Numerical Analysis Of Lorenz Manifold

In the past four decades, the aim of most studies on the Lorenz attractor is touncover the chaotic complexity of the Lorenz system from the perspective of dynamics.Although the fractal structure of the Lorenz attractor can embody the chaotic dynamicsfrom the perspective of geometry, only the Lorenz manifold can be regarded as theskeleton underpinning the phase space, organizing the dynamics therein globally anddeciding the fate of the orbit of any initial condition. The Lorenz manifold is the setof the points which approach the origin when time goes to infinity. It is unbounded,and can only be approximated numerically. It has been regarded as a common testcase for most manifold algorithms. However, these algorithms can not compute theLorenz manifold large enough to help one discover its real geometrical structure. Forexample, a geodesic distance based method, which neglecting the dynamics on theLorenz manifold, will stop where the local curvature is larger than it allows. Hence,there is still a large lack of comprehension of the Lorenz manifold. The major aimof this thesis is to get a picture of the Lorenz manifold to help one understand thechaotic dynamics of the Lorenz system. A series of algorithms are proposed to providenumerical and geometrical supports. These algorithms are not confined to the studieson the Lorenz system. They should be applicable to other types of systems, such as highdimensional and fast-slow, to help explain their dynamic essence from the perspectiveof geometry.Due to the diversity between the available manifold algorithms in the sense of theirdifferent implementation environments, it is hard to compare their results directly. Inthis thesis, they are classified into four groups, based on whether they are arclengthbased or geodesic distance based, and whether their implementations are flow-drivenor not. Then four algorithms selected from the four groups respectively, are realizedin the same environment Matlab. Their results are superposed utilizing the powerfulvisualizing tool in Matlab. A profile of a part of the real Lorenz manifold is constructed to make the Lorenz system more concrete and practical for testing on the manifoldalgorithms.Most manifold algorithms start from an initial circle of points near the origin andon the stable eigenplane E~s(O), and grow the manifold circle by circle. Hence theaccuracy of the initial selection plays an important role in determining the accuracyof the whole computation process. However, if these initial points are set too nearthe origin to increase the accuracy, the algorithms will slow down due to the smallvelocity |f(x)|. In order to fix the problem, higher order polynomial approximation ofthe local Lorenz manifold is needed. Semi-tensor product based method can providean approximation in arbitrary order of a manifold without involving the coordinationtransformation as required by others. Hence, it is utilized to compute the local Lorenzmanifold in this thesis to feed accurate initial points to the manifold algorithms.Guckenheimer and Vladimirsky proposed a fast method to approximate theLorenz manifold, which is locally described by an triangle. Every triangle is describedin the form of partial differential equations, which can be efficiently solved in an Eu-lerian framework. However, triangles are computed and organized in a messy wayto construct the surface of the manifold. Hence, the realization of the algorithm be-comes complex due to the malign structure. In this thesis a geodesic counterpart of thismethod is proposed to illustrate the flexibility of the fast core. Triangles are organizedin geodesic level sets to form the manifold surface in a benign way. The benign ge-ometrical structure makes the implementation quite easy from the perspective of datastructure.Due to the resource bottleneck of the computation system, or the limitation of thealgorithm itself, available methods can not achieve an enough large amount of data toreveal the intriguing geometrical structure of the Lorenz manifold. An arclength based,flow-driven method (AF06BIG) is proposed in this thesis. It adopts the local curvatureadaption mechanism. It is scalable and can describe the bigger Lorenz manifold witha relatively smaller amount of data without compromising the whole quality. Based onits results, the fact that the Lorenz manifold becomes attractive in the backward timedirection is found and confirmed. Based on the intriguing geometrical structure of the computed Lorenz manifold, the Lorenz manifold is of fractal is conjectured.Due to the unboundness of the Lorenz manifold, those methods which growingthe surface in radial direction are quite suitable. However, for those invariant mani-folds, which are bound, the orbit continuation algorithm (AN91 ORB) is more suitable.Its idea is to describe the manifold using orbits. The algorithm is fast, accurate andstable. But since the original implementation environment AUTO is not easy to use, itis substituted with Matlab in this thesis utilizing the continuation tool MatCont. Hencethere are five methods are realized in Matlab in this thesis to make a strong foundationfor the future release of a soft package on manifold computation.The validity of these five manifold algorithms realized during the numerical anal-ysis of the Lorenz manifold, needs confirmation on other types of systems. On the otherhand, they should be applied to help explain the dynamics of all kinds of systems. Forthese purposes, two systems are selected as example cases. One is a four dimensionalHamiltonian system that arises when studying the optimal control problem of balanc-ing an inverted pendulum on a cart subject to a quadratic cost function, the other isa fast-slow neuron model. The results on the former demonstrate the applicability ofthe algorithm AF06BIG on high dimensional systems. The study on the latter is aninitiative step in searching for the uniformly hyperbolic Plykin attractor in a neuronmodel. Although it is not found under the present parameter configuration, algorithmAN91ORB is confirmed to be competent for this searching.