Nonlinear dynamics and chaos

Applied Mathematics 147

(Fall 2004)

Instructor: Eli Tziperman;

TFs: Dorian Abbot <>, Laure Zanna <>

Day and time of course: Tue Thu 11:30-1;

Location: Cruft 319

Regular section time: Monday 17:00-18:00. Cruft 319

TF office hours for Dorian: Wednesday 14:00-15:00, Museum Building room 101, or call/ email him (Dorian Abbot <>)

Teaching notes Textbooks Syllabus Requirements

Announcements Last updated: Jan 15, 2005.
Final exam: will be held on Tue 01/18 at 2:15
Room: Sever Hall 102
Exam Aids Allowed: ONLY the following: 1. your notes from class; 2. notes from course website; 3. section handouts; 4. problem sets and solutions (your own and those posted on the course home page); 5. calculators/ graphing calculators.

Feel free to write or call me with any questions:
Eli Tziperman; eli AT

Teaching notes online:
1_intro, 2_intro, 3_bif1d, 4_bif1d, 5_bif1d, 6_bif2d, 7_bif2d, 8_bif2d, 9_bif2d, 10_chaos, 11_chaos, 12_chaos, 13_chaos, 14_chaos, 15_fract, 16_hamilt, 17_hamilt,

Sample Matlab programs: bakers_map.m, circle_map.m, cobweb_func.m, driven_pendulum.m, euler_course.m, feigenbaum.m, glider.m, henon.m, logistic_map.m, lorenz1.m, lorenz2.m, mandelbrot_Setsv.m, modified_euler_1d.m, my_quiver.m, pendulum.m, pendulum_self_sustained.m, perturbation_series_example.m pplane6.m, ppn6out.m, recon_lorenz_phase_space.m, sea_ice_switch.m, spectrum_harmonics.m, shilnikov.m, standard_map.m, standard_map_interactive.m, van_der_pol.m,

Homework: (due in class one week from date given) 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12

What's the point about optional/ extra credit problems: apart from the fun of doing them, they will count against homework problems in which you may have missed an answer. If you don't do the challenge problems, make sure you understand their solutions once posted.

homework solutions: 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11 12

Section notes from Dorian: 01, section1, section2, section3, section4, section5, section6, section7, section8, Gaspard_letter.pdf, section9, section10, section11,


Additional reading:


The course will introduce the students to the basic concepts of nonlinear physics, dynamical system theory, and chaos. These concepts will be demonstrated using simple fundamental model systems based on ordinary differential equations and some discrete maps. Additional examples will be given from physics, engineering, biology and major earth systems. The aim of this course is to provide the students with analytical methods, concrete approaches and examples, and geometrical intuition so as to provide them with working ability with non-linear systems.

The following detailed outline is very preliminary, will likely change and include less material.


(1 week) (St 1-37,+)

Bifurcations in one dimensional systems

(3 weeks)

Two-dimensional systems and some more basics

(4 weeks)

Chaos, transition to chaos

(4 weeks)

The Lorentz model as an introduction to chaotic systems (examples briefly motivating it from atmospheric dynamics and as a model of Magnetic field reversals of the Earth); and then a more systematic characterization of chaotic systems (examples from fluid dynamics and mantle convection) (St §9). Some preliminaries: Poincare maps.

Universal routes to chaos:


Chaos in Hamiltonian systems

(time permitting)


(time permitting)

Course requirements

Homeworks will be given throughout the course. The best 80% of the assignments will constitute 50% of the final grade. A final exam (possibly a take home) will constitute another 50%.


  1. An interesting account of Poincare's entry for King Oscar II 3-body problem competition, his error, and his discovery of sensitivity to initial conditions: here.
  2. A nice on-line chaos course is at
  3. Also nice: an interactive on line demo of a driven pendulum.
  4. Devil's staircase in circle map and Farey tree:
  5. For some interesting details about the KAM theorem, check here.
  6. On the Mandelbrot set