Multicorrelation analysis and state space reconstruction

Constructing a mathematical model of a nonlinear system involves developing methods for determining a set of nonlinear differential equations. Based on Floris Takens\u27 theory, the delayed-time space with a given time-series is created, where the first inflection of multicorrelation function is an approximation of the optimal delay time. The multicorrelation function is the generalization of the autocorrelation function into a higher dimension of the system. The standard Grassberger-Proccia algorithm computes the correlation dimension of an artificially generated data set, which involves measuring the distances between all pairs of points, and estimates the dimensionality of the nonlinear system. Finally, the governing differential equations are generated by using a polynomial least squares method. The generated state equations provide the possibility of predicting the system. The practical aspects of attractor reconstruction is discussed in this investigation, by using nonlinear ordinary differential equations with low degrees of freedom as examples