The Malkus-Robbins dynamo with a nonlinear motor

In a recent paper Moroz \cite{m02} considered a simplified version of the third class of self-exciting Faraday-disk dynamo model, introduced by Hide \cite{h97}, in the limit in which the Malkus-Robbins dynamo \cite{m72,r77} results as a special case. In that study a linear series motor was incorporated which led to an enriching of the range of possible behaviour that the original Malkus-Robbins dynamo could support. In this paper, we replace the linear motor by a nonlinear motor and consider the consequences on the dynamics of the dynamo

Unstable periodic orbits of perturbed Lorenz equations

The extended Malkus-Robbins dynamo [Moroz, 2003] reduces to the Lorenz equations when one of the key parameters, $\beta$, vanishes. In a recent study [Moroz, 2004] investigated what happened to the lowest order unstable periodic orbits of the Lorenz limit as $\beta$ was increased to the end of the chaotic regime, using the classic Lorenz parameter values of r = 28; $\sigma$ = 10 and b = 8=3. In this paper we return to the parameter choices of [Moroz, 2003], reporting on two of the cases discussed therein

Unstable periodic orbits of perturbed Lorenz equations

The extended Malkus-Robbins dynamo [Moroz, 2003] reduces to the Lorenz equations when one of the key parameters, $\beta$, vanishes. In a recent study [Moroz, 2004] investigated what happened to the lowest order unstable periodic orbits of the Lorenz limit as $\beta$ was increased to the end of the chaotic regime, using the classic Lorenz parameter values of r = 28; $\sigma$ = 10 and b = 8=3. In this paper we return to the parameter choices of [Moroz, 2003], reporting on two of the cases discussed therein

The extended Malkus-Robbins dynamo as a perturbed Lorenz system

Recent investigations of some self-exciting Faraday-disk homopolar dynamo ([1-4]) have yielded the classic Lorenz equations as a special limit when one of the principal bifurcation parameters is zero. In this paper we focus upon one of those models [3] and illustrate what happens to some of the lowest order unstable periodic orbits as this parameter is increased from zero

The Malkus–Robbins dynamo with a linear series motor

Hide [1997] has introduced a number of different nonlinear models to describe the behavior of n-coupled self-exciting Faraday disk homopolar dynamos. The hierarchy of dynamos based upon the Hide et al. [1996] study has already received much attention in the literature (see [Moroz, 2001] for a review). In this paper we focus upon the remaining dynamo, namely Case 3 of [Hide, 1997] for the particular limit in which the Malkus–Robbins dynamo [Malkus, 1972; Robbins, 1997] obtains, but now modified by the presence of a linear series motor. We compare and contrast the linear and the nonlinear behaviors of the two types of dynamo

Lie point symmetries and the geodesic approximation for the Schr\"odinger-Newton equations

We consider two problems arising in the study of the Schr\"odinger-Newton equations. The first is to find their Lie point symmetries. The second, as an application of the first, is to investigate an approximate solution corresponding to widely separated lumps of probability. The lumps are found to move like point particles under a mutual inverse-square law of attraction

When are projections also embeddings?

We study an autonomous four-dimensional dynamical system used to model certain geophysical processes.This system generates a chaotic attractor that is strongly contracting, with four Lyapunov exponents $\lambda_i$ that satisfy $\lambda_1+ \lambda_2+\lambda_3<0$, so the Lyapunov dimension is $D_L=2+|\lambda_3|/\lambda_1 < 3$ in the range of coupling parameter values studied. As a result, it should be possible to find three-dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior. We study mappings into $R^3$ to determine which can be used as embeddings to reconstruct the dynamics. We find dramatically different behavior in the two simplest mappings: projections from $R^4$ to $R^3$. In one case the one-parameter family of attractors studied remains topologically unchanged for all coupling parameter values. In the other case, during an intermediate range of parameter values the projection undergoes self-intersections, while the embedded attractors at the two ends of this range are topologically mirror images of each other

A Comparison of Tests for Embeddings

It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions

Early stages in the evolution of the atmosphere and climate on the Earth-group planets

The early evolution of the atmospheres and climate of the Earth, Mars and Venus is discussed, based on a concept of common initial conditions and main processes (besides known differences in chemical composition and outgassing rate). It is concluded that: (1) liquid water appeared on the surface of the earth in the first few hundred million years; the average surface temperature was near the melting point for about the first two eons; CO2 was the main component of the atmosphere in the first 100-500 million years; (2) much more temperate outgassing and low solar heating led to the much later appearance of liquid water on the Martian surface, only one to two billion years ago; the Martian era of rivers, relatively dense atmosphere and warm climate ended as a result of irreversible chemical bonding of CO2 by Urey equilibrium processes; (3) a great lack of water in the primordial material of Venus is proposed; liquid water never was present on the surface of the planet, and there was practically no chemical bonding of CO2; the surface temperature was over 600 K four billion years ago

Sequential inverse problems Bayesian principles and the\ud logistic map example

Bayesian statistics provides a general framework for solving inverse problems, but is not without interpretation and implementation problems. This paper discusses difficulties arising from the fact that forward models are always in error to some extent. Using a simple example based on the one-dimensional logistic map, we argue that, when implementation problems are minimal, the Bayesian framework is quite adequate. In this paper the Bayesian Filter is shown to be able to recover excellent state estimates in the perfect model scenario (PMS) and to distinguish the PMS from the imperfect model scenario (IMS). Through a quantitative comparison of the way in which the observations are assimilated in both the PMS and the IMS scenarios, we suggest that one can, sometimes, measure the degree of imperfection