Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations

This is the first on a series of articles that deal with nonlinear dynamical systems under oscillatory input that may exhibit harmonic and non-harmonic frequencies and possibly complex behavior in the form of chaos. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods because, most of the time, analytical solutions turn out to be difficult, if not impossible, since they are based on infinite series of trigonometric functions. The analytic matrix method reported here is a direct one that speeds up the solution processing compared to traditional series solution methods. In this method, we work with the invariant submanifold of the problem, and we propose a series solution that is equivalent to the harmonic balance series solution. However, the recursive relation obtained for the coefficients in our analytical method simplifies traditional approaches to obtain the solution with the harmonic balance series method. This method can be applied to nonlinear dynamic systems under oscillatory input to find the analog of a usual Bode plot where regions of small and medium amplitude oscillatory input are well described. We found that the identification of such regions requires both the amplitude as well as the frequency to be properly specified. In the second paper of the series, the method to solve problems in the field of large amplitudes will be addressed.

Difference, whiteness and the group analytic matrix: An integrated formulation

I am a black woman. This statement may trigger various responses and, perhaps even the urge to disengage. Nonetheless I write it as a social fact. Firstly, to forewarn the reader that the lifeworld they are about to enter may well challenge theirs and, to correct potential erroneous, normative, racial and gendered assumptions. I trust readers will stay with any potential discomfort. Read on. And, reflect upon it at the end of the article. I am too a psychologist and aspiring group analyst. This social and professional positioning means that I have heard many conversations on ‘difference’ where I, and others whose bodies look like mine, have been placed under the deforming microscope of the white gaze, for the alleged edification of my peers, one of the most objectifying encounter I continue to experience. There is a long history within western epistemic, ontological and other scholarly pursuits of normalizing whiteness, of regarding those ‘deviating’ from it as ‘different’ and, of subjecting them/us to investigation, curiosity and/or exoticisation. Ultimately, to consumption. Group analysis is no exception. Difference is a historically loaded term built on the brutality of white masculinist and heteronormative social constructions and thus, on the enactment of power related violence. Central to formulating the function of ‘difference’ and of such brutality between individuals and groups, is the group analytic concept of the matrix. Foulkes (1973), conceptualized it as a hypothetical web of communication and relationships providing the group a shared ground of meaning and significance. This article aims to critically examine the concept of the matrix with reference to race and specifically, to whiteness. It argues that fixating difference onto people of colour, serves fundamental functions for whiteness by linking this process to Foulkes’ concept of location of disturbance. In the second part, the group matrix and whiteness are considered. Finally, using various vignettes, a formulatory framework is suggested to illustrate how whiteness may be reproduced within different levels of the group matrix.