.
This will be corrected and resubmitted as a second draft. After corrections a final draft will be
submitted during finals weeks. The paper
will be written in the style of a scientific paper and the paper must use the
theory developed as the course proceeds.. This paper will comprise 40% of the
grade.
Topics:
The topics mostly follow the order of the primary text. They are:
1.
Outline of the course and familiarization with the texts and with the software.
2. Flows on the line.
3. Bifurcations.
4. Flows on the circle.
5. Linear systems.
6. Limit cycle.
7. Duffing oscillator solved with
approximations and with numerical ODE solvers (after most of the references
above). Multiple-valued solutions and
hysteresis (after the book by Hayashi) and subharmonics.
8. Van der Pol oscillator and limit cycles
and phase locking.
9. Lorenz oscillations and the onset of
chaos with examples of chaotic transients, intermittency, and period doubling.
10. Bifurcations revisited.
11. One-dimensional maps.
12. Strange attractors.
13. Nonlinear input-output functions and
spectra.
14. Nonlinear response of the inner ear and
it many manifestations. There is
considerable recent research activity on the role of Hopf oscillations in
setting hearing sensitivity.