國立清華大學數學系
Math 5312 (Fall 2005)
動態系統導論一
Introduction to Dynamical Systems (I)
General
Information
Hours and Days: W3W4F4 (i.e. 10:10 ~ 12:00 W,
11:10 ~ 12:00 F)
Classroom: 綜三 606
Instructor: 陳 國 璋
Office: 綜三 609
Office Hours: T3T4, or by appointments
Phone Number: (03) 5715131 Ext: 3067
Email: kchen@math.nthu.edu.tw
Prerequisite: Differential equations (existence and uniqueness of solutions,
solving linear systems), point-set topology (compactness, connectivity, etc),
complex analysis. Real analysis (measure theory) is required in the second
semester.
Textbooks/References:
1. Alligood, Sauer, Yorke: Chaos: An Introduction to Dynamical Systems,
Springer, 1996.
2. Brin, Stuck: Introduction to Dynamical Systems,
3. Robinson: Dynamical Systems - Stability, Symbolic Dynamics, and Chaos,
CRC Press, 1999.
4. Pollicott, Yuri: Dynamical Systems and Ergodic Theory,
Course Description
The theory of dynamical systems is a major
mathematical discipline connecting with various fields, including geometry,
probability, number theory, physics, biology, economics,
among many others. Its main goal is to understand the evolution and long term
behavior of time dependent systems. The time variable is either discrete or
continuous, and the law of evolution can be deterministic or stochastic. Many
simple nonlinear systems are known to exhibit surprisingly chaotic dynamics
that cannot be anticipated by linear analysis.
In this introductory course we will introduce basic concepts in dynamical
systems and focus on low dimensional systems. Topics to be covered in the first
semester include:
1. Basic concepts and
examples
2. One dimensional
maps (interval maps, circle maps)
3. Two dimensional
maps (horseshoe map, Hénon map,
twist maps, toral automorphisms, etc.)
4. Dynamics near fixed
and periodic points (Hartman-Grobman theorem, stable
and unstable manifolds, etc.)
In the second semester we will cover (tentative):
1. Fractals
2. Bifurcations
3.
Measurements of chaos
4. Hyperbolic
invariant sets
In the first semester, we will loosely
follow Alligood-Sauer-Yorke's book with selected materials
from Robinson's book. In the second semester, the
course materials will be selected from Robinson, Brin-Stuck, and Pollicott-Yuri's books.
Grading
Homework Assignments (60%): There will be about 5 homework assignments.
Final Exam (40%): The final exam will be on Friday,
January 6 from