Reconstruction of the Bifurcation Structure of a Dynamical System from Time Series Data
Bagarinao Epifanio · Institutional Repositories DataBase (IRDB) · 2000
any kriowledge of tlie explicit form of the dynamical system (differential/diff'erence equation).Instead, tiine se.ries at diff'erent parameter values are used to obt,ain a suitable t'amily of predictor functions, xvhich exhibits qualit,atively similar bifurcations as the given s>rsteni.The BD ofthis fainil)' of predictor funct,ions on soine parameter region, terined as prol'ect7jon reg'ion, is thenLTIie notation E[•] will be used to denote statistical expectation, that is, if X is a random variable and f(x) its deiisit,.v,then E[-X:] = .Lt".. :t;f(x)d•:i;• THELIKELIHOODFUNCTION 15and the conditional mean.For nonlinear cases, it is difficult to obtain these two values since the disti'ibution of the observed data is not always known.However, a method called the innovations approach will facilitate the computation of these two quantities from the observed data.This will be discussed in the next subsection. The innovations approachSuppose that observations (data) ofthe form z(t) == y(t) +v(t), OStS T, (3.4) where y(t) and v(t) are m-dimensional statistically independent vector processes, are given.Furthermore, v(t) is a ivhite Gaussian noise with the following properties: E[v(t)] -O (3.5) E[v(t)vT(s)] = I6(t-s). (3.6).iXlso, the signal process, y(t), which is not necessarily Gaussian, is characterized by the following:As mentioned earlier, the estimation problem involves the approximation of a random process, say x(t), sat,isfying Eq. (3.1) from the observation z(t).Also, x(t) is related to the signal process y(t.) vja y(t) == h[x(s), sS t] (3.10) and obeys the following properties:Tl}e probleni is to find the optimal estimat,e 5Z (tlT) of x(t) given {z(,s), O E{ .g< r :E{{ T} xvhich is readily obt,ained froin the conditional mean of x(t), that is, 5t (tl7) = E[Å~ (t) lz(s), O s{ s< 7].(:3 .13)