Global and local diffusion in the standard map

We study the global and the local transport and diffusion in the case of the standard map, by calculating the diffusion exponent $\mu$. In the global case, we find that the mean diffusion exponent for the whole phase space is either $\mu =1$, denoting normal diffusion, or $\mu =2$ denoting anomalous diffusion (and ballistic motion). The mean diffusion of the whole phase space is normal when no accelerator mode exists and it is anomalous (ballistic) when accelerator mode islands exist even if their area is tiny in the phase space. The local value of the diffusion exponent inside the normal islands of stability is $\mu =0$, while inside the accelerator mode islands it is $\mu =2$. The local value of the diffusion exponent in the chaotic region outside the islands of stability converges always to the value of 1. The time of convergence can be very long, depending on the distance from the accelerator mode islands and the value of the nonlinearity parameter $K$. For some values of $K$, the stickiness around the accelerator mode islands is maximum and initial conditions inside the sticky region can be dragged in a ballistic motion for extremely long times of the order of ${10}^7$ or more but they will finally end up in normal mode diffusion with $\mu =1$. We study, in particular, cases with maximum stickiness and cases where normal and accelerator mode islands coexist. We find general analytical solutions of periodic orbits of accelerator type and we give evidence that they are much more numerous than the normal periodic orbits. Thus, we expect that in every small interval $\mathrm{\Delta }K$ of the nonlinearity parameter $K$ of the standard map there exist smaller intervals of accelerator mode islands. However, these smaller intervals are in general very small, so that in the majority of the values of $K$ the global diffusion is normal.

Figure 1

The diffusion coefficient $D\left(K\right)$ derived from Eq. (4) as a function of the nonlinearity parameter $K$ [gray (red) solid curve for $n=5\times {10}^3$ iterations and black dotted curve for $n={10}^4$ iterations]. The presence of accelerator modes in the intervals defined by Eq. (2) generates anomalous diffusion where the diffusion coefficient is no longer constant but it goes to infinity with the number of iterations.

Figure 2

(a) The diffusion exponent $\mu$ as a function of the number of iterations $n$ for several values of the nonlinearity parameter $K$ for a number of iterations up to $2\times {10}^4$. The values of $\mu$ converge fast to $\mu =1$ in the case of normal diffusion, e.g., for $K=5.0$ and 10.0. On the other hand, in the cases of anomalous diffusion (accelerator modes) the values of $\mu$ converge to $\mu =2$ after a time of the order of $n=2\times {10}^4$ for $K=6.4$ and 12.69, while the other cases of accelerator modes have not yet converged. (b) Same as in (a) for a longer time up to $5\times {10}^5$. We observe that the values of $\mu$ seem to converge around the value of 2 after $n\approx 3\times {10}^5$ for all the values of $K$ related to the accelerating modes while they converge to 1 for values of $K$ related to the normal mode. (c) The case of $K=18.96$ calculated for even larger values of $n$, where it is obvious that $\mu$ finally converges to the value 2.

Figure 3

The diffusion coefficient $D\left(K\right)=\langle {(y-y_0)}^2\rangle /n$ [gray (red) curve] calculated for $n={10}^4$ iterations superimposed with the relative area $A$ of the stability islands corresponding to the accelerator modes (group II in [15]) for values of $K$ in the interval $6.3\le K\le 7.7$ multiplied by a factor $2\times {10}^4$ (black curve). The dashed black curve on the left refers to normal mode islands (group I in [15]). For an interval of $K$ ($6.3\le K\le 6.5$), a coexistence of normal and accelerator mode islands takes place.

Figure 4

(a) The evolution of $\langle {(y-y_0)}^2\rangle$ as a function of the number of iterations $n$, in a logarithmic scale for a set of initial conditions inside the large chaotic sea [gray (red) curve] and inside the sticky region of the normal mode island (black curve) for $K=5$. (b) The slope of the curves in (a) as a function of the number of iterations $n$, corresponding to the diffusion exponent $\mu$. The initial conditions inside the large chaotic sea converge fast to $\mu =1$, while the initial conditions inside the sticky region begin with $\mu =0$ during the time of stickiness and, then, after a transient time of abrupt increase of the slope it converges to $\mu =1$ which corresponds again to normal diffusion.

