Determination of the Dynamical Properties of Ferroelectrics Using Nonlinear Time Series Analysis

Abstract

We investigate the behavior of an experimental electric resonance circuit with a nonlinear capacitor. Triglycine sulfate (TGS) was used as nonlinear dielectric material. This is the most thoroughly investigated ferroelectric with a second-order phase transition. Its static dielectric small signal behavior is well described in the framework of the Landau theory of phase transitions. The dynamic behavior however is strongly related to relaxation phenomena due to switching of domains. Here the theoretical situation is still unclear. By mere data analysis we construct an ordinary differential equation which describes the dynamical behavior of the circuit, and which is simple in the sense of Ockham's razor. The structure of this equation is closely related to the dynamical properties of the ferroelectric. We are therefore able to gain some insights into the dynamical behavior of ferroelectrics, which may give some hints for a more exhaustive analysis of nonequilibrium properties of ferroelectrics.

What Can We Learn About Ferroelectrics Using Methods of Nonlinear Dynamics?

“…These signals are recorded by a digital storage oscilloscope and can be used for direct observation of the phase portrait as demonstrated in figure 2. Alternatively, it is possible to record only the signal U C and to calculate the temporal derivatives numerically [1,5].…”

“…From equation (1) one gets a second order differential equation [1,5,6], that is the equation of motion in the sense of the theory of dynamic systems:…”

“…What follows is a short description of the fitting procedure. For a detailed outline see [1,5]. Having recorded a long time series of U C ∼ D (129907 data points D n per time series in our experiments) it is possible to estimate the temporal derivativesḊ n andD n from adjacent data points of the time series D n according to the following formulas:…”

“…It could be shown ( [5]) that the relative error N saturates at approximately 8.5% for k > 12 and l > 5. A further increase of the degree of the polynomial F (D,Ḋ) to k = 15 and l = 7 gave no significant reduction of the error.…”

“…The result of this reduction was a small increase of the relative errors N odd ≈ N all + 0.5% for the three different cases. Further reduction of the number of the remaining 65 α ij was achieved by a modified backward elimination algorithm (for details see [5]). Nevertheless, it was necessary to keep about 20-30 basis functions to avoid a dramatic increase of the error.…”