Effective intermittency and cross correlations in the standard map

We define auto- and cross-correlation functions capable of capturing dynamical characteristics induced by local phase-space structures in a general dynamical system. These correlation functions are calculated in the standard map for a range of values of the nonlinearity parameter $k$. Using a model of noninteracting particles, each evolving according to the same standard map dynamics and located initially at specific phase-space regions, we show that for $0.6<k\le 1.2$ long-range cross correlations emerge. They occur as an ensemble property of particle trajectories by an appropriate choice of the phase-space cells used in the statistical averaging. In this region of $k$ values the single-particle phase space is either dominated by local chaos $(k\le k_c$ with $k_c\approx 0.97)$ or it is characterized by the transition from local to global chaos $\left(k_c<k\le 1.2\right)$. Introducing suitable symbolic dynamics we demonstrate that the emergence of long-range cross correlations can be attributed to the existence of an effective intermittent dynamics in specific regions of the phase space. Our findings support the recently established relation of intermittent dynamics and cross correlations [F. K. Diakonos, A. K. Karlis, and P. Schmelcher, Europhys. Lett. 105, 26004 (2014)] in simple one-dimensional intermittent maps, suggesting its validity also for two-dimensional Hamiltonian maps.

Figure 1

The mixed PS of the SM for $k=0.95$. The colored (gray) regions correspond to chaotic while the black regions to regular dynamics. The lines indicate the locations of $\mathrm{\Gamma }$ (dashed) and $\mathrm{\Omega }$ (solid) invariant spanning curves. The red colored region defines zone 1, as indexed by the numbers in the column at the right-hand side of the plot, while the blue and the green colored regions define zones 2 and 3, respectively (see discussion in Sec. 3).

Figure 2

The cumulative distribution functions for the probability of a chaotic trajectory to cross up to time $t$ the second to last spanning curve (red dashed line) starting from a PS cell in the first zone (region 1 in Fig. 1) and the last spanning curve (blue solid line) starting from a PS cell in the second zone (region 2 in Fig. 1). Both functions are calculated using $k=1$. The dotted line at $F_T\left(t\right)=1$ is plotted to guide the eye.

Figure 3

Numerical demonstration of the cancellation of the correlations in the regular domain. Shown is $A_p^{\left(r\right)}$ of orbits in a cell of size $0.5\times 0.5$ within the stability island centered around $(\pi ,0)$ when (a) a single trajectory and (b) ${10}^5$ trajectories are used in the averaging. The $C_p^{\left(r\right)}\text{s}$ (not shown) have similar form. Analogous results are obtained also for the LFTCFs of the $\theta$ variable, for all ${\mathfrak{C}}^{\left(d\right)}$ within the regular domain.

Figure 4

Left subfigures depict the average $A_x^{\left(c\right)}$ for the cell $[0,0.25]\times [0,0.25]$ in $x=p$ (a) and $x=\theta$ (b) variables for $k=0.95$. Right subfigures are typical time-series for $p$ and $\theta$ with initial conditions in this cell. Subfigure (c) corresponds to $p_n$ and (d) to ${\theta }_n$ versus $n$ (number of iterations).

Figure 5

(a) $A_p^{\left(c\right)}$ in the cell $[0,0.25]\times [0,0.25]$ for a wide range of $k$ values plotted in linear scale. An ensemble of $S_d=5\times {10}^5$ orbits each of length $N=5\times {10}^3$ is used. (b) The same plot in semilog scale. The $A_p^{\left(c\right)}\text{s}$ are normalized to one at $m=0$ by dividing the autocorrelation function of each trajectory in the ensemble with the corresponding standard deviation.

Figure 6

The inverse decay rates $\tau \left(k\right)$ of the $A_p^{\left(c\right)}\text{s}$ (calculated using initial conditions in the PS cell $[0,0.25]\times [0,0.25])$ shown versus $k$ (black circles). The red solid line is an interpolating sigmoidal curve to guide the eye. The vertical dotted line indicates the location $k=k_c$.

Figure 7

The $A_p^{\left(c\right)}\text{s}$ belonging to the different zones of the chaotic domain at $k=0.95$.

Figure 8

$C_p^{\left(c\right)}$ for an ensemble of $5\times {10}^5$ chaotic trajectories with initial conditions in ${\mathfrak{C}}^{\left(c\right)}=[0,0.25]\times [0,0.25]$ (zone 1) obtained using $k=0.8$ (black line, squares), $k=k_c$ (orange line, circles), $k=1.2$ (blue line, triangles) and $k=1.5$ (green line, stars). (a) Linear-linear, (b) linear-log, and (c) log-log scale. The cross-correlation function of each pair of trajectories in the ensemble is normalized by dividing with the root of the product of the standard deviations of those trajectories.

Figure 9

The $C_p^{\left(c\right)}\text{s}$ calculated for $k=0.95$ using PS cells located in different zones of the chaotic domain. For zone 1 we use ${\mathfrak{C}}^{\left(c\right)}=[0,0.25]\times [0,0.25]$, for zone $2{\mathfrak{C}}^{\left(c\right)}=[0.02,0.17]\times [1.66,1.8]$, and for zone $3{\mathfrak{C}}^{\left(c\right)}=[4.82,5.03]\times [3.57,3.68]$. Each cell is numbered with a violet-colored number. The blue lines above cells 1 and 4 correspond to the fixed point's $(0,0)$ eigenvectors (see end of Sec. 4). The blue line above cell 1 represents the unstable eigenvector, whereas the blue line above cell 4 represents the stable eigenvector.

Figure 10

Time-series of the variables $p$ (a, b, c) and $\theta$ (d, e, f) of the standard map, for a typical orbit within the chaotic domain, for $k=0.95$ and initial conditions $(0.1,0.2)$ $(n$ is the number of iterations). Shown are (a, d) the first ${10}^4$ iterations, (b, e) a zoom-in in the region $[0,{10}^3]$, and (c, f) a zoom-in in the region $[400,500]$ of this orbit.

Figure 11

A typical trajectory $\mathrm{[}{p}_n$ shown in (a), ${\theta }_n$ shown in (b)] for $k=1.8$ and initial conditions $(0.2,0.1)$ $(n$ counts the number of iterations). In the zoom-in of the $p_n$ component, shown in (c), it is clearly seen that the triangular substructure in the time series of $p$ is not sustained and a laminar phase (as defined in Sec. 4) does not exist. In (d) we also show for completeness a zoom-in of the ${\theta }_n$ component.

Figure 12

Fixed points of the standard map for $k=1.0$ found by the method introduced in Ref. [31].

Figure 13

Laminar length distributions $P\left(\lambda \right)$ for various values of $k$, shown in log-log scale. We used $5\times {10}^5$ trajectories with initial conditions in the cell $[0,0.25]\times [0,0.25]$ and verified the independence of the cell location in zone 1.