Mathematical theory for quasi-periodic connections in Hamiltonian systems: applications to space trajectory design and water waves

Quasi-periodic (QP) connections connect different quasi-periodic orbits and are used in inter-planetary space trajectory design and are found in hydraulic jumps in water waves. Both systems are Hamiltonian and conserve energy and while there are some numeric investigations in the circular restricted 3 body problem, there is no mathematical theory for the emergence of quasi-periodic connections. This PhD project will develop a rigorous mathematical theory for the emergence of QP connections focusing on unfolding a special Hamiltonian point in parameter space, which allows the use of mathematical modulation theory to develop a mathematical theory for the emergence of QP connections, as well as initial conditions which can be used in numerical continuation methods. The PhD will then develop a novel numerical method to systematically explore QP connections far away from the special Hamiltonian point unfolded in the modulation theory. The results of this PhD will allow for a deeper understanding of how and where QP connections occur in a wide range of Hamiltonian systems which will have applications in space trajectory design and fluid mechanical systems. Quasi-periodic (QP) connections connect different quasi-periodic orbits and are used in inter-planetary space trajectory design and are found in hydraulic jumps in water waves. Both systems are Hamiltonian and conserve energy and while there are some numeric investigations in the circular restricted 3 body problem, there is no mathematical theory for the emergence of quasi-periodic connections. This PhD project will develop a rigorous mathematical theory for the emergence of QP connections focusing on unfolding a special Hamiltonian point in parameter space, which allows the use of mathematical modulation theory to develop a mathematical theory for the emergence of QP connections, as well as initial conditions which can be used in numerical continuation methods. The PhD will then develop a novel numerical method to systematically explore QP connections far away from the special Hamiltonian point unfolded in the modulation theory. The results of this PhD will allow for a deeper understanding of how and where QP connections occur in a wide range of Hamiltonian systems which will have applications in space trajectory design and fluid mechanical systems.

You will need to meet the minimum entry requirements for our Mathematics PhD programme.

Open to candidates who pay UK/home rate fees. See UKCISA for further information.