Variability is a critical aspect of a dynamical systems analysis. Because there are a number of numerical techniques that can be used in such an analysis, the calculation of variability has several issues that must be addressed. The purpose of this paper is to present a variety of quantitative methods for investigating variability from a dynamical systems perspective. The paper is divided into two major sections covering discrete and continuous methods. Each of these sections is subdivided into two sections. Within discrete methods, we discuss, first, the calculation of the discrete relative phase from a time-series history of two parameters and, second, the use of return maps. Using continuous methods, we present procedures for using angle-angle plots in the evaluation of relative phase. We then discuss the use of phase plots in the calculation of the continuous relative phase. Each of these methods presents unique problems for the researcher and the method to be used is determined by the nature of the question asked.
Joseph Hamill, Jeffrey M. Haddad and William J. McDermott
Richard E.A. Van Emmerik, Michael T. Rosenstein, William J. McDermott and Joseph Hamill
Nonlinear dynamics and dynamical systems approaches and methodologies are increasingly being implemented in biomechanics and human movement research. Based on the early insights of Nicolai Bernstein (1967), a significantly different outlook on the movement control “problem” over the last few decades has emerged. From a focus on relatively simple movements has arisen a research focus with the primary goal to study movement in context, allowing the complexity of patterns to emerge. The approach taken is that the control of multiple degrees-of-freedom systems is not necessarily more difficult or complex than that of systems only comprising a few degrees of freedom. Complex patterns and dynamics might not require complex control structures. In this paper we present a tutorial overview of the mathematical underpinnings of nonlinear dynamics and some of its basic analysis tools. This should provide the reader with a basic level of understanding about the mathematical principles and concepts underlying pattern stability and change. This will be followed by an overview of dynamical systems approaches in the study of human movement. Finally, we discuss recent progress in the application of nonlinear dynamical techniques to the study of human locomotion, with particular focus on relative phase techniques for the assessment of coordination.