Figure 5

(a) The phase space for $K=6.8$ with accelerator mode islands (calculations are made for the sticky, black region, around the islands of stability). (b) The evolution of $\langle {(y-y_0)}^2\rangle$ as a function of the number of iterations $n$, in a logarithmic scale for a set of initial conditions inside the extreme sticky region of (a). The diffusion is anomalous (ballistic) for the time interval of stickiness around the accelerator mode island ($n_{st}\approx {10}^3$) where the diffusion exponent is equal to 2 ($\mu =2$), but after a transient time it becomes normal with diffusion exponent $\mu =1$. (c) Same as in (b) but for a set of initial conditions in the outer sticky region. Here, the ballistic motion lasts for a shorter time ($n_{st}\approx {10}^2$). (d) Same as in (b) for a set of initial conditions far from the sticky region and in the large chaotic sea. Here, the diffusion is normal almost from the beginning of time.

Figure 6

The displacements of $y$ for three initial conditions for $K=6.8$. The black curve corresponds to an initial condition outside the sticky zone of Fig. 5 with distance $\mathrm{\Delta }y=0.012$ from the last $KAM$ curve of the island of stability which diffuses normally, while the other initial conditions (red and blue) which are closer to the sticky zone of Fig. 5 (with distance $\mathrm{\Delta }y=0.002$ and 0.001, respectively) exhibit ballistic motion for a while, dragged by the accelerator mode island and then continue with normal diffusion.

Figure 7

(a) A part of an accelerator mode island for $K=6.6$ around a periodic orbit with $x\approx 0.7,y=0$. This island is surrounded by four accelerator mode islands surrounding a period-4 periodic orbit (only two of the four islands are shown here) that has bifurcated from the central orbit. These orbits lie inside the last $KAM$ curve and do not communicate with the large chaotic sea (b) The same as in (a) for $K=6.608$. Here, the period-4 periodic orbits are inside the large chaotic sea and extreme sticky regions surround them. The area inside the last $KAM$ curve has decreased abruptly. (c) A zoom of the sticky region around the accelerator mode islands outside the last $KAM$ curves [thick gray (red) curves] of the islands of stability. (d) The extreme sticky region [gray (red)] outside the last $KAM$ curve (thick black curve) of one of the four islands.

Figure 8

(a) The diffusion coefficient $D$ as a function of time for the case of $K=6.608$ and a set of initial conditions inside the large chaotic sea far away from the sticky region of the accelerator mode islands. The value of $D$ tends towards a constant value after more than $3\times {10}^7$ iterations. (b) The evolution of $\langle {(y-y_0)}^2\rangle$ as a function of the number of iterations $n$, in a logarithmic scale. The slope of this curve corresponds to the diffusion exponent $\mu$, which has a mean value of $\approx 2.0$ until $5\times {10}^6$ iterations while it converges to the value $\mu =1$, corresponding to normal motion, after $3\times {10}^7$ iterations.

Figure 9

The displacements of $y$ for several initial conditions of the phase space for $K=6.608$. The dragging of the accelerator mode island can last for very long times (the black curve corresponds to ballistic motion for a time of $4\times {10}^6$ iterations, during which the displacement of $y$ is proportional to the number of iterations $n$), depending on how close the initial condition is to the island, but all initial conditions outside the accelerator mode island finally exhibit normal diffusion.

Figure 10

Three different unstable periodic orbits of multiplicity 23 (point $A$), 25 (point $B$), and 28 (point $C$) with their corresponding unstable manifolds [dark and light gray (blue and red) curves] in the extreme sticky region of the accelerator mode island for $K=6.608$. The light gray (green) thick curve is the last $KAM$ curve surrounding the island of stability. Initial conditions on these manifolds have different times of escape from the sticky zone and different times of Poincaré recurrence time (see text). The Poincaré recurrence times are in general much longer than the corresponding stickiness times.

Figure 11

(a) The accelerator mode island and the sticky region around it and (b) normal mode islands. The sticky region around one of them is shown. Both (a) and (b) are for the same nonlinearity parameter $K=6.4$ where the normal mode and the accelerator mode islands coexist.

