  In a tripartite system comprising a $\Lambda$-atom interacting with two
radiation fields in the presence of field nonlinearities and an
intensity-dependent field-atom coupling, striking features have been shown to
occur in the dynamics of the mean photon number $\aver{N_{i}(t)}$ ($i = 1,\,2$)
corresponding to either field. In this Letter, we carry out a detailed
time-series analysis and establish an interesting correlation between the
short-time and long-time dynamics of $\aver{N_{i}(t)}$. Lyapunov exponents,
return maps, recurrence plots, recurrence-time statistics, as well as the
clustering coefficient and the transitivity of networks constructed from the
time series, are studied as functions of the intensity parameter $\kappa$.
These are shown to carry signatures of a special value $\kappa = \bar{\kappa}$.
Our work also exhibits how techniques from nonlinear dynamics help analyze the
behavior of observables in multipartite quantum systems.
