Nonlinear dynamics and chaos

Applied Mathematics 203

(Fall 2003)

Instructors: Eli Tziperman; Jeremie Charles Korta.

Day and time of course: Mon, Wed, Fri, 10:00-11:00. Location: Pierce 307

Teaching notes Textbooks Syllabus Requirements

Announcements Last updated: Jan 10, 2003.
Final exam will be a one day take home, Jan 13. Please come take the exam from my office at 10am, and return the following day at 16:00. Allowed material: course notes (yours and from the web) and HW plus their solutions. Date is flexible, so that if you have conflicts with other exams, you can do the take home a few days earlier or later. Please contact me regarding any such conflicts or other difficulties with this.
Matlab may be used but wont be necessary for the final. You'll need to plot one or two things, but can also do this by hand and using a pocket calculator.

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

Teaching notes online:
intro1, intro2, bif1d1, bif1d2, bif2d1, bif2d2, bif2d3, chaos1, chaos2, chaos3, chaos4, fract, Ham1, Ham2,

Sample Matlab programs: logistic_map.m, pendulum.m, pendulum_self_sustained.m, euler _course.m, circle_map.m, lorenz.m, henon.m, lorenz2.m, standard_map.m, standard_map_interactive.m, shilnikov.m, my_quiver.m, bakers_map.m, cobweb.m, feigenbaum.m,

Homework: 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11

Jeremie's homework solutions: 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11

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. . .


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 nice on-line chaos course is at
  2. Also nice: an interactive on line demo of a driven pendulum.
  3. Devil's staircase in circle map and Farey tree:
  4. For some interesting details about the KAM theorem, check here.
  5. Links to some climate related papers using misc nonlinear dynamics tools (of mine at this stage, will try to add more later): El Nino's (quasi-periodicity route to) chaos, here and here. Glacial cycles and a climate bifurcation that happened one million years ago: here. Controlling El Nino's chaos (don't take this seriously): here. El Nino as a weakly nonlinear oscillation and its amplitude-period relation: here. Is the oceanic circulation/ climate close to an instability threshold? here and here.