Course Details

The scope of this course is to present the basic principles, analytic and numerical methods of the theory of non-linear dynamical systems. The term ‘dynamical system’ describes any physical phenomenon evolving in time. Since a physical system can be described by a set of variables, dynamical system is a physical system where one or more variables change in time. If the dynamical system is non-linear, i.e. can be represented by a set non-linear equations, the behavior can be static, periodic or even chaotic. Many phenomena observed in nature, related to Engineering (chemical and biochemical kinetics, mechanical systems, mass transport etc) evolve in time and thus it is very important for an Engineer to become familiar with the methods of study and analysis of such systems.  A special case concerns physical phenomena evolving in different space and/or time scales (e.g. molecular dynamics, systematic biology, meteorology etc). In these systems larger/slower time scales usually prevail. Thus, the mathematical modeling of such systems has to be performed in those time scales, taking into account also small/fast time scales.

Contents:

Part A

1. Basic concepts of non-linear dynamical systems: Types of dynamical behavior, trajectories, stability, attractors.
2. Linear dynamical systems: Eigenvalues, eigenvectors, solution of systems of linear ordinary differential equations (ODEs), autonomous ODEs in two dimensions
3. Linear stability analysis: Linearization, linearized stability
4. Bifurcations: Center manifold theory, static bifurcations, Hopf bifurcation
5. Numerical method for bifurcation analysis: The  AUTO-07P and XPPAUT package.
6. Periodicity and Chaos: Periodic oscillations, relaxation oscillations, bursting oscillations. Transition to Chaos, the chaotic attractor, the characterization of the chaotic attractor

Part B

1. Boundary Layer Behavior
2. Steady BLs (spatial multiscale): 2-point BV problem
3. Time dependent BLs (temporal multiscale): O’Malley-Vasil’eva  expansion, Fenichel’s theorems
4. Evolution equations: (temporal and spatial multiscale)
5. Averaging
6. WKB methods
7. Algorithmic asymptotic methods.

Notes:

1. Introduction of non-linear dynamical systems and physiochemical applications: A. Karantonis

Suggested bibliography:

1. Nonlinear oscillations, dynamical systems and bifurcation of vector fields, J. Guckenheimer and Ph. Holmes, Springer-Verlag (1983)
2. Introduction to applied nonlinear dynamical systems and chaos, S. Wiggins, Springer-Verlag (1990)
3. Methods and Applications of Singular Perturbations, F. Verhulst, Springer 2000.
4. Matched Asymptotic Expansions, P.A. Lagerstrom, Springer-Verlag 1988.