Dynamical Systems

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Course Code: 
Course Credit Units: 
Semester 1
Year of Study: 
Year 3
Undergraduate or Graduate Level: 
Undergraduate Level
Course Discipline: 
Course Description & Objectives: 

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish in a lake is examples of dynamical systems. A dynamical system has a state determined by a collection of real numbers. Small changes in the state of the system correspond to small changes in the numbers. The course describes the theory of dynamical systems in one and two dimensions. The main areas include bifurcation theory, chaos, attractors, limit cycles, non-linear dynamics.

Learning Outcomes: 

At the end of this course the student should be able to:

  • Identify fundamental differences between linear and nonlinear dynamical  systems.
  • Construct and interpret phase portraits of maps and flows in one and two dimensions.
  • Identify fixed points and periodic points and determine their stability.
  • Understand elementary bifurcations.
  • Understand characterizations and measurements of chaos such as sensitive dependence on initial conditions and Lyapunov exponents.
  • Use symbolic dynamical systems and conjugacy to analyse maps.
  • Explain  how fractals arise from dynamical systems.
  • Use potential functions to analyse flows.
  • Understand limit sets and attractors.
  • Use software to simulate and study dynamical systems in one and two dimensions.


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