Physics Department
Rockefeller Bldg.
Case Western Reserve University
Cleveland, Ohio 44106-7079
rwb@po.cwru.edu

NOVICE: Nonlinear Viewpoints for Introducing a Computational Experience

First, let us briefly describe the ten lectures and problem sets that lead to the final computer project. (Copies of representative lectures are scanned into a web site whose URL address is given above.) The first lecture is an explanation of generalized dimensions, with practice problems in calculating the dimension for those fractals generated by simple rules and for approximate fractals generated by real data. The second lecture introduces iteration maps in simple population studies and the third revisits these maps using a graphical approach. A fourth lecture includes an introduction into the basic elements of cooperative learning with groups of three or four students. Bifurcations, period doubling and Feigenbaum universality as discovered with the iteration maps are the subject of the fifth lecture. The linear simple harmonic oscillator is presented in the sixth lecture, providing an analytical solution against which the finite-differencing computer methods of the seventh lecture can be validated. In the eighth lecture, these numerical methods are used to discover the extreme sensitivity to initial conditions present for certain nonlinear differential equations. A ninth lecture is comprised of the introduction to the project. The concept of a strange attractor is revealed in higher-dimensional maps, along with an explanation of the Mandelbrot set, in the tenth lecture.

In the NOVICE project module, the students use both the computer programming and graphics skills derived in the associated homework and the learned concepts of fractals, period-doubling paths to chaos, initial-condition sensitivity, and strange attractors, to analyze their computer data and their experimental data described previously.

SUMMARY: The NOVICE module has proven to be an attractive tool for meeting the challenge of building essential freshman skills and engaging the new, hopeful students who come to us with high expectations. We flesh out the mathematics behind the popular knowledge freshman students already possess about the emergence of chaos as a new science. And we respond to the desires of more and more students who already have an increasingly sophisticated high-school experience in computational science problems.