Many physical, chemical, biological, and electrical systems are well described by ordinary differential equations.
For instance, one may have an electrical circuit that in theory is described by a system of ordinary differential equations, but due to the tolerance of resistors, variations of the supply voltage, or interference from outside influences, they do not know the exact parameters of the system.
Therefore, it may be necessary to construct more exact differential equations by building them up based on the actual system performance rather than a theoretical model.
In the best possible case, one has data streams of measurements of all the system variables, equally spaced in time, say for beginning at several different initial conditions.
Some mathematicians have not only used radial basis functions and polynomials to reconstruct a vector field, but they have also used Lyapunov exponents and singular value decomposition.
[2] Gouesbet and Letellier used a multivariate polynomial approximation and least squares to reconstruct their vector field.
[4] In this case, the modification of the standard approach in application gives a better way of further development of global vector reconstruction.
Frequently, one has only a single scalar time series measurement from a system known to have more than one degree of freedom.
A comprehensive review of the topic is available from [6] A global mathematical model along with numerical simulations with applications to wind velocity fields using is given in.