Figure 1

(a) Lyapunov exponents vs $ϵ\sim \kappa {\omega }^{-1/2}$. Shown are results of numerical simulations of an Euler discretization of the linearized equations of motion (4), replacing the elements of $\mathbf{F}(t)$ by random numbers with appropriate statistics (symbols). Also shown are Padé-Borel summations of (17), with Padé approximants of orders $P_{22}$ (dash–double-dotted line), $P_{23}$ (dash-dotted line), $P_{33}$ (dashed line), and $P_{34}$ (solid lines). (b) shows $\Delta$ vs $ϵ$ [obtained from (2) using the data in (a)]. (c) $-\sum _i{\lambda }_i/\gamma$ [circles, solid red line; see (a) and (b)] compared to the rate of caustic formation determined from simulations as described above (squares) and a fit to $C\exp (-S/{ϵ}^2)$ with $C=0.21$ and $S=1/8$. (d) Schematic phase diagram in the $\kappa \mathrm{\text{−}}\omega$ plane (note the logarithmic scales). The red line indicates where the quantity $\Delta$ is zero. Above the red line $\Delta$ is positive implying clustering. In the under- and overdamped advective limits, $\Delta$ is positive and small (weak clustering), while below the red solid line $\Delta$ is negative. Here clustering occurs by caustics.Reuse & Permissions