Nonlinear dynamics and chaos

Applied Mathematics 203

(Fall 2005)

Instructor: Eli Tziperman;

TF: Ian Eisenman; tel: 617-496-6352, office: Geol. Mus. 101. Office hours: Monday 2-3pm; section hours: Tue 4:30-5:30pm; section location: Geological Museum 103A

Day and time of course: Tue-Thu 11:30-1:00; Location: Cruft 319. First meeting: Sept 20.

Teaching notes Textbooks Syllabus Requirements

Announcements Last updated: Jan 17, 2005.
Take home final: please come pick up the exam from outside of my office, museum building room 202b, Thursday Jan 19, 11am. It will be due 24 hours later at the same place. Allowed material: only your class notes and all materials from the course home page.

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

Teaching notes online:
first lecture, intro1, intro2, bif1d1, bif1d2, bif2d1, bif2d2, bif2d3, cntr-mnfld, chaos1, chaos2, chaos3, chaos4, fract, Ham1, Ham2,

Sample Matlab programs: bakers_map.m, circle_map.m, cobweb.m, euler_course.m, feigenbaum.m, henon.m, global_bif_cycle_saddle.m, logistic_map.m, lorenz.m, lorenz2.m, my_quiver.m, pendulum.m, pendulum_self_sustained.m, shilnikov.m, standard_map.m, standard_map_color.m, standard_map_interactive.m, van_der_pol.m,

Homework: 01, 02, 03, 04, 05, 06, 07, 08, 09, 10,

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

sections: Ian's notes may be found here,

What's the point of optional/ extra credit problems: apart from the fun of doing them, they will count instead of homework problems in which you may have missed an answer. . .


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.


(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

(2 weeks)


(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 take home exam will constitute another 50%.


  1. A Java applet demonstrating sensitivity to initial conditions in the Lorenz system: here.
  2. An on-line chaos course is at
  3. An interactive on line demo of a driven pendulum.
  4. Devil's staircase in circle map and Farey tree: and also this paper.
  5. For some interesting details about the KAM theorem, check here.
  6. Nonlinear resonance and bridge collapsing: The Tacoma Narrows Bridge and the millennium pedestrian bridge in the UK.