Cycle-expansion method for the Lyapunov exponent, susceptibility, and higher moments

Lyapunov exponents characterize the chaotic nature of dynamical systems by quantifying the growth rate of uncertainty associated with the imperfect measurement of initial conditions. Finite-time estimates of the exponent, however, experience fluctuations due to both the initial condition and the stochastic nature of the dynamical path. The scale of these fluctuations is governed by the Lyapunov susceptibility, the finiteness of which typically provides a sufficient condition for the law of large numbers to apply. Here, we obtain a formally exact expression for this susceptibility in terms of the Ruelle dynamical $\zeta$ function for one-dimensional systems. We further show that, for systems governed by sequences of random matrices, the cycle expansion of the $\zeta$ function enables systematic computations of the Lyapunov susceptibility and of its higher-moment generalizations. The method is here applied to a class of dynamical models that maps to static disordered spin chains with interactions stretching over a varying distance and is tested against Monte Carlo simulations.

Figure 1

Temperature dependence of the Lyapunov observables obtained by Monte Carlo simulations for $N_{\mathrm{n}}=2$ (blue), 3 (red), 4 (cyan), and 5 (magenta), along with the exact results for $N_{\mathrm{n}}=1$ (black). (a) Subtracting the high-temperature limit, $ln\left(2\right)$, from $\lambda$ shows it to scale as ${\tilde{\beta }}^2$ for $\tilde{\beta }\ll 1$ and as $\tilde{\beta }$ for $\tilde{\beta }\gg 1$. (b) The normalized free-energy density $\tilde{f}\equiv f/J$ as a function of the normalized temperature $\tilde{T}\equiv T/J$. For $N_{\mathrm{n}}\ge 3$, the normalized free-energy monotonically varies as the interaction range increases. It is expected to asymptote to the mean-field Sherrington-Kirkpatrick limit with $\pm J/\sqrt{N}$ disorder distribution, but such a disordered model has not yet been solved. (c) The Lyapunov susceptibility, ${\chi }_{\mathrm{L}}$, also shows two regimes. Its growth as ${\tilde{\beta }}^2$ for $\tilde{\beta }\gg 1$ reveals the finiteness of ${lim}_{N\to \infty }N\left(\mathbb{E}\left[f_N^2\right]-\mathbb{E}{\left[f_N\right]}^2\right)$, which governs sample-to-sample fluctuations in the thermal free energy. (d) The correlation length, $\xi$, diverges exponentially toward $T=0$ for $N_{\mathrm{n}}=1$, 2, and 5, but is maximal at $\tilde{T}\sim 0.5$ for $N_{\mathrm{n}}=3$ and 4, thus demonstrating the nonuniversality of the appropriate order parameter across models of disordered spin chains. Note that for all the temperatures considered, $\xi <N_{\mathrm{equil}}=0.1N$. Note also that the error on the various quantities is of order $\sqrt{{\chi }_{\mathrm{L}}/\left(N_{\mathrm{D}}N\right)}$, and is thus much smaller than the width of the lines in this figure.

Figure 2

Generalized Lyapunov exponents evaluated at integer values of $n$ for the NNN model at $\tilde{\beta }=1$ (orange) and $\tilde{\beta }=10$ (navy blue). The diamonds at $n=0$ denote the values for ${lim}_{n\to 0}\frac{L\left(n\right)}{n}=\lambda$ obtained by Monte Carlo simulations. The values of the generalized Lyapunov exponent are obtained through the tensorial replication technique described in Refs. [12, 57] for $n=1,...,7$. A sextic polynomial fit (dashed lines) gets more or less the correct result for $\tilde{\beta }=1$, but fails dramatically for $\tilde{\beta }=10$.

Figure 3

Lyapunov observables obtained through the cycle-expansion method for $N_{\mathrm{n}}=2$ (blue) and 3 (red) at order $k=3$ (dashed), 7 (dotted), and 11 (dash-dotted), along with the Monte Carlo simulations results (solid). (a) The normalized free-energy $\tilde{f}$ as a function of the normalized temperature $\tilde{T}$. (b) The Lyapunov susceptibility, ${\chi }_{\mathrm{L}}$, as a function of $\tilde{T}$. (c) The correlation length, $\xi$, as a function of $\tilde{T}$. Convergence of the cycle expansion for the subleading eigenvalue is slower than for the leading eigenvalue, but nonetheless suffices to recover qualitative features including the nonmonotonic temperature evolution of $\xi \left(T\right)$ for $N_{\mathrm{n}}=3$.

Figure 4

(a) The skewness for the NNN model obtained by the cycle expansion at order $k=3$ (dashed), 7 (dotted), 11 (dash-dotted), and 15 (solid) quickly converges with increasing $k$. (b) The skewness at $\tilde{\beta }=1$ evaluated directly through Monte Carlo simulations as a function of the number of spins $N$ with a fixed number of disorder realization $N_{\mathrm{D}}={10}^1$ (dashed line with triangles), ${10}^3$ (dotted line with squares), and ${10}^5$ (dash-dotted line with circles) fluctuates around the cycle-expansion result for $k=15$ (solid line). The skewness and higher moments are associated with fine features of the distribution that otherwise approaches a sharply peaked Gaussian distribution for large $N$, making their reliable estimates by simple Monte Carlo numerically challenging.