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MMath Mathematics and Statistics / Course details

Year of entry: 2021

Course unit details:Dynamical Systems

Unit code MATH44041 15 Level 4 Semester 1 Department of Mathematics No

Overview

Dynamical systems theory is the mathematical theory of time-varying systems; it is used in the modelling of a wide range of physical, biological, engineering, economic and other phenomena. This module presents a broad introduction to the area, with emphasis on those aspects important in the modelling and simulation of systems. General dynamical systems are described, along with the most basic sorts of behaviour that they can show. The dynamical systems most commonly encountered in applications are formed from sets of differential equations, and these are described, including some practical aspects of their simulation. The most regular kinds of behaviour---equilibrium and periodic---are the most easy to analyze theoretically; linearization about such trajectories are discussed (for periodic behaviour this is done using the PoincarÃ© map.)

Much more complex behaviours, including chaos, may be found; these are described by means of their attractors. The linearization approach can be extended to these, and leads to the concept of Lyapunov exponents.

In applications it is often important to know how the observed behaviour changes with changes in the system parameters; such changes can often be sudden, but frequently conform to one of a relatively small number of scenarios: the study of these forms the subject of bifurcation theory. The simplest bifurcations are discussed.

Pre/co-requisites

Students are not permitted to take MATH44041 and MATH64041 for credit in an undergraduate programme and then a postgraduate programme.

Aims

To develop a basic understanding dynamical systems theory, particularly those aspects important in applications. To describe and illustrate how the basic behaviours found in dynamical systems may be recognized and analyzed.

Learning outcomes

On successful completion of this course unit students will be able to:

• solve simple systems of ODEs or maps, and deduce the long term behaviours of the solutions,
• derive qualitative properties of solutions to systems of ODEs or maps by semi-group property, conserved quantities, invariant sets or change of variables in polar coordinates,
• calculate fixed points of system of ODEs, determine their linear types and sketch the phase portrait,
• construct Lyapunov function to show the stability of the solutions,
• apply the Poincaré-Bendixson theorem to show the existence of periodic solutions and apply Floquet theory to periodic linear system,
• calculate stable and unstable manifolds,
• calculate the centre manifold and classify the bifurcation types from the reduced dynamics,
• calculate fixed points or periodic orbits of maps, locate and classify bifurcation points.

Syllabus

1. Basics. Basic concepts of dynamical systems: states, state spaces, dynamics. Discrete and continuous time systems. [1 lecture]
Some motivating examples: (discrete): simple population models, numerical algorithms; (continuous): chemical and population kinetics, mechanical systems, electronic and biological oscillators. [1]
2. Basic features of dynamical systems. Trajectories, fixed points, periodic orbits, attractors and basins. Autonomous and non-autonomous systems. Phase portraits in the plane and higher dimensions; examples of phase portraits of 2-d and 3-d systems. [2]
3. Ordinary differential equations. Systems of first order ordinary differential equations; initial value problems, existence and uniqueness of solutions. Flows. [2]
4. Equilibria and linearization. Fixed and equilibrium points and their linearization; classification and the Hartman-Grobman theorem; invariant manifolds; examples in 2-d and 3-d. Computing equilibrium points. Stability and Liapounov functions. [5]
5. Periodic orbits and linearization. Poincaré sections and the Poincaré map. Linearization and characteristic multipliers of periodic orbits, and stability; examples. Computing periodic orbits. [3]
6. Attractors and long-term behaviour. ω-limit sets and long term behaviour. Chaotic attractors; illustrative examples. Lyapunov exponents and their computation. One-dimensional maps and simple routes to chaos (unimodal maps and Lorenz maps). Crises, chaotic transients [8]
7. Bifurcations of flows. Local bifurcations and centre manifolds, global bifurcations; examples. Computing bifurcation diagrams by continuation.[8]

Assessment methods

Method Weight
Other 20%
Written exam 80%
• Mid-semester coursework: 20%
• End of semester examination: weighting 80%

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Core: Meiss, James D. Differential dynamical systems. Vol. 14. Siam, 2007

Essential: Stephen H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, Cambridge, MA, USA, 1994.

Essential: Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, NY, USA, second edition, 2003.

Recommended: Lynch, Stephen. Dynamical systems with applications using Matlab. Springer, 2014.

Further reading: Kathleen T. Alligood, Tim D. Sauer and James A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York, NY, USA, 1996.

Further reading: Y.A. Kuznetsov Elements of Applied Bifurcation Theory, Springer Applied Math. Sci. 112, 1995.

Study hours

Scheduled activity hours
Lectures 24
Tutorials 12
Independent study hours
Independent study 114

Teaching staff

Staff member Role
Yanghong Huang Unit coordinator