Chaotic Systems with Absorption

Motivated by applications in optics and acoustics we develop a dynamical-system approach to describe absorption in chaotic systems. We introduce an operator formalism from which we obtain (i) a general formula for the escape rate $\kappa$ in terms of the natural conditionally invariant measure of the system, (ii) an increased multifractality when compared to the spectrum of dimensions $D_q$ obtained without taking absorption and return times into account, and (iii) a generalization of the Kantz-Grassberger formula that expresses $D_1$ in terms of $\kappa$, the positive Lyapunov exponent, the average return time, and a new quantity, the reflection rate. Simulations in the cardioid billiard confirm these results.

Figure 1

Billiards naturally incorporate both partial reflection at the boundary and nontrivial return times between collisions. (a) Cardioid billiard, whose boundary in polar coordinates is $r(\varphi )=1+\cos (\varphi )$ [24]. The intensity $J$ of the rays decays due to $R(\overrightarrow{x})=R^{*}<1$ in the gray boundary interval at the top ($R(\overrightarrow{x})=1$ otherwise). (b) Return time distribution $\tau (\overrightarrow{x})$ in the cardioid billiard (velocity modulus is chosen to be unity). Birkhoff coordinates $\overrightarrow{x}=(s,p=\sin \theta )$ are used where $s$ is the arc length along the boundary and $\theta$ is the collision angle.

Figure 2

Open baker map with absorption and return times. Intensity $J$ decays due to $R<1$. Trajectories leave the system (unit square) when $(x,y)\in E$. The extended map [Eq. (1)] is $(x^{\prime },y^{\prime })=(bx,ay)$ for $y<1/2$, $(x^{\prime },y^{\prime })=1-b(1-x),1-a(1-y)$ for $y\ge 1/2$, $(\tau ,R)=({\tau }_1,R_1)$ if $y<1/a$ and $({\tau }_2,R_2)$ if $y>1-1/a$. $b\le 1/2$, $a\ge 2$; ${\mathcal{D}}_f=ab$.

Figure 3

Conditionally invariant density ${\rho }_c$ for the cardioid billiard with absorption. As shown in Fig. 1, $R=1$ everywhere except in $\{s\in [0.4,0.6],p\in [-1,1]\}$ where $R=R^{*}<1$. (a) $R^{*}=0.1$; and (b) $R^{*}=0.75$. Structures in (b) amount to 0.2% difference between $D_{0,c}$ and $D_{1,c}$; see Table .