Time-series analysis of monopole moments

There are two basic modes of the Giant monopole resonances: isoscalar motion, when proton and neutron densities oscillate in phase, and isovector motion, with two densities oscillating in opposite phases. Here we investigate the corresponding time-dependent monopole moments [2] for a particular case of of $^{208}$Pb and NL3 effective interaction. The isoscalar giant monopole resonance displays regular undamped oscillations, while for the isovector mode oscillations are strongly damped and anharmonic, but the main peaks obtained by Fourier analysis in both cases are close to the experimental values [2]. In order to obtain more information about the underlying nuclear dynamics we start with reconstruction of the phase-space from the time-series of monopole moments. In general, vector in the d-dimensional reconstructed phase space from the time series x(n) is $\vec{y}(n) = \left[ x(n), x(n+T), x(n+2T), ... ,x(n+(d-1)T)\right]$, where T is the time lag. The time delay should be chosen in such a way that $x(n+jT)$ and $x(n+ (j+1)T)$ present two independent coordinates, while the dimension of the reconstructed phase space $d_E$ correspond to the phase space on which the attractor unfolds. The most appropriate approach for determination of the time-lag T for nonlinear systems is by using the method of average mutual information [1]

$$I(T)= \sum\limits_{n=1}^N P(x(n),x(n+T))~{\rm log_2}~ \left[ {{ P(x(n),x(n+T))}\over {P(x(n)) P(x(n+T))}}\right] .$$ (2)

From this function we can conclude how much information can be learnt about a measurement at one time from a measurement taken at another time. For the time-lag we choose the value for which I(T) displays the first minimum [1], and as a result we obtain time-lags 27 fm/c for the isoscalar, and 13 fm/c for the isovector mode. The embedding dimension is determined by the method of false nearest neighbors  [1]. For each vector in the phase space of dimension d, the nearest neighbor is found and distance between them is calculated,

$$R_d^2(n) = [x(n) - x^{NN}(n)]^2 + ... + [x(n+(d-1)T) - x^{NN}(n+(d-1)T)]^2.$$ (3)

Two vectors are declared nearest neighbors if this distance is small, but if the two points are nearest neighbors in dimension $d$, while $R_{d+1}^2(n)$ is large compared to $R_d^2(n)$, then we declare the two points as the "false nearest neighbors". In order to determine the embedding dimension we simply count the number of nearest neighbors for all vectors, for the particular dimension $d$, and we estimate the percentage of false nearest neighbors when going to dimension $(d+1)$. Minimal necessary embedding dimension $d_E$ is the one for which the percentage of false nearest neighbors goes to zero. For the case of monopole moments for $^{208}$Pb we evaluate the embedding dimension: $d_E=3$ for the isoscalar mode, and $d_E=4$ for the isovector mode. To present reconstructed phase spaces in the reccurence plots for isoscalar and isovector case, we define a sequence of vectors $\vec {y}(n) = \left[ x(n), ... ,x(n+(d_E - 1)T) \right]$ and evaluate the distance between any two points in the phase space $\delta(m,n) = \vert\vec{y}(m) - \vec{y}(n)\vert$ . If $\delta(m,n) < r$, where r is appropriate distance, then we plot a dot at the coordinate $(m,n)$ [1]. In the case of isoscalar mode the recurrence plot displays a pattern representative for regular oscillations, while for the isovector mode it indicates non-stationarity. If one is dealing with deterministic system, the set of phase space trajectories converges into the attractor. Convenient measure of the density of dots in the reccurence plot is provided by the correlation integral from which one can estimate correlation dimension for the attractor [1]. In the case of isoscalar mode, for $d\geq 3$, the correlation dimension saturates at $D_2 = 2$; it does not saturate for the isovector mode, but slowly increases to some fractional value between 2 and 3 which correspond to chaotic or stochastic dynamics.

Nils Paar, 1999.