Finding the effective dynamics to make rare events typical in chaotic maps

Abstract

Dynamical fluctuations or rare events associated with atypical trajectories in chaotic maps due to specific initial conditions can crucially determine their fate, as the may lead to stability islands or regions in phase space otherwise displaying unusual behavior. Yet, finding such initial conditions is a daunting task precisely because of the chaotic nature of the system. In this work, we circumvent this problem by proposing a framework for finding an effective topologically-conjugate map whose typical trajectories correspond to atypical ones of the original map. This is illustrated by means of examples which focus on counterbalancing the instability of fixed points and periodic orbits, as well as on the characterization of a dynamical phase transition involving the finite-time Lyapunov exponent. The procedure parallels that of the application of the generalized Doob transform in the stochastic dynamics of Markov chains, diffusive processes and open quantum systems, which in each case results in a new process having the prescribed statistics in its stationary state. This work thus brings chaotic maps into the growing family of systems whose rare fluctuations -- sustaining prescribed statistics of dynamical observables -- can be characterized and controlled by means of a large-deviation formalism.

Figures (5)

• Figure 1: Rare trajectories due to the repulsive effect of an unstable fixed point are made typical. Fluctuations of the time-averaged indicator function, $A=N^{-1}\sum_{n=0}^{N-1} \mathbb{I}_{[x^*\pm0.05]}(x_n)$, of the tent map around the unstable fixed point $x^*=2/3$. (a) Cobweb plot for $N=100$ iterations. The support of the indicator function is highlighted in light blue. (b) Trajectory illustrated in (a). (c) Histogram, $P(A=a)$, based on $10^5$ trajectories, with mean $\langle A \rangle=a_1=0.1$. (d) Cobweb plot for $N=100$ iterations of the Doob effective map with $s_0=-1$, making typical the rare fluctuation highlighted in (c). (e) Trajectory illustrated in (d). (f) Histogram, $P_{s_0}(A=a)$, based on $10^5$ trajectories of the map in (d), with mean $\langle A \rangle=a_2\approx 0.78$.
• Figure 2: Rare trajectories due to the repulsive effect of unstable period-2 orbits are made typical. Fluctuations of the time-averaged indicator function, $A=N^{-1}\sum_{n=0}^{N-1} (\mathbb{I}_{[x_-^*\pm 0.025]}(x_n)+ \mathbb{I}_{[x_+^*\pm 0.025]}(x_n))$, of the logistic map around the period-2 orbit formed by $x_{\pm}^*=(5\pm\sqrt{5})/8$. (a) SCGF $\theta(s)$ and biased average $\langle A \rangle_s=-\theta'(s)$. The three points highlighted correspond to $s=-1$ (square), $s=0$ (circle), $s=1$ (triangle). (b) Rate function $I(a)$, and Gaussian fluctuations around its average $\langle A \rangle$. (c) Cobweb plot of the Doob effective map for $s_0=-1$. The support of the indicator function is highlighted in light blue. (d) Trajectory corresponding to the cobweb in (c). (e, f) Cobweb plot and trajectory of the (unbiased) logistic map ($s_0=0$). (g, h) Cobweb plot and trajectory of the Doob effective map for $s_0=1$.
• Figure 3: Characterization of phases in a DPT for the Lyapunov exponent of the logistic map. Main panel: SCGF $\theta(s)$ and biased average $\langle A \rangle_s=-\theta'(s)$. The three points highlighted correspond to $s=-3$ (square), $s=-2$ (star), $s=0$ (circle). The latter corresponds to the logistic map, shown in Fig. \ref{['fig2']}(e) with a typical trajectory displayed in Fig. \ref{['fig2']}(f). Lower inset: Doob effective map and representative trajectory for $s_0=-3$. Upper inset: Same as lower inset but at the critical point $s_0=-2$, exhibiting coexistence between both dynamical phases. In both insets the original (logistic) map is also shown (see dashed lines)
• Figure S1: Comparison between analytical results and numerical results obtained with the cloning algorithm and with the numerical method, given by \ref{['itpr2']}-\ref{['itpr5']}, used in the main text. We consider here the fluctuations of the time-averaged position $A=N^{-1}\sum_{n=0}^{N-1} x_n$ for the doubling map $x_{n+1}=2x_n\!\!\!\mod 1$. (a) SCGF: The blue solid line is the analytical result while the red dots have been obtained with the numerical method. The black square is the value obtained with the cloning algorithm for $s=-1$. The inset shows the convergence to the analytical result (black dashed line) of the natural logarithm of the largest eigenvalue associated with the right and left eigenfunctions using the numerical method for $s=-1$. (b) Analytical result (blue solid line) for the right eigenfunction, together with the results based on the cloning algorithm (red solid line) and the numerical method (black solid line), all for $s=-1$. A perfect overlap is observed. (c) Left eigenfunction for $s=-1$. In the absence of an explicit analytical expression, results with the cloning algorithm (red solid line) and the numerical method (black solid line) are displayed.
• Figure S2: Original and Doob effective map, together with the typical and atypical trajectories for the doubling map. Fluctuations of the time-averaged position, $A=N^{-1}\sum_{n=0}^{N-1} x_n$, of the doubling map $x_{n+1}=2x_n\!\!\!\mod 1$. (a) Cobweb plot for $N=100$ iterations of the doubling map. (b) Trajectory illustrated in (a). (c) Cobweb plot for $N=100$ iterations of the Doob effective map for $s_0=-1$. (d) Trajectory illustrated in (c).