Stability of Differential Dynamical Systems

Stability of Differential Dynamical Systems

and Numerical Methods

TU Berlin, Spring 2013

Instructor: Dr. VU HOANG LINH, Associate Professor

Current address:     TU Berlin, MA 4-5, Room 446

                     E-mail: linh@math.tu-berlin.de

Permanent address:   Faculty of Mathematics, Mechanics, and Informatics

                     Vietnam National University, Hanoi

                     E-mail: linhvh@vnu.edu.vn

Course objective & topic: This course aims to provide fundamental concepts and techniques for studying the stability of differential equations and differential dynamical systems. It is designed for advanced undergraduate or beginning graduate students in mathematics/applied mathematics. The course is divided into three parts. The first part covers fundamental knowledge for both autonomous and nonautonomous systems of linear differential equations. The second part is concerned with techniques for studying qualitative properties of smooth dynamical systems described by nonlinear systems of differential equations. In the last part, some numerical aspects such as numerical methods for solving differential equations, stability concepts of numerical methods, and numerical methods for the stability analysis will be discussed.

Grading:The final grade will be given based on homeworks and the final exam (50% of the homework points are required to be admitted to the final exam).

References:

1. L.Ya. Adrianova, Introduction to Linear Systems of Differential Equations, AMS, 1995.

2. James Meiss, Differential Dynamical Systems, SIAM, 2007.

3. J.L. Daleckii and M.G. Krein, Stability of solutions of differential equations in Banach spaces, AMS, 1974.

4. U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, 1998.

Course Outline (tentative):

1.     Introduction (4 hours)

1.1         Modeling and differential equations

1.2         One dimensional dynamics

1.3         Real-life examples

1.4         Two-dimensional dynamics

2.     Linear Autonomous Systems (8 hours)

2.1         Matrix ODEs

2.2         Two dimensional linear systems

2.3         Exponentials of operators

2.4         Fundamental solution

2.5         Linear stability

2.6         Nonautonomous systems and Floquet theory

3.     Lyapunov characteristic exponents (10 hours)

3.1         Definition and main properties

3.2         The spectrum of a linear system

3.3         Normal bases

3.4         Lyapunov transformations

3.5         Reducibility

3.6         Regularity

3.7         Stability of Lyapunov exponent

3.8         Chaotic dynamics and Lyapunov exponents

4.     Stability of linear systems with perturbations (8 hours)

4.1         Stability of linear systems

4.2         Almost constant systems

4.3         Uniform stability and uniform asymptotic stability

4.4         Bohl exponent

4.5         Bounded solutions of nonhomogenous systems

4.6         Exponential dichotomy and Sacker-Sell spectrum

5.     Nonlinear Systems: Existence and Uniqueness (4 hours)

5.1         Existence and uniqueness theorem

5.2         Dependence on initial conditions and parameters

5.3         Maximal interval of existence

6.     Dynamical Systems (14 hours)

6.1         Definitions

6.2         Flows

6.3         Global existence of solutions

6.4         Linearization

6.5         Stability

6.6         Lyapunov functions

6.7         Topological conjugacy and equivalence

6.8         Hartman-Grobman theorem

6.9         Omega-limit sets

6.10    Attractors and basins

6.11    Stability of periodic orbits

6.12    Poincaré maps

7.     Numerical methods(8 hours)

7.1         Numerical methods for solving ODEs

7.2         Stability concepts

7.3         Numerical methods for approximating Lyapunov and Sacker-Sell spectra