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.
Demonstrate a deep understanding of the fundamental concepts of dynamical systems, including state variables, time evolution, and stability.
Apply non-linear differential equations to model complex systems and recognize how they deviate from linear behavior.
Identify and analyze bifurcations in dynamic systems, understanding their significance in predicting sudden qualitative changes as parameters vary.
Describe the principles of chaos theory, including the concept of deterministic chaos, sensitive dependence on initial conditions, and the existence of strange attractors.
Utilize phase space analysis as a tool to visualize and interpret the behavior of dynamical systems, including limit cycles, periodic orbits, and chaotic trajectories.
Investigate discrete-time systems using difference equations, evaluating stability and periodicity in these systems.
Incorporate the concepts of uncertainty and stochasticity into dynamical systems, including the ability to model random noise and apply probabilistic techniques.
Apply asymptotic theory to analyze the behavior of dynamical systems, including stability near equilibria and periodic orbits.
Recognize and apply dynamical systems theory to real-world applications across various disciplines, including physics, biology, economics, and engineering.
Critically evaluate the assumptions and limitations of dynamical models, especially in scenarios involving complex or uncertain systems.