Dynamical Systems module (MA51003)

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This module takes an advanced look at dynamical systems. You will study the dynamics governing several biological, chemical, or physical processes and systems considered in engineering and economics. You will focus on difference and differential equations and other equations such as delay or fractional differential equations. By studying these systems of equations, you can infer information about the qualitative behaviour of the corresponding biological, chemical or physical systems. You will explore the existence and stability of solutions, how the long-time behaviour of solutions changes depending on the system parameters, and the existence of chaotic behaviour and strange attractors in the system.  

MATLAB will be extensively used in the seminars and the assessment of this module.  

Topics include: 

  • Discrete-time dynamical systems: one and two-dimensional maps, stability of fixed points and periodic orbits, Lyapunov exponents and chaos  
  • Examples of physical, biological and technological processes described by ordinary differential equations (ODEs). 
  • Existence and uniqueness results for ODEs, Banach fixed point theorem.   
  • Qualitative theory of ODEs: linearised stability analysis, Grobman-Hartman theorem, Lyapunov functions and Lyapunov stability.  
  • Two-dimensional dynamical systems: Hamiltonian and gradient systems, Poincare-Bendixson theorem.  
  • Periodic solutions: Floquet theory, Poincare map and stability of periodic orbits.  
  • Bifurcation theory: implicit function theorem, one-parameter bifurcation, Hopf bifurcation.  
  • Chaotic systems: strange attractors for the forced double well oscillator, Rossler and Lorenz systems, routes to chaos  
  • Examples of systems described by partial differential equations (PDEs)  
  • Linearised stability analysis for some PDEs. 


This module is available on following courses: