We provide an analytical approximation of a periodic solution of the three body problem in celestial mechanics, the so-called figure eight solution, discovered 1993 by C. Moore. This approximation has the form of a Fourier series whose components are in turn Taylor series w. r. t. some parameter. The method is first illustrated by application to two other problems, (1) the problem of oscillations of a particle in a cubic potential that has a well-known analytic solution in terms of elliptic functions and (2) periodic solutions and corresponding eigenvalues of a generalized Mathieu equation that cannot be solved analytically. When applied to the three body problem it turns out that the Fourier-Taylor series, evaluated up to 30th order, represents un-physical solutions except for a particular value of the series parameter. For this value the series approximates the numerical solution known from the literature up to a relative error of 1.6 × 10 −3 .

No abstract

Personality Systems Interactions (PSI) theory explains differences in personality based on the properties of four cognitive systems—object recognition (OR), intuitive behaviour (IB), intention memory (IM) and extension memory (EM). Each system is associated with characteristic modes of perception and behaviour, so personality is determined by which systems are primarily active. According to PSI theory, the activities of the cognitive systems are regulated by positive and negative affect (reward and punishment). Thus, differences in personality ultimately emerge from four parameters—the sensitivities of up- or downregulating positive and negative affect. The complex interactions of affect and cognitive systems have been represented in a mathematical model based on a system of differential equations. In this study, the environment of a person represented by the mathematical model is modelled by a time series of perturbations with positive and negative affect that are generated by a stochastic process. Comparing the average activities of the cognitive systems for different parameter sets exposed to the same time series of affect perturbations, we observe that different dominant cognitive systems emerge. This demonstrates that different sensitivities for positive and negative affect lead to different modes of cognition and, thus, to different personality types such as agreeable, conscientious, self-determined or independent. Varying the relative frequencies of negative and positive affect perturbations reveals that the average activities of all cognitive systems respond linearly. This observation enables us to predict that conscientious and independent personalities benefit from increased exposure to positive affect, whereas agreeable and self-determined personalities achieve a better balance of their cognitive systems by increased negative affect.