2.2.3 The Fractal Dimension of the Attractor Tells Us the Number of Independent Variables Needed to Generate the Time Series
The fractal dimension of the attractor in the phase space provides very useful information about the nature of the process that generated the sequence of values measured in time.
In a deterministic system, the present values of the measured properties are related to their previous values. The dimension of the attractor tells us the number of independent variables in this relationship. The number of independent variables is the smallest integer that is greater than or equal to the fractal dimension of the attractor.
For example, the fractal dimension of the attractor for the logistic system is slightly less than 1. Thus 1 equation with 1 independent variable can generate the sequence of values in time of the logistic system.
For example, the fractal dimension of the attractor for the Lorenz system is equal to 2.03. Thus a set of 3 equations with 3 independent variables can generate the sequence of values in time of the Lorenz system.