Figure 12

(a) The diffusion coefficient $D$ as a function of time for the case of $K=6.4$ and a set of initial conditions inside the large chaotic sea far away from the normal or accelerator mode islands. The value of $D$ tends towards a constant value after more than ${10}^7$ iterations. (b) The evolution of $\langle {(y-y_0)}^2\rangle$ as a function of the number of iterations $n$, in a logarithmic scale. The slope of this curve corresponds to the diffusion exponent $\mu$, which has a mean value of $\approx 1.44$ until ${10}^6$ iterations while it converges to the value $\mu =1$, corresponding to normal motion, after ${10}^7$ iterations.

Figure 13

(a) Characteristics of accelerator mode periodic orbits of period 2, i.e., $x$ as a function of $K$ derived from Eq. (11) for $l=-1$, (b) $\overline{y}$ as a function of $K$ derived from Eq. (9) for $l=-1$. Black solid curves correspond to Eq. (11) with the $+$ sign, while (blue) dotted curves correspond to Eq. (11) with the $-$ sign.

Figure 14

(a) Characteristics of accelerator mode periodic orbits of period 2, i.e., $x$ as a function of $K$ derived from Eq. (11) for $l=-2$, (b) $\overline{y}$ as a function of $K$ derived from Eqs. (9) and (11) for $l=-2$. Black solid curves correspond to Eq. (11) with the $+$ sign, while (blue) dotted curves correspond to Eq. (11) with the $-$ sign. (The orbits with constant $\overline{y}=0.5$, i.e., $y=1$, are of period 1.)

Figure 15

Four accelerator double periodic orbits for $K=4.0844$. The two black dots represent a stable periodic orbit with modulo 1 both in $x$ and in $y$ (black squares correspond to the unstable periodic orbit). The two gray (red) dots represent a second stable orbit with $x^{\prime }=1-x$ and $y^{\prime }=1-y$ [gray (red) squares correspond to the unstable periodic orbit].

Figure 16

The stability diagram of the accelerator mode periodic orbits of Fig. 15. The orbits are stable for $-2<\alpha <2$.

Figure 17

Characteristics of accelerator mode periodic orbits of period 4, i.e., $x$ as a function of $K$ derived from Eqs. (13) and (14) for $\mathrm{\Lambda }=-1$.

Figure 18

(a) The diffusion coefficient $D\left(K\right)$ of accelerator modes for values of the nonlinearity parameter $2.9<K<3.2$. There are intervals of $K$ in which the diffusion coefficient $D$ increases with the number of iterations $n$ and small intervals that seem to stay constant with the number of iterations (but they still correspond to anomalous diffusion, see text). The black curve is derived from a grid of $N={10}^5$ initial conditions, while the gray (red) curve is derived from a grid of $N={10}^6$ initial conditions. (b) The time evolution of the global diffusion exponent $\mu$ as a function of the number of iterations $n$ for two nearby values of $K$. For $K=2.982$, the diffusion exponent is $\mu =2$ denoting anomalous diffusion, but for $K=2.995$ it has a mean value around $\mu =1$ denoting normal diffusion. However, this result is not true for the case of $K=2.995$ and it is due only to the small number of initial conditions taken in the whole phase space that do not populate the small accelerator mode islands of stability (see text). (c) Accelerator mode periodic orbits of period 4 [black, initially stable, and gray (red), unstable] and period 8 (blue) existing in the same interval as in (a). (d) The stability index $\alpha$ of the accelerator mode periodic orbits of (c).

Figure 19

(a) For $K=2.987$ the accelerator mode unstable periodic orbit of period 4 is inside the last $KAM$ curve (black curve) although this orbit is unstable. Its asymptotic curves [gray (red)] surround two points representing a bifurcated periodic orbit of period 8. (b) For $K=2.995$ the period-4 orbit is unstable, but there is a stable periodic orbit of period 8 that has bifurcated from the period-4 orbit, which is surrounded by an island (one of the eight islands). (c) The accelerator mode stable periodic orbit of period 4 (one of four islands of stability) for $K=3.305$. In all three cases, the area of the periodic orbits corresponds to anomalous diffusion and gives a global diffusion exponent $\mu =2$.