School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2019/20 Last Updated: 20 November 2019

MAT-30002 - Non-linear Differential Equations

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MAT-20008 Differential Equations

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Systems that evolve in time can often be modelled by differential equations. There are countless examples of such systems from the physical world including the weather, climate change, stock markets, the economy, population dynamics, mechanical systems, etc. The great variety of behaviours exhibited by these systems is reflected in the solutions to the corresponding differential equations. This module introduces a number of methods for identifying and classifying various types of behaviour in various types of differential equation. While linear differential equations model some processes, the majority are described by nonlinear equations, and it is these that display the greatest diversity of behaviour. However, very few nonlinear differential equations have exact solutions. Nevertheless, a great deal of insight can be obtained from qualitative methods. This module focuses on geometric methods for constructing phase plane representations of dynamics and perturbation methods for obtaining approximate solutions. With these tools it is then possible to examine the changes in behaviour that can occur when a parameter is varied, and bifurcation theory is introduced to describe this. The relation between the evolution of differential equations and the evolution of maps is explained, and more exotic behaviour, like period doubling and chaos, are then studied in terms of the dynamics of maps.

The module develops the following Keele Graduate attributes:

1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought.

2. An appreciation of the development and value of Mathematics and the links between different areas of the subject.

4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand.

6. The ability to communicate clearly and effectively in written form.

The module develops the following Keele Graduate attributes:

1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought.

2. An appreciation of the development and value of Mathematics and the links between different areas of the subject.

4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand.

6. The ability to communicate clearly and effectively in written form.

The aim of this module is to study the qualitative behaviour of solutions to ordinary differential equations through the use of phase plane analysis.

analyse the qualitative behaviour of solutions to ordinary differential equations through the use of the phase plane: analyse the qualitative behaviour of solutions to ordinary differential equations through the use of the phase plane: 1,2

construct the phase plane by locating and classifying fixed points: construct the phase plane by locating and classifying fixed points: 1,2

demonstrate knowledge of the theory of steady bifurcations: demonstrate knowledge of the theory of steady bifurcations: 1,2

demonstrate knowledge of the theory of self-excited oscillations, their representation by a limit cycle, and the generation of limit cycles at change in stability of a fixed point: the Hopf bifurcation: demonstrate knowledge of the theory of self-excited oscillations, their representation by a limit cycle, and the generation of limit cycles at change in stability of a fixed point: the Hopf bifurcation: 1,2

demonstrate knowledge of the theory of forced oscillations and the use of perturbation methods to discover the effect of nonlinearity in producing harmonics and the consequences for resonance: demonstrate knowledge of the theory of forced oscillations and the use of perturbation methods to discover the effect of nonlinearity in producing harmonics and the consequences for resonance: 1,2

demonstrate knowledge of the theory and application of Poincaré Maps: demonstrate knowledge of the theory and application of Poincaré Maps: 1,2

construct the phase plane by locating and classifying fixed points: construct the phase plane by locating and classifying fixed points: 1,2

demonstrate knowledge of the theory of steady bifurcations: demonstrate knowledge of the theory of steady bifurcations: 1,2

demonstrate knowledge of the theory of self-excited oscillations, their representation by a limit cycle, and the generation of limit cycles at change in stability of a fixed point: the Hopf bifurcation: demonstrate knowledge of the theory of self-excited oscillations, their representation by a limit cycle, and the generation of limit cycles at change in stability of a fixed point: the Hopf bifurcation: 1,2

demonstrate knowledge of the theory of forced oscillations and the use of perturbation methods to discover the effect of nonlinearity in producing harmonics and the consequences for resonance: demonstrate knowledge of the theory of forced oscillations and the use of perturbation methods to discover the effect of nonlinearity in producing harmonics and the consequences for resonance: 1,2

demonstrate knowledge of the theory and application of Poincaré Maps: demonstrate knowledge of the theory and application of Poincaré Maps: 1,2

Lectures: 30 hours

Independent study: 118 hours

Unseen examination : 2 hours

Independent study: 118 hours

Unseen examination : 2 hours

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Two class tests

Continuous assessment will consist of two class tests which are held in weeks 5 and 10 approximately, each lasting 40 minutes.

2 HOUR UNSEEN EXAM

The examination paper will consist of no less than five and not more than eight questions all of which are compulsory.