PHY 380.03 NONLINEAR SCIENCE

A course developed under the ISU Undergraduate Computational Science Lab Project <

Look at the detailed syllabus for this course.

• Richard Martin
• Moulton Hall 312F
• Email: rfm@entropy.phy.ilstu.edu
• Office hours: to be announced

Course Content:

The course will cover the basics of nonlinear science, an interdisciplinary paradigm for investigating problems in many fields which share the property that their mathematical models are nonlinear. We will begin in familiar territory with Newton's second law of motion, then progress through new tools and ideas for looking at chaotic systems with no dissipation, chaotic systems with dissipation, and end up applying some of these ideas and techniques of nonlinear science to begin to understand spontaneous self-organization and the behavior of complex systems. Selected "archetypal" examples will build intuition for each stop along the way, while examples from physics and chemistry will keep us grounded in the real world.

Here is a rough roadmap of what I'd like to cover:

I. Introduction: Nonlinear Dynamical Systems

• Linear and Nonlinear: What's the difference? Why does it matter?
• Everything "interesting" is nonlinear: some examples
• Newton's 2nd law: dynamical systems described by differential equations
• Some basic ideas: dynamical systems and the geometry of phase space
• Solution of dynamical systems by Integrals of motion
• Archetypal Example: the plane Pendulum
• Free pendulum: phase portraits
• Driven and damped pendulum: First return maps and chaos
• General Example: Bouncing ball on an oscillating table: dynamical systems described by discrete maps

II. Nondissipative Systems

• Hamiltonian systems: a different approach than Newton
• Finding Integrals of Motion: symmetry and luck/experience
• Fixed Points, stability and instability
• Effective Potentials
• Physics Example: Charged Particles in Magnetic Field Reversals - observability of the structure of phase space
• Types of orbits: transient, periodic, quasiperiodic, chaotic
• Poincaré surface-of-section
• Chemistry Example: Molecular Binding - chaos and dissociation
• Perturbation theory: approximate solutions
• A route to nondissipative chaos: KAM theory
• The chaotic hierarchy: simple harmonic motion to systems as random as a coin toss
• General Example: What makes time go forward? Does chaos imply irreversibility?

III. Dissipative Systems

• Dissipation and attractors
• Archetypal Example: A predator-prey equation: the logistic map
• Approaches to dissipative chaos: period doubling, quasiperiodicity, intermittency
• Physics/Engineering Example: The bouncing ball electronic circuit
• Correlation (fractal) dimension from experimental data
• Chemistry Example: Chemical kinetics of chemical oscillators
• Dissipative structures

IV. Chaos and Order: The road toward understanding complex systems

• Low dimensional complex systems: chaos versus randomness
• Self-organization: bifurcations and symmetry breaking
• Chemistry Example: Bifurcation-induced dissipative structures
• Self-organized criticality: the edge of instability
• Archetypal Example: the steady state angle of a sand pile
• Physics Example: the onset of convection
• Life far from equilibrium

Course Structure and Grading:

The class meets MWF 2:00 - 2:50 p.m. in Moulton 202. I will assume a level of educational maturity commensurate with senior standing; i.e. you all should be aware by now that learning science is an interplay between the professor's (hopefully useful) summary of basic material in class, reading the ideas of other scientists (in books, e.g.) and working with the tools and methods yourself (by doing sample problems and computational exercises). However, I realize that not everyone has the self-discipline to follow such a regimen by themselves, so I will assist you.

First, to encourage folks to keep up with the class, I will assign each day one person to take especially good notes, fill in gaps in derivations as needed, and writeup the notes (my class notes will be available for reference). These class notes will be kept on-line on the Entropy server. At the end of the semester the combined daily notes will be assembled and edited (by me) and hard copies made available to each student. Lecture transcriptions will not be graded, per se, but your participation will be recorded.

Of course, there will be also be homework problems: the teacher's way of easing you into "hands-on" practice with the new material. Problems will be designed fill in gaps in the lecture and go beyond the basics to apply your new tools and methods. These will be graded.

A great deal of what we know about nonlinear systems has come from numerical computations and simulations. These can provide physical intuition into the behavior of a complicated system, can yield new insights and ideas for subsequent analytical investigations, and sometimes are the only way of elucidating the physics contained in a mathematical model. To give you a taste of this method of discovery, I will assign computer projects. The full physics department network (Macs, terminals, RS6000s) will be available for these programming exercises, although you may use other hardware that you have may have access to. I will use FORTRAN 77 and Mathematica for my solutions, but will accept other languages; however, if you choose another language you will be required to overdo the comments explaining your code, so I can make sense of it.

As usual in upper division courses, collaboration and exchanging ideas on the homework problems is allowed (even encouraged), but copying is not; each student must hand in original homework solutions and computational projects.

I do not currently plan any mid-term exams, but will give an overview final exam (officially scheduled for Thursday, May 11 at 1:00 p.m.).

Your total grade will be determined as follows:

• Homework..............................45%
• Computational projects................45%
• Final Exam..............................10%

Late homework

Since I consider the homework of benefit to you, I'd like each student to finish it. Hence, I am somewhat flexible on hand-in date. However, to avoid total procrastination and to be fair to those who do turn in homework (including computational projects) on time, I will use the following rules:

(1) I will take off points if: (a) your paper is turned in after I have graded the bulk of the homeworks, or (b) if an "unreasonable" amount of time has passed since the due date.

(2) The homework will not be graded (i.e. your grade will be a zero) if the solutions have already been posted (for obvious reasons).

Note that you should always hand in any work you have done, to receive partial credit - e.g. don't feel that if you didn't get a computer program to work perfectly that your efforts are worth nothing: 40% is a lot better than zero...

To account for possible absences or illness, the lowest homework score will be dropped. No computer assignment will be dropped, however.

Miscellaneous:

I will put some materials on reserve in the library for your reference. Ask for the Physics 380.03 materials at the reserve desk. Other texts are recommended for their alternative viewpoints, by the way. Often looking at a topic from different points of view can clarify it in your own mind.

You are encouraged to come for help or general discussion during office hours. In particular, don't beat your head against the wall on a homework or computer problem you are stuck on. More often than not a few words of explanation can clear up the block. If you cannot make the scheduled hours, feel free to make an appointment with me or stop by and try your luck.

Note on Computer assignments:

Computer assignments are not unlike labs in other courses: you are given a small project with some instructions and will be expected to use the computer to solve the problem. You will hand in a writeup explaining what you did and how, and showing the results - often graphically. Hand in a copy of the source code also, with brief comments describing what each part of the program does. Note that only a program listing with a graph will earn few points - English words of explanation are also needed.