From Complexity in the Natural Sciences to Complexity in Operations Management Systems

Appendix 2Time Series Analysis with a View to Deterministic Chaos

When time-ordered series of measurements are analyzed to help forecast the future behaviors of all sorts of ecosystems, it is important to detect whether there is some underlying equation for the phenomenon observed or it is a stochastic phenomenon.

What is called the phase space of a system is the set of all instantaneous states available to a system. The attractor of a dynamical system is the subset of phase space towards which the system evolves as time elapses. It can be just a point or a limited set of points.

When an attractor exists, the trajectories of the time-ordered data points with connecting line segments are treated as fractals in the associated phase space. Fractals are geometric forms with irregular patterns that repeat themselves at different scales. The forms consist of fragments of varying size and orientation but similar shape. The fractal dimension of an attractor is a parameter which characterizes a part of its properties.

Dimension is the way to measure the effect of enlargement (scaling) on length, area, volume and fractal object. Scaling by a factor n of length is length multiplied by n, of area is area multiplied by n², of volume is volume multiplied by n³ and of d-dimensional object is object multiplied by n^d.

While dimension provides information about the scaling properties of an object, it does not deliver information about the very structure of the object. However, its value gives ...