Single-Course English 5 ECTS

# Introduction to Dynamical Systems

## Overall Course Objectives

To be able to analyze nonlinear dynamical systems and understand their complexity. Many models in science and engineering are differential equations describing the evolution of some variables. Dynamical systems theory is the study of such evolution aiming to answer questions like: What is the long term dynamics? How does the system depend on parameters? These are very important questions in applications and hence dynamical systems theory is a very important topic in science and engineering. Dynamical systems is also a fascinating field of mathematics as it combines most of the core mathematical disciplines, in particular, analysis and geometry.
The course will lay the mathematical foundation for the following more advanced courses: 01257 Advanced Modelling – Applied Mathematics, where the students get to apply dynamical systems theory to concrete problems in modelling
01621 Advanced Dynamical Systems: Global Theory
01622 Advanced Dynamical Systems with Applications in Science and Engineering

Learning Objectives

• Determine phase portraits for linear ordinary differential equations.
• Analyze stability in systems.
• Determine and exploit hyperbolicity in the analysis of equilibrium points.
• Operate with stable, unstable and center manifolds.
• Topological equivalence and index theory.
• Apply Poincaré-Bendixon theorem and other results to decide the existence of limit cycles.
• Classify local bifurcations in general and find the possible local bifurcations in a specific system.
• Apply Hartman-Grobman theorem about linearisation to give a qualitative description of the dynamics in a vicinity of a hyperbolic equilibrium.
• Combine the aforementioned points to give a global phase portrait of certain dynamical systems.
• Poincaré maps and chaos.

Course Content

Phase portraits of linear systems. Stability analysis including the Hartman-Grobman theorem. Stable, unstable and center manifolds. Local bifurcation theory. Theory of planar systems including Poincaré-Bendixon theorem. The mathematical techniques will be applied to problems from physics, chemistry, and biology.

Recommended prerequisites

01025/01034/01035/01037, Basic knowledge of linear algebra and linear differential equations

Teaching Method

Lectures and classes.

Faculty

See course in the course database.

## Registration

Language English 13 weeks Compute DTU Lyngby Campus 01617 Candidate Week 35 Week 48 Fri 13-17
Price 7.500,00 DKK