How Well Can One Resolve the State Space of a Chaotic Map?

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching or contraction and the smearing due to noise. We propose to determine the “finest attainable” partition for a given hyperbolic dynamical system and a given weak additive white noise, by computing the local eigenfunctions of the adjoint Fokker-Planck operator along each periodic point, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. Numerical tests of such “optimal partition” of a one-dimensional repeller support our hypothesis.

Figure 1

$f_0$, $f_1$: branches of the deterministic map (9) for ${\Lambda }_0=8$ and $b=0.6$. The local eigenfunctions ${\tilde{\rho }}_a$ with variances given by (8) provide a state space partitioning by neighborhoods of periodic points of period 3. These are computed for noise variance ($D=\mathrm{\text{diffusion constant}}$) $2D=0.002$. The neighborhoods ${\mathcal{M}}_{000}$ and ${\mathcal{M}}_{001}$ already overlap, so ${\mathcal{M}}_{00}$ cannot be resolved further. For periodic points of period 4, only ${\mathcal{M}}_{011}$ can be resolved further, into ${\mathcal{M}}_{0110}$ and ${\mathcal{M}}_{0111}$.

Figure 2

Transition graph (graph whose links correspond to the nonzero elements of a transition matrix $T_{ba}$) describes which regions $b$ can be reached from the region $a$ in one time step. The 7 nodes correspond to the 7 regions of the optimal partition (10). Dotted links correspond to symbol 0, and the full ones to 1, indicating that the next region is reached by the $f_0$, respectively, $f_1$ branch of the map plotted in Fig. 1.

Figure 3

(left scale) the escape rate of the repeller (9) vs the noise strength $D$, calculated using (□) the “optimal partition,” and ($\times$) a uniform discretization (16) in $N=128$ intervals; (right scale) the Lyapunov exponent of the same repeller vs $D$, estimated using (•) the “optimal partition,” and (⋄) the average (17).