Dynamical Systems and Chaos
Primary references
- Nonlinear Oscillations, dynamical systems and bifurcations of vector fields - Guckenheimer and Holmes.
- Introduction to the modern theory of dynamical systems - Katok & Hasselblatt.
- Dynamical systems and Chaos - Ed Ott.
References for special topics
- Topics in ergodic theory - Ya. G. Sinai.
- Topics in bifurcation theory and applications - Iooss and
Adelemeyer.
- Mathematical methods of classical mechanics - V. I. Arnold.
List of topics I hope to discuss in the course of the year
- Differential equations, stability and linearization.
- Maps
- Examples of chaos and the dynamics of chaotic
systems. Attractors, Measures of chaos, symbolic dynamics and
topological dynamics.
- Ergodic theory, mixing, information theory.
- Hyperbolic systems and Strucutral stability.
- Bifurcation theory, normal form analysis of local bifurcations,
global bifurcations, routes to chaos, renormalization analysis of the
period doubling route to chaos.
- Hamiltonian systems, Integrability and KAM theory.
The following are topics that are related to
dynamical systems, and are important applications of the methods/ideas
from the study of dynamical systems.
- Multiple scale methods and averaging methods for dynamical
systems.
- Extended (infinite-dimensional) dynamical systems.
- Applications of dynamical systems in Physics, Biology, Mechanics, etc.
Grading policy, etc.
- Homework will be assigned regularly, roughly once every two weeks.
- Here is a list of
potential projects. You can pick one of these topics or come talk
to me if you have some ideas you would like to investigate.
- Here is a link to the nonlinear
journal, a collection of papers by math557 students from years
past.
Homework 1
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Homework 2
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Homework 3
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Homework 4
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Homework 1
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Homework 2
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Homework 3
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Expanding maps of a circle
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The Arnold "Clown" map
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Hyperbolic Toral Automorphisms
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Numerical solution of the Peron-Frobenius
equation
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Shankar Venkataramani
2005-08-30