Chaos identification through the autocorrelation function indicator $({\textit{ACFI}})$

Abstract

Chaotic motion affecting small bodies in the Solar system can be caused by close encounters or collisions or by resonance overlapping. Chaotic motion can be detected using approaches that measure the separation rate of trajectories that starts infinitesimally close or changes in the frequency power spectrum of time series, among others. In this work, we introduce an approach based on the autocorrelation function of time series, the ${\textit{ACF}}$ index (${\textit{ACFI}}$). Autocorrelation coefficients measure the correlation of a time series with a lagged copy of itself. By measuring the fraction of autocorrelation coefficients obtained after a given time lag that are higher than the 5% null hypothesis threshold, we can determine how the time series autocorrelates with itself. This allows identifying unpredictable time series, characterized by low values of ${\textit{ACFI}}$. Applications of ${\textit{ACFI}}$ to orbital regions affected by both types of chaos show that this method can correctly identify chaotic motion caused by resonance overlapping, but it is mostly blind to close encounters induced chaos. ${\textit{ACFI}}$ could be used in these regions to select the effects of resonance overlapping.

Appendix A: dependence of \({\textit{ACFI}}\) on free parameters

In this appendix, we discuss how the choice of three free parameters of the ${\textit{ACFI}}$: the null hypothesis value for autocorrelation, the $i_{\mathrm{in}}$ and $i_{\mathrm{fin}}$ numbers of autocorrelation coefficients over which ${\textit{ACFI}}$ is computed, affects the results of our method. We start by considering different possible values for the null hypothesis level. Traditionally, the null hypothesis level is set at 5%. If a correlation coefficient is within $\pm 0.05$, the two times series are thought not to be correlated. But what happened if we altered this interval?

Fig. 14

Dependence of the mean value of ${\textit{ACFI}}$ on the choice of the null hypothesis level for 2500 regular and chaotic particles in the orbital region of the Veritas family. The shaded area displays the confidence interval, defined as plus or minus one standard deviation

Fig. 15

Dependence of the mean value of ${\textit{ACFI}}$ on the $i_{\mathrm{in}}$ (left panel) and $i_{\mathrm{fin}}$ (right panel) free parameters. As in Fig. 14, the shaded area displays the confidence interval

To test this hypothesis, we changed this level for 2500 test particles used for the dynamical map in the region of the Veritas asteroid family. Figure 14 shows our results. Increasing the null hypothesis level causes a decrease in the mean value of ${\textit{ACFI}}$, and vice versa. This is understandable, since lowering the value of the null hypothesis level means increasing the fraction of autocorrelation coefficients that satisfies the ${\textit{ACFI}}$ requirement, and vice versa. However, the net effect on the relationship between ${\textit{ACFI}}$ for regular and chaotic particles does not change: ${\textit{ACFI}}$ remains higher for regular particles and lower for chaotic ones. As long as we consistently use one single level for all test particles in a map, the choice of these parameters should, therefore, not affect the overall outcome. Considering the precedents found in the literature that normally associates a 5% level with the null hypothesis, in this work we will also use this value.

We then also checked the dependence of the mean value of ${\textit{ACFI}}$ on the $i_{\mathrm{in}}$ and $i_{\mathrm{fin}}$ parameters. We first fixed $i_{\mathrm{fin}}$ at 500 and let vary $i_{\mathrm{in}}$ in the range 100 to 300 and then fixed $i_{\mathrm{in}}$ at 200 and let vary $i_{\mathrm{fin}}$ between 300 and 1000. Our results are displayed in the two panels of Fig. 15. ${\textit{ACFI}}$ is not strongly dependent on either parameter. For our applications, we will use $i_{\mathrm{in}}=200$ and $i_{\mathrm{fin}}=500$.