Math  515   Spring 2005

	 Applied Dynamical Systems 

		   J. Duan
		duan@iit.edu
	   
     Class hours:  Tu & Th 11:25am -- 12:40pm
     Office Hours: 


This is an introductory course in nonlinear dynamical systems and applications,
for graduate students in applied mathematics, engineering and science.  
This course provides a coherent treatment of the basic ideas, methods and
techniques in this exciting area.  The emphasis is on understanding
and the ability to apply the theory to complex problems in
engineering and science.  Students will learn
nonlinear partial/ordinary differential equations along the way, i.e., in the context of
mathematical models. Throughout the course, many examples of
nonlinear differential equations will be used to illustrate basic ideas and methods.

Dynamical systems theory describes the behavior of solutions of nonlinear
evolutionary equations. Evolutionary equations are mathematical models of complex
phenomena representing the change of processes in time. Dynamical systems theory provides a unified
conceptual framework for utilizing general strategies to formulate
mathematical models, for investigating nonlinear phenomena described by
such models, for quantifying complex behavior, and for devising
strategies to control or exploit nonlinear phenomena.
Applying the ideas and methods of dynamical systems theory to
applications has been a thoroughly interdisciplinary effort.

We will expect you to acquire some of the material.  Our
responsibility is to point out where you should concentrate your
study and to help pace your progress through the course.


Topics include:

 
Autonomous Dynamics ---
Equilibrium solutions,  linearization and linear stability, 
asymptotic and Liapunov stability;
Phase plane analysis, stable and unstable manifolds, 
periodic orbits, homoclinic and heteroclinic orbits, bifurcations,
generating orbits and exploring dynamics via Matlab; 

Chaotic dynamics ---
Sensitive dependence on initial conditions,
unpredictability, symbolic dynamics, strange attractors,
Poincare maps, mechanisms for chaos, detection
and quantification of chaos;


Nonautonomous Dynamics ---
Cocycles, skew-product flows; peiodic,quasiperiodic and almost periodic motions;
recurrent motions; exponential dichotomy;
invariant manifolds; Lyapunov exponents; and Oseledets multiplicative ergodic theorem;

Infinite Dimensional Dynamics ---
Evolutionary partial differential equations (parabolic type); 
semiflows, invariant sets, absorbing sets and attractors; energy estimates and
dissipativity; asymptotic behavior; Liapunov exponents; Ruelle multiplicative ergodic theorem  
 
Pre-requisite:   Calculus, linear algebra, elementary differential equations,
Math 500, or consent of the Instructor.


 
Textbooks 
and Reference Books:

L. Perko, Differential Equations and Dynamical Systems. 2001, 3rd Edition.
W. Coppel: Dichotomies in Stability Theory. Springer, 1978.
G. Sell and Y. You, Dynamics of Evolutionary Equations. 2002.

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,  1990.
J. Guckenheimer and P. Holmes,  Nonlinear Oscillations,Dynamical Systems, and Bifurcations of Vector Fields. 1983
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics
C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations