This module provides an introduction to nonlinear systems and their applications in modelling. The aims of the module are:

• To introduce students to various aspects of the mathematical theory of nonlinear systems
• To illustrate the use of nonlinear systems in mathematical modelling of various phenomena, particularly those that involve physical oscillations 
• To describe the qualitative changes in the behaviour of solutions of nonlinear systems that can arise when a system parameter is varied 

Prerequisite Modules: MA0232 Modelling with Differential Equations

• Appreciate the structure of a range of nonlinear systems
• Understand how these nonlinear systems can arise from mathematical models of various phenomena
• Describe and analyse the behaviour of nonlinear systems in both quantitative and qualitative terms
• Appreciate the significance of bifurcations in the analysis of nonlinear systems

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions for given exercises.

Skills:

Problem formulation and modelling.

Solution and qualitative analysis of systems of nonlinear differential equations using advanced mathematical techniques. 

Transferable Skills:

Mathematical modelling of continuous nonlinear systems, including the parametric dependence of equilibrium solutions and oscillatory behaviour.

Analysis and interpretation of the results obtained from a nonlinear mathematical model.

Formative assessment is carried out by means of regular tutorial exercises. Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

Summative assessment by written examination at the end of the module. 

The examination gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

• Bifurcations
  • Equilibrium solutions of ordinary differential equations
  • Saddle-node, transcritical, pitchfork and Hopf bifurcations
  • Subcritical and supercritical instability
  • Bifurcation tests. Topological equivalence. Descriptive usage of normal forms.

• Flows on the circle and the torus
  • Uniform and non-uniform oscillators
  • quasi-periodicity
  • coupled oscillators
  • phase-locking and entrainment
  • saddle-node bifurcations of cycles
  • Forced pendulum motion, including illustration of chaotic behaviour.

• Limit cycles
  • Application of Poincare-Bendixson theorem and trapping regions
  • Lienard systems
  • Low dimensional bifurcations embedded in higher-dimensional systems

• Various applications (mostly involving biological models) such as
  • Insect outbreaks, including hysteresis
  • firefly flashing
  • genetic control switches
  • simple mechanical systems

Nonlinear Dynamics and Chaos (with applications to Physics, Biology, Chemistry & Engineering), Strogatz, Westview

Nonlinear Systems, Drazin, Cambridge University Press