Computation Sequences for Series and Polynomials

Approximation to the solutions of non-linear differential systems is very useful when the exact solutions are unattainable. Perturbation expansion replaces the system with a sequences of smaller problems, only the first of which is typically nonlinear. This works well by hand for the first few terms, but higher order computations are typically too demanding for all but the most persistent. Symbolic computation is thus attractive; however, symbolic computation of the expansions almost always encounters intermediate expression swell, by which we mean exponential growth in subexpression size or repetitions. A successful management of spatial complexity is vital to compute meaningful results. This thesis contains two parts. In the first part, we investigate a heat transfer problem where two-dimensional buoyancy-induced flow between two concentric cylinders is studied. Series expansion with respect to Rayleigh number is used to compute an approximation of a solution, using a symbolic- numerical algorithm. Computation sequences are used to help reduce the size of intermediate expressions. Up to 30th order solutions are computed. Accuracy, validity and stability of the computed series solution are studied. In the second part, Hilbert’s 16th problem is investigated to find the maximum number of limit cycles of certain systems. Focus values of the systems are computed using perturbation theory, which form multivariate polynomial sys- tems. The real roots of such systems leads to possible limit cycle conditions. A modular regular chains approach is used to triangularize the polynomial systems and help to compute the real roots. A system with 9 limit cycles is constructed using the computed real roots

The Computational Complexity of Symbolic Dynamics at the Onset of Chaos

In a variety of studies of dynamical systems, the edge of order and chaos has been singled out as a region of complexity. It was suggested by Wolfram, on the basis of qualitative behaviour of cellular automata, that the computational basis for modelling this region is the Universal Turing Machine. In this paper, following a suggestion of Crutchfield, we try to show that the Turing machine model may often be too powerful as a computational model to describe the boundary of order and chaos. In particular we study the region of the first accumulation of period doubling in unimodal and bimodal maps of the interval, from the point of view of language theory. We show that in relation to the ``extended'' Chomsky hierarchy, the relevant computational model in the unimodal case is the nested stack automaton or the related indexed languages, while the bimodal case is modeled by the linear bounded automaton or the related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of manuscrip