Vector field reconstruction

Vector field reconstruction [ [http://prola.aps.org/pdf/PRE/v51/i5/p4262_1 Global Vector Field Reconstruction from a Chaotic Experimental Signal in Copper Electrodissolution.] Letellier C, Le Sceller L , Maréchal E, Dutertre P, Maheu B, Gouesbet G, Fei Z, Hudson JL. Physical Review E, 1995 May;51(5):4262-4266] is a method of creating a vector field from experimental data, usually with the goal of finding a differential equation model of the system.

Formulation

In the best possible case, one has data streams of measurements of all the system variables, equally spaced in time, say

:s₁(t), s₂(t), ... , s_k(t)

for

:t = t₁, t₂,..., t_n,

beginning at several different initial conditions. Then the task of finding a vector field, and thus a differential equation model consists of fitting functions, for instance, a cubic spline, to the data to obtain a set of continuous time functions

:x₁(t), x₂(t), ... , x_k(t),

computing time derivatives dx₁/dt, dx₂/dt,...,dx_k/dt of the functions, then making a least squares fit using some sort of basis functions (orthogonal polynomials, radial basis functions, etc.) to each component of the tangent vectors to find a global vector field. A differential equation then can be read off the global vector field.

Applications

Vector field reconstruction has several applications, and many different approaches. Some mathematicians have not only used radial basis functions and polynomials to reconstruct a vector field, but they have used Lyapunov exponents and singular value decomposition [ Global vector-field reconstruction of nonlinear dynamical system from a time series with SVD method and validation with Lyapunov exponent. Wei-Dong L, Ren F K, Cluzel G M, Gouesbet G. Chin. Phys. Soc, 2003 December; Vol 12 No 12:1366-1373] . Gousebet and Letellier used a multivariate polynomial approximation and least squares to reconstruct their vector field. This method was applied to the Rössler system, and the Lorenz system, as well as thermal lens oscillations.

The Rossler system, Lorenz system and Thermal lens oscillation follows the differential equations in standard system as

:X'=Y, Y'=Z and Z'=F(X,Y,Z)

where F(X,Y,Z) is known as the standard function [Global vector field reconstruction by using a multivariate polynomial "L"₂ approximation on nets.Gousebet G and Letellier C. Physical Review E, 1994 June; Vol 49, No 6: 4955-4972 ] .

Implementation issues

In some situation the model is not very efficient and difficulties can arise if the model has a large number of coefficients and demonstrates a divergent solution. For example, nonautonomous differential equations give the previously described results [Constructing nonautonomous differential equations from experimental time series. Bezruchko B.P and Smirnov D.A. Physical Review E, 2000; Vol 63, 016207:1-7 ] . In this case the modification of the standard approach in application gives a better way of further development of global vector reconstruction.

Usually the system being modeled in this way is a chaotic dynamical system, because chaotic systems explore a large part of the phase space and the estimate of the global dynamics based on the local dynamics will be better than with a system exploring only a small part of the space.

Frequently, one has only a single scalar time series measurement from a system known to have more than one degree of freedom. The time series may not even be from a system variable, but may be instead of a function of all the variables, such as temperature in a stirred tank reactor using several chemical species. In this case, one must use the technique of delay coordinate embedding, [Embedology, Tim Sauer, James A. Yorke, and Martin Casdagli, Santa Fe Institute working paper] where a state vector consisting of the data at time t and several delayed versions of the data is constructed.