Figure 1
(a) Sketch of a stochastic process in time governed by a cubic drift and quadratic diffusion contributions and (b) its corresponding probability density function (PDF). Though the series shows a bistable dynamics (cubic drift) the PDF follows a Gaussian function, equivalent to an Ornstein-Uhlenbeck process (see text).
Figure 2
One-dimensional Langevin Approach: (a) drift coefficient, D(1)(x) = –x3 + x, and (b) diffusion coefficient, D(2)(x) = x2 + 1. Circles indicate the numerical results while the red dashed line indicates the theoretical coefficient, used when generating the synthetic data. Here 107 data points from the series illustrated in Figure 1a, were used for computing the averaged conditional moments.
Figure 3
PDFs of the increments for τ =1, τ =10, τ = 100 and τ = 1000 time lags (from top to bottom). Solid lines show the results for the original time series, broken lines the result for the reconstructed time series.
Figure 4
(a) Trajectory (X1(t), X2(t)) from Equations 15 with a = 0 and (b) the same trajectory integrating the same equations with non-zero stochastic terms (a = 0.05). For plotting 106 resp. 105 data points where used.
Figure 5
Drift coefficient of (a) the X1 component, , and (b) the X2 component, , together with all diffusion coefficients, namely (c) , (d) , (e) . See Equation 9. Estimated with added noise, i.e., a = 0.05 in Equation 15.