Scarring in classical chaotic dynamics with noise

We report the numerical observation of scarring, which is enhancement of probability density around unstable periodic orbits of a chaotic system, in the eigenfunctions of the classical Perron-Frobenius operator of noisy Anosov (“perturbed cat”) maps, as well as in the noisy Bunimovich stadium. A parallel is drawn between classical and quantum scars, based on the unitarity or nonunitarity of the respective propagators. For uniformly hyperbolic systems such as the cat map, we provide a mechanistic explanation for the classical phase-space localization detected, based on the distribution of finite-time Lyapunov exponents, and the interplay of noise with deterministic dynamics. Classical scarring can be measured by studying autocorrelation functions and their power spectra.

Figure 1

Magnitude of the second eigenfunction of the noisy Perron-Frobenius operator, numerically evaluated for: (a) Bunimovich quarter stadium billiard (in the inset with the bouncing-ball orbit) with noise of amplitude $D={10}^{-2}$, using the Ulam matrix (3) with $N=2^{14}$. Here $(\varphi ,sin\theta )$ are the coordinates of the dynamics on the boundary of the billiard: $\varphi$ is the polar angle locating the bounce, and $sin\theta$ is the angle of incidence with the normal to the boundary; (b) perturbed cat map (periodic point at the origin) with $\epsilon =0.1$, $\nu =1$, using the scheme (4) with diffusivity $\mathrm{\Delta }=5\times {10}^{-3}$ and $M=100$.

Figure 2

Classical scars of short periodic orbits: (a)–(c) Bunimovich quarter stadium billiard. Here the Ulam matrix (3) has $N=2^{14}$, and $D={10}^{-2}$. (a) Bowtie orbit (corresponding scar pointed to by an arrow); (b) rectangular orbit; (c) triangular orbit; (d) scar of a period-3 orbit [marked by $(\times )$] of the perturbed cat map ($\epsilon =0.1,\nu =2$) on the unit torus, obtained as the second eigenfunction of the transfer matrix (4), and $M=50$, $\mathrm{\Delta }={10}^{-5}$. The fixed point at the origin $(+)$ is “antiscarred”.

Figure 3

The perturbed cat map: magnitudes of the (a) left- and (b) right-second eigenfunctions of the Perron-Frobenius operator, realized through the Ulam matrix (3), with $N={10}^4$, $D={10}^{-2}$; Husimi distributions of (c) right- and (d) left-eigenfunctions of the spectrum of the subunitary quantum propagator coupled to a single-channel opening (dashed circle). Details of the quantization in Ref. [46].

Figure 4

(a) Unwrapping: the operator (6) is applied for $t=3$ iterations to the second eigenfunction of the noisy Perron-Frobenius operator for the perturbed cat map [parameters as in Fig. 5]; (b) Husimi distribution of a scarred eigenfunction of the unitary quantum propagator of the same map.

Figure 5

(a) The second eigenfunction of the noisy Perron-Frobenius operator (4), numerically evaluated for a perturbed cat map with $\mathrm{\Delta }={10}^{-2}$ and cutoff $M=100$; (b) the phase-space distribution of the finite-time Lyapunov exponents ($t=5$, sampling: ${10}^5$ initial points) for the same map.

Figure 6

(a) An initial Gaussian density centered at the fixed point of the perturbed cat map is mapped by the Ulam matrix (3); (b) t=1 iterations; (c) t=5; (d) t=20. (e) Magnitude of the power spectrum of the autocorrelation function minus the steady state: numerics for (dots) the density in (a)–(d), (diamonds) the same initial density centered at random in the phase space, (solid line) prediction (12).