MAT3101: Dynamical Systems

This course is a foundation course that introduces learners to the basic mathematical concepts. It covers the following topics: Brief Review of Differential Equations (first Order and Linear Systems); Introduction to Discrete and Continuous Dynamical Systems; Classification of Fixed points of Discrete and Continuous Non-linear Systems; and Periodicity and Chaos in Non-linear Systems.

Course Objectives  
On successful completion of this course unit, the learners should be able to: 

• Describe various models used in characterization of the swinging pendulum, the flow of water in a pipe, or the number of fish in a lake or pond as examples of dynamical systems. 
• Describe the concept of sensitivity analysis in which small changes in the state corresponds to small changes in the numbers in its neighbourhood and such should be applied to areas such as bifurcation theory, chaos, attractors, limit cycles, and non-linear dynamics. 
• Identify fundamental differences between linear and non-linear dynamical systems. Construct and interpret phase portraits of maps and flows in one and two dimensions.   
• Identify fixed points and periodic points and characterize their stability as attractors, unstable saddles, and others.  
• Understand the concept of characterizations and measurements of chaos such sensitive dependence on initial conditions, and Lyapunov exponents. 
• Use Computer Software and mathematical packages such Maple, Matlab and Fortran to simulate and study dynamical systems in one and two dimensions. 

Expected Learning Outcome
This course unit is meant: 

• To discuss the basic competence in the concepts, principles, and procedures of dynamical systems and their applications to mathematical modeling and biological processes and their time dependence of a point in a geometrical space. 
• To encourage orderliness, speed and accuracy in the presentation of mathematical expressions in chaotic differential systems. 
• To help learners acquire the skills of expression in proper mathematical language and using mathematical symbols correctly. 
• To provide instruction that contributes to the learners’ abilities to think critically and solve real life problems regarding sensitivity analysis, to reason mathematically and apply computational skills. 
• To build a strong foundation in mathematical presentation as preparation for subsequent courses in applied mathematics and biomathematics.