Robert W. Easton

Isolated invariant sets and isolating blocks

Closed invariant sets of smooth flow on compact manifold involving homoclinic or heteroclinic point theory of Poincare

REGULARIZATION OF VECTOR FIELDS BY SURGERY

Abstract : Regularization is a common procedure in the study of differential equations. The usual method is to multiply the vector field by a suitably chosen positive scalar function which vanishes on the set of singularities of the vector field. The new vector field or differential equation thus obtained may have no singularities and thus may be easier to study. The qualitative behavior of solutions off the set of singularities is the same for both vector fields. Another method of regularization involves surgery and roughly, the idea is to excise a neighborhood of the singularity from the manifold on which the vector field is defined and then to identify appropriate points on the boundary of the region. The regularization of the planar 2-body problem on surfaces of constant negative energy was first discussed by Levi-Civita. The 2-body problem is regularized using surgery. This method gives a new way of looking at the classical result and makes apparent the geometrical reasons for its success. It is possible to regularize the three body problem on surfaces of nonzero angular momentum using the present methods. This result will be treated in a subsequent paper. (Author)

Some topology of the 3-body problem

Abstract The purpose of this paper is to topologically characterize the surfaces of constant momentum, angular momentum, and energy occuring in the planar 3-body problem. For most values of these parameters the integral surfaces are manifolds. We have assumed for simplicity that the three bodies have equal masses. Independently, Stephen Smale [1] has recently characterized the integral surfaces occuring in the planar n -body problem for arbitrary masses using different methods.

Isolating blocks and epsilon chains for maps

Abstract The relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored. Epsilon chains are used as a mathematical representation of computer generated pseudo-orbits. The question “When is a pseudo-orbit shadowed by a true orbit” is discussed. Isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks. The Conley index is extended and applied to maps.

Transport through chaos

Certain orbits of area preserving maps of the plane appear to wander randomly and to densely fill regions of the plane having positive Lebesgue measure. These regions are called ergodic zones. Trellises (or homoclinic tangles) embedded within these zones are shown to guide the transport of ensembles of points. Trellises or tangles can be localized within what are called resonance zones. Transport through resonance zones is calculated using the MacKay-Meiss-Percival action principle.

Stability of levitrons

Abstract The Levitron is a magnetic spinning top which can levitate in the constant field of a repelling base magnet. An explanation for the stability of the Levitron using an adiabatic approximation has been given by Berry. In experiments the top eventually loses stability at a critical spin rate which cannot be predicted by Berry’s approach. The present work develops an exact theory of the Levitron with six degrees of freedom which allows for the calculation of critical spin rates. The main result is a complete classification of possible Levitrons that allow for an interval of stable spin rates. Stability of the relative equilibrium is lost in Hamiltonian Hopf bifurcations if either the spin rate is too large or too small.

Climate Change and Damage from Extreme Weather Events

The risks of extreme weather events are typically being estimated, by federal agencies and others, with historical frequency data assumed to reflect future probabilities. These estimates may not yet have adequately factored in the effects of past and future climate change, despite strong evidence of a changing climate. They have relied on historical data stretching back as far as fifty or a hundred years that may be increasingly unrepresentative of future conditions. Government and private organizations that use these risk assessments in designing programs and projects with long expected lifetimes may therefore be investing too little to make existing and newly constructed infrastructure resistant to the effects of changing climate. New investments designed to these historical risk standards may suffer excess damages and poor returns. This paper illustrates the issue with an economic analysis of the risks of relatively intense hurricanes striking the New York City region.

Parabolic Orbits in the Planar Three Body Problem

It is not known whether there exist oscillatory and capture orbits in the planar 3-body problem. Sitnikov [6] proved such orbits exist for the restricted 3-body problem. Alekseev [I] extended this work and related it to the existence of homoclinic orbits. McGehee and Easton ]3] studied a model problem closely related to the planar 3-body problem and proved the existence of oscillatory and capture orbits near certain orbits biasymptotic to an invariant 3-sphere. The goal of this paper is to construct an invariant 3-sphere “at infinity” in the planar 3-body problem and to prove that the stable and unstable manifolds of this sphere are Lipschitz manifolds. To complete the study of oscillatory and capture orbits one must investigate the way in which the stable and unstable manifolds of the 3-sphere intersect each other. This is a difficult question which is not treated here. However, using methods of Melnikov and using one of the masses as a perturbation parameter such an investigation might be possible. The stable manifold theorem which is proven here is not standard since the invariant 3-sphere is not hyperbolic. Geometric methods of McGehee [ 4 ] are extended and applied to obtain the theorem. A special case of the system of equations that we study has the form J, =x: dY)lX, + E,(x, Y)L f, = 4 dY>l-x* + E,(x, Y)L (0.1) 4’ =

On the existence of invariant sets inside a submanifold convex to a flow

Invariant set existence inside submanifold convex to flow established from vector field properties on submanifold boundary

Combining Analog Method and Ensemble Data Assimilation: Application to the Lorenz-63 Chaotic System

Nowadays, ocean and atmosphere sciences face a deluge of data from space, in situ monitoring as well as numerical simulations. The availability of these different data sources offers new opportunities, still largely underexploited, to improve the understanding, modeling, and reconstruction of geophysical dynamics. The classical way to reconstruct the space-time variations of a geophysical system from observations relies on data assimilation methods using multiple runs of the known dynamical model. This classical framework may have severe limitations including its computational cost, the lack of adequacy of the model with observed data, and modeling uncertainties. In this paper, we explore an alternative approach and develop a fully data-driven framework, which combines machine learning and statistical sampling to simulate the dynamics of complex system. As a proof concept, we address the assimilation of the chaotic Lorenz-63 model. We demonstrate that a nonparametric sampler from a catalog of historical datasets, namely, a nearest neighbor or analog sampler, combined with a classical stochastic data assimilation scheme, the ensemble Kalman filter and smoother, reaches state-of-the-art performances, without online evaluations of the physical model.