Alexander, Nicholas Andrew; (1990) Computational algorithms for the global stability analysis of driven oscillators. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

To develop an effective process for analysis and description of global instability phenomena such as capsizes of boats and other marine structures there is a need to investigate the intimately involved invariant manifold tangencies of unstable saddle cycles. For an engineer information about the final instability of stable resonant solutions and quantitative changes in the regions of stability of that solution are vital in any safety specification of a nonlinear dynamic model. This thesis provides an computational algorithm for the systematic location of certain heteroclinic and homoclinic manifold tangencies under variation of the system parameters of the dynamic under observation. The manifold tangency algorithm is based on the following threefold division. Chapter 2 develops the ideas of a manifold tangency criterion which is based on a geometric formulation of a distance idealization based on a multi-variant piece-wise cubic approximation to the manifolds under observation. This distance function is grounded in concepts of tangent signing and a need to introduce various conditional criteria extend the definitions of certain computationally standard numerical functions. Chapter 3 develops an automotive procedure for the description of invariant manifolds. This effective procedure is designed to locate the saddle cycle(s) accurately and evaluate the eigenvectors. The extrapolation of the invariant manifolds occurring by repeated mapping of a line of points along the eigenvectors. Storing the resultant data in a singly linked list and making use of stack reference structure and negatively signed pointers, ordering and insertion/deletion of points can be achieved while keeping a bound on computational time and space. Chapter 4 develops the standard ideas of path following local bifurcation and local solutions in parameter/phase space. Thus chapter 3 develops portraits of the manifolds, chapter 2 then evaluates the manifold tangency expressions and modifies the system parameter vector to locate tangency. Chapter 4 deal with the problem to following the saddle(s) under such a variation in the system parameter vector and indicates the requirements for continuation along the manifold tangency path in system parameter space.

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