Universal exponent for transport in mixed Hamiltonian dynamics

We compute universal distributions for the transition probabilities of a Markov model for transport in the mixed phase space of area-preserving maps and verify that the survival probability distribution for trajectories near an infinite island-around-island hierarchy exhibits, on average, a power-law decay with exponent $\gamma =1.57$. This exponent agrees with that found from simulations of the Hénon and Chirikov-Taylor maps. This provides evidence that the Meiss-Ott Markov tree model describes the transport for mixed systems.

Figure 1

An elliptic island of Eq. (2) and a Markov tree. Each node is labeled by the state $S$. Several transition probabilities $p_{S\to S^{\prime }}$ related to Eqs. (4) and (5) are also indicated. The illustrative transition probabilities correspond to state $S=10$.

Figure 2

Distribution densities of (a) area scalings $a$ and (b) flux scalings $w$ for the Hénon map with $-0.25<\kappa <0.75$. Distributions for levels are shown in (dark) gray, and for classes, in red (light gray).

Figure 3

Distribution densities of (a) class and (b) level area scalings for the Hénon map with parameters chosen in $-0.25<\kappa <0.25$, in gray, and in $0.25<\kappa <0.75$, in red (light gray).

Figure 4

Histograms of the joint probability distributions for $w$ and $a$. (a) $f^{\left(\text{Class}\right)}(a,w)$ taken from 2629 class transitions. (b) $f^{\left(\text{Level}\right)}(a,w)$ taken from 3608 level transitions.

Figure 5

Plot of the survival probability $P_{\text{sur}}$ vs time for 200 realizations of the sum Eq. (7). The heavy line is the average, decaying asymptotically with the slope $\gamma \approx 1.5$.

Figure 6

Plot of the survival probability $t^{{\gamma }_{\pm }}{P}_{\text{sur}}$ (a) ${\gamma }_{+}=1.6$ and (b) ${\gamma }_{-}=1.4$, vs time for 200 realizations of the sum Eq. (7).

Figure 7

Empirical verification of the power laws Eq. (A2) for a single realization of the matrix $\mathcal{W}$ in Fig. 5. (a) Plot of ${\sum }_{S\ne ⌀}{A}_n{\rho }_{n_S}$ vs $n$, and a fit with slope ${\delta }_2=8.8$. (b) Plot of the eigenvalues ${\lambda }_n$ of $\mathcal{W}$ vs $n$, leading to the slope ${\delta }_1=5.1$.

Figure 8

The survival probability Eq. (B1) for simulations of Eq. (1). The heavy line is the average over 45 parameter values (see text) resulting in $\gamma \approx 1.573$.