Figure 1

Fidelity decay. To compute fidelity we divide the phase space in $100\times 100$ cells. We then take ${10}^5$ points in one cell and evolve these points with the map (1) up to time $n$. Then we compute the reverse evolution with the map (3) and compute the fraction of points which fall again in the initial cell, after time $2n$. The result is then averaged over initial distributions in 49 different randomly chosen cells. (a) $\log [1-F_{\delta }(n)]$ versus $\log ({\delta }^{2/5}n)$ magnifying the behavior for short times. The broken curves refer to $\delta ={10}^{-5},\dots ,{10}^{-10}$ (from top to bottom). The full line is the theoretical expression (9). (b) $\log [\mathbf{(}{F}_{\delta }(n)-u\mathbf{)}/(1-u)]$ versus ${\delta }^{2/5}n$ magnifying asymptotic (long-time) behavior. Here $u={10}^{-4}$ is the relative area of the initial set $\mathcal{A}$ giving the asymptotic value of fidelity. The meaning of broken curves is the same as in (a). The full curve gives the numerical solution of the random Gaussian model (12), while the dashed line has slope $-1$ to indicate asymptotic exponential decay.Reuse & Permissions

Figure 2

(a) Average distance $d_n=|\Delta z_n|$ versus time $n$ for two nearby initial orbits of the unperturbed map (1). The initial distance is $d_0={10}^{-9}$; the average is taken over $2.5\times {10}^8$ different initial conditions. The dotted line has slope 1. (b) Average distance versus time for two orbits starting from the same initial condition and evolving under the unperturbed and perturbed maps, (1, 3), respectively. The values of perturbation $\delta$, for the curves (from top to bottom), are ${10}^{-9}$, ${10}^{-10}$, ${10}^{-11}$, ${10}^{-13}$, and ${10}^{-15}$. The data are averaged over ${10}^5$ different initial conditions. The dotted lines have slope 1.5.Reuse & Permissions

Figure 3

Spreading of phase-space points of the echo dynamics. We consider 5000 initial points in the central cell of a $3\times 3$ grid (a). We then evolve these points up to time $n$ and then reverse the motion with the perturbed dynamics, with $\delta ={10}^{-6}$, up to the echo time $2n$. The density of points, for $n=200$, 400, 600 is shown in (b),(c),(d), respectively.Reuse & Permissions

Figure 4

The time evolution of the coarse-grained entropy for the triangle map, computed by taking $5\times {10}^7$ points initially distributed randomly in one cell of a $N\times N$ phase-space grid with $N=700$. We plot $\ln [S(n)-S(1)]$ versus $\ln \ln n$. The straight line has slope $3/2$. Obviously, the entropy will eventually saturate at $S(\infty )=\ln N^2$. In the inset we show the entropy computed for the uniquely ergodic and nonmixing dynamics with $\alpha =0$. The dotted line has slope 1.Reuse & Permissions