MATH36121 - 2010/2011

 

Specification

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.

Brief Description of the unit

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 Poincaré 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.

Learning Outcomes

On successful completion of this course unit students will

• understand the general concept of a dynamical system, and the significance of dynamical systems for modelling real world phenomena;
• be able to analyze simple dynamical systems to find and classify regular behaviour;
• appreciate some of the more complex behaviours (including chaotic), and understand some of the features of the attractors characterizing such behaviour;
• be familiar with some of the simpler bifurcation scenarios, and how they can be analyzed.

Future topics requiring this course unit

None.

Syllabus

1. Basics. Basic concepts of dynamical systems: states, state spaces, dynamics. Discrete and continuous time systems. [1]
   Some motivating examples: (discrete): simple population models, numerical algorithms; (continuous): chemical and population kinetics, mechanical systems, electronic and biological oscillators. [2]
2. Basic features of dynamical systems. Trajectories, equilibria, 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. Equilibrium points and their linearization; classification and the Hartman-Grobman theorem; examples in 2-d and 3-d. [3]
5. Periodic orbits and linearization. Poincaré sections and the Poincaré map. Linearization and characteristic multipliers of periodic orbits, and stability; examples. [3]
6. Invariant manifolds. Stable and unstable manifolds for equilibrium points [2]
7. Attractors and long-term behaviour. ω-limit sets and long term behaviour. Chaotic attractors; illustrative examples. Lyapunov exponents. [3]
8. Bifurcations of flows. Fold, transcritical and pitchfork bifurcations; bifurcation diagrams, examples. The Hopf bifurcation. Structural stability. [4]

Textbooks

• James D. Meiss. Differential Dynamical Systems. SIAM, Philadelphia, PA, USA, 2007. ISBN 0898716357. xxi+434 pp.
• Stephen H. Strogatz. Nonlinear Dynamics and Chaos. Perseus Books, Cambridge, MA, USA, 1994. ISBN 0-201-54344-3. xi+498 pp.
• Kathleen T. Alligood, Tim D. Sauer and James A. Yorke. Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York, NY, USA, 1996. ISBN 0-387-94677-2. xvii+603 pp.
• Thomas S. Parker and Leon O. Chua. Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York, NY, USA, 1989. ISBN 0-387-96688-9. x+348 pp.
• Lawrence Perko. Differential Equations and Dynamical Systems. Springer-Verlag, New York, NY, USA, third edition, 2001. ISBN 0-387-95116-4. xiv+555 pp.
• Paul Glendinning. Stability, Instability and Chaos. CUP, Cambridge, UK, 1994. ISBN 0-521-42566-2. xiii+388 pp.

Teaching and learning methods

22 lectures (two lectures per week), with a fortnightly examples class. In addition should expect to spend at least four hours each week on private study.

Assessment

Mid-semester coursework: 20%

End of semester examination: two hours weighting 80%

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