MAT230

MAT230#

Dynamical Systems and Chaos

Objectives and Content#

Objectives#

This course explores dynamical systems with a focus on non-linear dynamics, chaos, uncertainties, and asymptotic theory. Students will understand fundamental concepts, model complex systems, analyse deviations from linearity, and apply phase space analysis. The course covers ordinary differential equations, discrete-time systems, uncertainty, and practical applications, fostering critical thinking and hands-on skills.

Learning Outcomes#

On completion of the course, the student should have the following learning outcomes defined in terms of knowledge, skills, and general competence:

Knowledge#

The student:

  • Demonstrates a deep understanding of the fundamental concepts of dynamical systems, including state variables, time evolution, and stability.

  • Understands the principles of chaos theory, including deterministic chaos, sensitivity to initial conditions, and strange attractors.

  • Recognizes how non-linear systems deviate from linear behavior and how bifurcations indicate qualitative changes in system dynamics.

  • Understands how uncertainty and stochasticity are incorporated into dynamical models.

Skills#

The student:

  • Can apply non-linear differential equations to model complex systems.

  • Can identify and analyze bifurcations in dynamic systems.

  • Can utilize phase space analysis to interpret system behavior, including limit cycles, periodic orbits, and chaos.

  • Can investigate discrete-time systems using difference equations and evaluate their stability and periodicity.

  • Can model random noise in systems and apply probabilistic tools to uncertain dynamics.

  • Can apply asymptotic theory to analyze stability near equilibria and periodic orbits.

General Competence#

The student:

  • Recognizes how dynamical systems theory applies across disciplines such as physics, biology, economics, and engineering.

  • Has the ability to critically evaluate assumptions and limitations of mathematical models in the presence of complexity or uncertainty.

  • Understands the real-world relevance of theoretical models and develops a systems-thinking mindset.

Semester of Instruction#

Spring

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