Computational Nonlinear Dynamics - 2019 entry

Most mathematical problems in engineering and science lead to systems of nonlinear equations that cannot be solved with pencil and paper, and where a numerical approach does not give a complete answer. In this module you will use theory and mathematical methods from Stages 1 and 2 (calculus, dynamics, differential equations, numerics and scientific computing) to solve realistic problems as they occur in nonlinear dynamics in engineering and science.

You will gradually assemble a toolbox of small programs (in MATLAB) and then use these programs to study nonlinear dynamical systems with complicated behaviour (for example, chaos). In the end you will have solved some fairly complex problems from scratch, using tools developed and written by yourself during the module. For example, you will have proved (or, at least given robust numerical evidence for) chaos in the forced pendulum. Half of the contact time will be supervised lab sessions during which you will get support to get started on the problems.

Pre-requisites: MTH1003 Mathematical Modelling or equivalent programming experience.

You will learn to combine your previously acquired knowledge from Stages 1 and 2 (specifically calculus and modelling) and your programming skills to solve nonlinear problems as they occur in real-world applications, for example, in mechanics, lasers, climate, ecology, chemistry, biology, or electronic circuitry.

The problems will all come from applications encountered in academic research (e.g., lasers, mechanical systems, population dynamics, fluid dynamics). The module does NOT intend to teach you  how to use particular state-of-the-art research tools (such as AUTO), but will rather guide you to develop idealized versions of these tools from scratch. The module will give you the opportunity to solve problems that are beyond the reach of exam-based assessment but short of individual research projects.

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1. solve high-dimensional nonlinear systems that depend on parameters

2. apply mathematical and computational methods previously learned to study dynamical systems from applications

Discipline Specific Skills and Knowledge

3.  solve mathematical problems of medium complexity (that is, requiring combination of a range of computational and mathematical techniques)

Personal and Key Transferable / Employment Skills and Knowledge

4. apply computational and programming skills to problem-solving

5. develop a project independently and with appropriate time management.

- Implicit function theorem and Newton iteration in arbitrary dimensions;

- Numerical differentiation and numerical solution of initial-value problems of ODEs in arbitrary dimensions;

- Solution of parameter-dependent nonlinear problems;

- Computation and visualisation of phase portraits and their structural stability;

- Finding and tracking singularities (bifurcations: saddle-node and Hopf bifurcations);

- [*] Regularity and discretisation of ODE boundary-value problems;

- [*] Tracking of periodic orbits, starting from a Hopf bifurcation, and some of their bifurcations;

- [*] computation of basins of attraction for equilibria of autonomous systems or periodic points of periodically forced systems;

- [*] Lyapunov exponents, stable and unstable manifolds of periodic points and detection of their homoclinic tangles in periodically forced systems.
 

[*] only a selection of these topics will be covered, varying each year

All referred/deferred assessments will normally be by assignment.

Basic reading:

ELE: http://vle.exeter.ac.uk/

Web based and Electronic Resources:

http://www.dynamicalsystems.org/tu/cm/

Other Resources:

Reading list for this module: