A unified approach to compute foliations, inertial manifolds, and tracking initial conditions

Several algorithms are presented for the accurate computation of the leaves in the foliation of an ODE near a hyperbolic fixed point. They are variations of a contraction mapping method in [25] to compute inertial manifolds, which represents a particular leaf in the unstable foliation. Such a mapping is combined with one for the leaf in the stable foliation to compute the tracking initial condition for a given solution. The algorithms are demonstrated on the Kuramoto-Sivashinsky equation

Global error analysis and inertial manifold reduction

Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to include rescaling of time. To determine the amplificatioh of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations. We combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction. These global error concepts are exercised to illustrate their utility on the Lorenz equations and inertial manifold reductions of the Kuramoto-Sivashinsky equation. (C) 2016 Elsevier B.V. All rights reserved

(Un)Stable Manifold Computation via Iterative Forward-Backward Runge-Kutta Type Methods

I present numerical methods for the computation of stable and unstable manifolds in autonomous dynamical systems. Through differentiation of the Lyapunov-Perron operator in [Casteneda, Rosa 1996], we find that the stable and unstable manifolds are boundary value problems on the original set of differential equation. This allows us to create a forward-backward approach for manifold computation, where we iteratively integrate one set of variables forward in time, and one set of variables backward in time. Error and stability of these methods is discussed

Consistent Young Earth Relativistic Cosmology

We present a young earth creationist (YEC) model of creation that is consistent with distant light from distant objects in the cosmos. We discuss the reality of time from theological/philosophical foundations. This results in the rejection of the idealist viewpoint of relativity and the recognition of the reality of the flow of time and the existence of a single cosmological “now.” We begin the construction of the YEC cosmology with an examination of the “chronological enigmas” of the inhomogeneous solutions of the Einstein field equations (EFE) of General Relativity (GR). For this analysis we construct an inhomogeneous model by way of the topological method of constructing solutions of the EFE. The topological method uses the local (tensorial) feature of solutions of the EFE that imply that if (M, g) is a solution then removing any closed subset X of M is also a solution on the manifold with MA = M - X and the restriction gA = g|MA. Also if (MA, gA) and (MB, gB) are solutions of the EFE in disjoint regions then the “stitching” together of (MA, gA) and (MB, gB) with continuous boundary conditions is also a solution. From this we show conceptually how an approximate “crude” model with a young earth neighborhood and an older remote universe can be constructed. This approximate “crude” model suffers from having abrupt boundaries. This model is an example of a spherically symmetric inhomogeneous space-time. We discuss the class of exact spherically symmetric inhomogeneous universes represented by the Lemaître-Tolman (L-T) class of exact solutions of the EFE. A more realistic model refines this technique by excising a past subset with an asymptotically null spacelike surface from the Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology. We build the model from the closed FLRW solution by selecting a spacelike hyperboloidal surface as the initial surface at the beginning of the first day of creation. This surface induces, by way of embedding into FLRW space-time, an isotropic but radially inhomogeneous matter density consistent with the full FLRW space-time. The resulting space-time is a subset of the usual FLRW space-time and thus preserves the FLRW causal structure and the observational predictions such as the Hubble law. We show that the initial spacelike surface evolves in a consistent manner and that light from the distant “ancient” galaxies arrives at the earth within the creation week and thereafter. All properties of light arriving from distant galaxies retain the same features as those of the FLRW space-time. This follows from the fact that the solution presented is an open subset of the FLRW space-time so that all differential properties and analysis that applies to FLRW also applies to our solution. Qualitatively these models solve the distant star light problem and from a theological point of view, in which God advances the (cosmic) time of the spacelike hypersurfaces at a non-uniform rate during the miraculous creation week, solve the distant light problem. We conclude by briefly discussing possible objections of some of our key assumptions and showing that a relativist cannot consistently object to our assumptions based on the merely operationalist point of view that an absolute spacelike “now” cannot be empirically determined

Aspects of Invariant Manifold Theory and Applications

Recent years have seen a surge of interest in "data-driven" approaches to determine the equations governing complex systems. Yet in spite of modern computing advances, the high dimensionality of many systems --- such as those occurring in biology and robotics --- renders direct machine learning approaches infeasible. This dissertation develops tools for the experimental study of complex systems, based on mathematical concepts from dynamical systems theory. Our approach uses the fact that parsimonious assumptions often lead to strong insights from dynamical systems theory; such insights can be leveraged in learning algorithms to mitigate the “curse of dimensionality” and make these algorithms practical. Our first contribution concerns nonlinear oscillators. Oscillators are ubiquitous in nature, and usually associated with the existence of an "asymptotic phase" which governs the long-term dynamics of the oscillator. We show that asymptotic phase can be expressed as a line integral with respect to a uniquely defined closed differential 1-form, and provide an algorithm for estimating this "ToF" from observational data. Unlike all previously available data-driven phase estimation methods, our algorithm can: (i) use observations that are much shorter than a cycle; (ii) recover phase within the entire region for which data convergent to the limit cycle is available; (iii) recover the phase response curves (PRC-s) that govern weak oscillator coupling; (iv) show isochron curvature, and recover nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series. Our next contribution concerns locomotion systems which are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that in the ``Stokesian'' (viscous; zero Reynolds number) limit, the motion is governed by a reduced order "connection'' model that describes how body shape change produces motion for the body frame with respect to the world. In the "perturbed Stokes regime'' where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer connection-like. We derive this model using results from noncompact NHIM theory in a singular perturbation framework. Using a normal form derived from theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a "gait") directly from observational data of shape and body motion. Our algorithm has applications to the study of optimality of animal gaits, and to hardware-in-the-loop optimization to produce gaits for robots. Finally, we make fundamental contributions to NHIM theory: we prove that the global stable foliation of a NHIM is a $C^0$ disk bundle, and we prove that the dynamics restricted to the stable manifold of a compact inflowing NHIM are globally topologically conjugate to the linearized transverse dynamics at the NHIM restricted to the stable vector bundle. We also give conditions ensuring $C^k$ versions of our results, and we illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.PHDElectrical and Computer EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147642/1/kvalheim_1.pd