Finding unstable periodic orbits for nonlinear dynamical systems using polynomial optimisation

Computing unstable periodic orbits (UPOs) for systems governed by ordinary differential equations (ODEs) is a fundamental problem in the study of nonlinear dynamical systems that exhibit chaotic dynamics. Success of any existing method to compute UPOs relies on the availability of very good initial guesses for both the UPO and its time period. This thesis presents a computational framework for computing UPOs that are extremal, in the sense that they optimise the infinite-time average of a certain observable. Constituting this framework are two novel techniques. The first is a method to localise extremal UPOs for polynomial ODE systems that does not rely on numerical integration. The UPO search procedure relies on polynomial optimisation to construct nonnegative polynomials whose sublevel sets approximately localise parts of the extremal periodic orbit. Points inside the relevant sublevel sets can then be computed efficiently through direct nonlinear optimisation. Such points provide good initial conditions for UPO computations with existing algorithms. The second technique involves the addition of a control term to the original polynomial ODE system to reduce the instability of the extremal UPO, and, in some cases, to provably stabilise it. This control methodology produces a family of controlled systems parametrised by a control amplitude, to which existing UPO-finding algorithms are often more easily applied. The practical potential of these techniques is demonstrated by applying them to find extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow, an extended version of the Lorenz system, and two different three-dimensional chaotic ODE systems. Extensions of the framework to non-polynomial and Hamiltonian ODE systems are also discussed.Open Acces

Quantum Control Landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry, Vol. 26, Iss. 4, pp. 671-735 (2007

Algorithmic Algebraic Geometry and Flux Vacua

We develop a new and efficient method to systematically analyse four dimensional effective supergravities which descend from flux compactifications. The issue of finding vacua of such systems, both supersymmetric and non-supersymmetric, is mapped into a problem in computational algebraic geometry. Using recent developments in computer algebra, the problem can then be rapidly dealt with in a completely algorithmic fashion. Two main results are (1) a procedure for calculating constraints which the flux parameters must satisfy in these models if any given type of vacuum is to exist; (2) a stepwise process for finding all of the isolated vacua of such systems and their physical properties. We illustrate our discussion with several concrete examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure