More information regarding dynamical systems via convergence to correlation dimension

Abstract

It is a well-known fact that a fractal number calculated from time-series data cannot fully describe the complex structure of a strange attractor in dynamical systems. Hence, it is necessary to find a procedure through which fractal dimension provides more information about the underlying dynamical system. In order to serve such a purpose, in this study correlation integrals have been calculated at every row or off-diagonal from the distance matrix. Subsequently, the slope of each integrals has been plotted versus the number of the row or off-diagonal considered in the distance matrix. This gives an asymptotic convergence to correlation dimension and has close relation with lacunarity and types of behavior, i.e., periodic or aperiodic.