Energy localization in two chaotically coupled systems

We set up and analyze a random matrix model to study energy localization and its time behavior in two chaotically coupled systems. This investigation is prompted by a recent experimental and theoretical study of Weaver and Lobkis on coupled elastomechanical systems. Our random matrix model properly describes the main features of the findings by Weaver and Lobkis. Due to its general character, our model is also applicable to similar systems in other areas of physics—for example, to chaotically coupled quantum dots.

Figure 1

Results from the experiment by Weaver and Lobkis. The energy ratio between the different rooms is plotted versus time (in ms). We note the different scales on the $y$ axes. Reprinted from Ref. 23.

Figure 2

Energy ratio $E_2∕E_1$ versus (dimensionless) time. Data from N1 of Ref. 23, bin 16, plotted as a dashed line. The result from Eq. (16) for $\lambda \to \infty$, $B$ set by property 2 of Eq. (18) and a rescaling of time to make the two curves fit for early times, is plotted as a solid line.

Figure 3

Comparison between the $2\times 2$ RMT model and the low-frequency bins of N2 of Ref. 23. The energy ratio $E_2∕E_1$ is plotted versus (dimensionless) time.

Figure 4

Energy ratio $E_2∕E_1$ versus (dimensionless) time. Data from N1 of Ref. 23 plotted as dashed lines and from refolded numerical simulations using the RMT model as solid lines. We notice the different $E_2∕E_1$ scale in plot (a).

Figure 5

Energy ratio $E_2∕E_1$ versus (dimensionless) time. Data from N2 of Ref. 23 plotted as dashed lines and from refolded numerical simulations using the RMT model as solid lines. We notice the different $E_2∕E_1$ scale in plot (a).