Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites for this course are 8.02 Electricity and Magnetism, and 18.03 Differential Equations. Knowledge of ordinary differential equations is essential. Some linear algebra (knowledge of eigenvectors and eigenvalues) is also necessary. Having some experience with numerical computation is helpful but not necessary. This is an undergraduate course. Graduate students are reminded that this course carries no graduate credit and are encouraged to take 18.385J Nonlinear Dynamics and Chaos instead.
This course provides an introduction to nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in engineering and science. The course concentrates on simple models of dynamical systems, and their relevance to natural phenomena. The emphasis is on nonlinear phenomena that may be described by a few variables that evolve with time. The theory of nonlinear continuum systems is covered in the sequel to this course, Nonlinear Dynamics II: Continuum Systems (12.207J/18.354J).
To promote the notion of numerical experiments, we assign several laboratory-like problem sets that require the use of a computer. The computer exercises usually use MATLAB®, but students are free to use whatever software tools and computers they desire. No previous experience with numerical computation is necessary.
Requirements and Grading
There will be problem sets assigned weekly (or nearly weekly). Some problems will be analytical while others will require use of a computer.
There will be a midterm examination. The midterm exam will have a classroom written portion and a take-home computational portion.
At the end of the semester there will be a major assignment (in lieu of a final exam) due at the time of the last scheduled class. This assignment will require the assimilation of the semester's work.
Problem sets should represent the student's own work but cooperation in the form of, say, comparison of one's numerical results with another's, or helpful hints, is welcome. The final assignment, however, must be an entirely individual effort. Students are not permitted to consult previously corrected problem sets, or solution sets or exams from previous years.
The final grade will be based approximately as follows:
|General Problem Sets||50%|
|Final Problem Set||25%|
The most important book, which we recommend that you purchase, is:
Strogatz, S. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering. Westview Press, 2001. ISBN: 9780738204536.
Strogatz will be especially valuable for our discussion of stability of ordinary differential equations.
Additional book recommendations are found in the readings section.
List of Course Topics
- Free oscillators: Linear and nonlinear pendulum. Phase space and phase portraits. Fixed points. Stability. Liouville's theorem and conservation of areas in phase space. Damped pendulum: At-tractors. Contraction of areas in phase space.
- Forced oscillators: Van der Pol equation. Limit cycles. Forced pendulum. Resonance. Stability.
- Introduction to bifurcation theory.
- Methods for analyzing periodic, quasiperiodic, and aperiodic systems. Poincare sections. Floquet matrices and stability. Maps. Reduction of ﬂows to maps. Reconstruction of phase space from one-dimensional signals.
- Strange attractors. Dissipation, attraction, and reduction of dimensionality. Minimum dimension of deterministic aperiodic systems. Sensitivity to initial conditions. Stretching and folding. Rossler attractor. Derivation of Lorenz attractor. Stability of Lorenz equations. Henon attractor.
- Quantitative analysis of strange attractors. Lyaponov exponents. Fractal dimension.
- Transitions to chaos. Period doubling: logistic map, Feigenbaum numbers, scaling, and universality. Quasiperiodicity. Intermittency. Illustrations from experiments.
- If time permits, various applications of the course material in geosciences, math biology, etc.