About the relationships among variables observed in the real world

Since a stationary chaotic system is determined by nonlinear equations connecting its components, the appurtenance of two variables to such a system has been considered a sign of nontrivial relationships between them including also other quantities. These relationships could remain hidden for the approach usually employed in the research analyses, which is based on the extent of the correlation that characterises the dependence of one variable on the other. The appurtenance to the same system can be hypothesized if the topological features of the attractors reconstructed from two time series representing the evolution of the corresponding variables are close to each other. However, the possibility that both attractors represent different systems with similar behaviour cannot be excluded. For that reason, an approach allowing the reconstruction of the attractor by using jointly two time series was proposed and the conclusion about the common origin of the variables under study can be made if this attractor is topologically similar to those built separately from the two time series. In the present study, the features of the attractors were presented by the correlation dimension and the largest Lyapunov exponent and the proposed algorithm has been tested on numerically generated sequences obtained from various maps. It is believed that this approach could be used to reveal connections among the variables observed in experiments or field measurements.