Engineering topics
Publications and source records attributed to Ott, Edward.
We propose a novel hybrid modeling approach that combines a physics-based, numerical model of the Earth system with a machine learning model. We also envision to integrate the hybrid model training with a data assimilation cycle, so that new observations update both the state estimate used as initial conditions and the machine learning model parameters.
In this paper, we study dynamical systems in which a large number N of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors. These behaviors include time periodic cluster states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as a behavior in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic. We argue that this second type of behavior is “extensive” in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with N and that the number of positive Lyapunov exponents of the attractor also scales linearly with N. An important focus of this paper is the transition between cluster states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe discontinuous transitions between the cluster states (which correspond to low dimensional dynamics) and the extensively chaotic states. Furthermore, examining the cluster state, as the system approaches the discontinuous transition to extensive chaos, we find that the oscillator population distribution between the clusters continually evolves so that the cluster state is always marginally stable. This behavior is used to reveal the mechanism of the discontinuous transition. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors.
In this paper, we propose, discuss, and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered, including attractors, repellers, and non-periodically forced systems. This definition is based on an entropy-like quantity, which we call “expansion entropy,” and we define chaos as occurring when this quantity is positive. We relate and compare expansion entropy to the well-known concept of topological entropy to which it is equivalent under appropriate conditions. We also present example illustrations, discuss computational implementations, and point out issues arising from attempts at giving definitions of chaos that are not entropy-based.
In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.
The evolution of the magnetic field in models that incorporate chaotic field line stretching, field cancellation, and finite magnetic Reynolds number is examined analytically and numerically. Although the models used here are highly idealized, it is claimed that they display and illustrate typical behavior relevant to fast magnetic dynamic behavior. It is shown, in particular, that consideration of magnetic flux through a finite fixed surface provides a simple and effective way of deducing fast dynamo behavior from the zero resistivity equation. Certain aspects of the fast dynamo problem can thus be reduced to a study of nonlinear dynamic properties of the underlying flow.
This paper considers the question of the existense of a steady fast kinematic magnetic dynamo for a conducting fluid with a steady velocity field and vanishingly small electrical resistivity. The analysis of examples of steady dynamos, found by considering the zero-resistivity dynamics, indicated that, for sufficiently small resistivity, dynamo action can indeed occur in steady smooth three-dimensional chaotic fluid flows and that fast dynamos should consequently be a typical occurrence for such flows.
The kinematic dynamo problem is considered in the R(m) approaching infinity limit. It is shown that the magnetic field tends to concentrate on a zero volume fractal set; moreover, it displays arbitrarily fine-scaled oscillations between parallel and antiparallel directions. Consideration is given to the relationship between the dynamo growth rate and quantitative measures of chaos, such as the Liapunov element and topological entropy.