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Title

Dynamical Systems
Author

Alfred Clark Jr.
Organization: University of Rochester
Department: Departments of Mechanical Engineering and Mathematics
URL: http://www.me.rochester.edu/~clark
Education level

Graduate
Objectives

The objective of the course is to give advanced undergraduates and graduate students a thorough introduction to systems of nonlinear ordinary differential equations, and a brief introduction to iterated maps.
Materials

The entire course, except the lecture notes is on the web site. There are 12 homework assignments and solutions, and several exams with solutions. There are 30 detailed examples in the form of Mathematica notebooks, and 25 short movies giving dynamic views of bifurcations and other events. Finally there is available for download a Mathematica tutorial, and a comprehensive dynamical systems package called DynPac. The text book and related references are given in the detailed course description on the web site.
Description

This graduate level course, which is offered every two years, is typically taken by about 15 students. The students are a mix of undergraduates and graduate students in science and engineering. There is a strong emphasis on computational exploration. For this computation, students use the package DynPac which runs under Mathematica, and which is available on the course web site.

Student response has generally been very positive. The main pedagogical issue is finding the right balance between fundamental mathematical theory and computation.

Topics:
  • Plane Autonomous Systems: phase plane; stability by linearization; Liapunov's method for stability; periodic solutions; stability of periodic solutions; global phase portraits; bifurcations
  • Forced Oscillations of Plane Systems: poincaré maps; resonances and subharmonics in Duffings equation; the forced pendulum; chaotic solutions
  • Higher Order Autonomous Systems: matrix methods for linear systems; Floquet theory and stability of periodic solutions; bifurcation study of Lorenz equations; tent map and the Lorenz equations; Liapunov exponents.
  • Iterated Maps: Logistic map; Feigenbaum numbers; Hénon map.
Subject

*Mathematics > Calculus and Analysis > Dynamical Systems
Related items

*DynPac: A Dynamical Systems Package   [in MathSource: Packages and Programs]
URL

http://www.me.rochester.edu/courses/ME406