Course outline: The course consists of four 30-lecture modules. Modules 2OD and 2ND are offered in the first semester and modules 2BP and 2NA in the second semester.
2NA NUMERICAL ANALYSIS: (coded as MAM2053S for Engineering students)
Solutions to non-linear equations and rates of convergence. Direct and iterative methods for solving linear systems, pivoting strategies, matrix factorization, norms, conditioning. Solutions to initial value problems including higher order ordinary differential equations. Interpolation and approximation theory, splines, discrete and continuous least squares. Numerical differentiation and integration. Error analysis and control.
2OD ORDINARY DIFFERENTIAL EQUATIONS:
First order linear and nonlinear equations; existence and uniqueness of solutions. Linear equations of the n-th order and systems of n linear first order equations. Nonhomogeneous linear equations and systems; variation of parameters; qualitative theory of nonlinear equations; phase plane analysis; externally and parametrically driven oscillators; resonances; application to the theory of nonlinear vibrations. Calculus of variations.
2BP BOUNDARY-VALUE PROBLEMS: (coded as MAM2050S for Engineering students)
Boundary-value problems, Green's functions, eigenvalue problems. Variational calculus, applications to Lagrangean and Hamiltonian mechanics. Diffusion, Laplace's and wave equation. Solution by separation of variables.
2ND NONLINEAR DYNAMICS:
Fixed points, bifurcations, phase portraits. Conservative and reversible systems. Index theory, Poincaré-Bendixson theorem, Liénard systems, relaxation oscillators. Hopf bifurcations, quasiperiodicity and Poincaré maps. Applications to oscillating chemical reactions and Josephson junctions. Chaos on a strange attractor, Lorentz map, logistic map, Hénon map, Lyapunov exponents. Fractals.
Tutorials: One tutorial per week, Thursday, 14h00-16h00.
DP requirements: A class record of 30% or more is required in each module of the course.
Assessment: For each module the class record counts 30% and one no longer than 2-hour examination paper counts 70%.