MX4555: NONLINEAR DYNAMICS & CHAOS THEORY II (2024-2025)

This is the second part of a two-part course on nonlinear dynamics and chaos theory.

Part I laid out the formal mathematical foundations required to understand nonlinear dynamical systems. The focus was on the types of dynamical behaviour exhibited by linear dynamical systems, how these describe the behaviour of nonlinear systems locally, and how these can in turn be used to model time-varying natural systems. It introduced concepts like the phase line and phase plane, equilibria, stability, perturbations, and linearisation of solutions around hyperbolic equilibria.

In Part II there is less emphasis on formal proof-building and more on applying the concepts to build on the foundation that Part I established. Part II follows a clear path through the analysis of 1D and 2D nonlinear systems, culminating in the Poincaré-Bendixson theorem, which shows that trajectories in 2D dynamical systems must converge to fixed points or closed orbits, or diverge to infinity. We also study bifurcations, where quantitative changes in a system’s control parameter result in qualitative changes in the system’s behaviour. Bifurcations are an essential part of understanding real nonlinear phenomena like population collapse and the catastrophic failure of mechanical systems.

About halfway through the course we consider 3D systems for the first time. We find some systems are bounded but with no stable equilibria or periodic orbits: they are chaotic. In chaotic systems such as those describing convection, population dynamics, and weather, tiny differences in initial conditions grow exponentially fast until the system’s predictability is lost.

Chaos theory is a broad and fascinating subject that has major implications for the predictability of natural systems. We study the Lorenz equations, an idealised model of fluid convection that is the prototype for chaotic behaviour. This leads on to related topics such as fractals, non-integer dimension, and maps, which are dynamical systems in discrete time. After relativity and quantum mechanics, chaos is often thought of as the third major advance in physics during the 20th century, as it challenges the idea that systems described by Newton’s laws of motion are always predictable.

Syllabus:

• Lagrangian and Hamiltonian mechanics
• Invariant manifolds
• Bifurcations in 1D and 2D systems
• Index theory
• Limit cycles
• Poincaré-Bendixson theorem
• Chaos
• Lorenz equations
• Lyapunov exponents
• Discrete maps
• Fractals
• Strange attractors

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

There are no assessments for this course.