Stochastic modeling of spreading and dissipation in mixed-chaotic systems that are driven quasistatically

We analyze energy spreading for a system that features mixed chaotic phase space, whose control parameters (or slow degrees of freedom) vary quasistatically. For demonstration purpose we consider the restricted three-body problem, where the distance between the two central stars is modulated due to their Kepler motion. If the system featured hard chaos, one would expect diffusive spreading with coefficient that can be estimated using linear-response (Kubo) theory. But for mixed phase space the chaotic sea is multilayered. Consequently, it becomes a challenge to find a robust procedure that translates the sticky dynamics into a stochastic model. We propose a Poincaré-sequencing method that reduces the multidimensional motion into a one-dimensional random walk in impact space. We test the implied relation between stickiness and the rate of spreading.

Figure 1

Poincaré landscape. The upper panel is a standard Poincaré section for Hill's Hamiltonian Eq. (1) with ${\mu }_1={\mu }_2=1/2$, $\varepsilon =0.2$, and $g=25$ at energy $E=-22.2$. The control parameter $\mathsf{R}$ is frozen at the value $\theta =0$. In the middle panel each row is a $p_x=0$ stripe of the Poincaré section at different $E$, while $\mathsf{R}$ is frozen at the value $\theta =0$. In the lower panel the Poincaré stripes are plotted for frozen $\mathsf{R}$ at different values of $\theta$. Initially, $E=-22.2$, and later we follow $E$ adiabatically. The color code is such that chaotic trajectories are red, while quasiregular chaotic trajectories are blue.

Figure 2

Poincaré pulses. Representative piece of the signal $f\left[\mathit{r}\right(t\left)\right]$. The vertical red lines indicate the moments $t_j$ that are selected by the Poincaré section. The signal is regarded as a sequence of rectangular pulses (green line). Each pulse has duration $T_j$ and average height ${\overline{F}}_j=F_j/T_j$. In the figure $\overline{F}$ is scaled vertically ($\times 3$) to improve resolution.

Figure 3

Variation of the energy. The energy for a cloud of chaotic trajectories is plotted as a function of time (upper panel). The spreading is displayed in the lower panel (black line). Here we define $S_E$ as the width of the central region that supports 50% of the distribution. The blue line is a moving average.

Figure 4

Spreading. The dynamics is generated by Eq. (1), with the same parameters as in Fig. 1. The time dependence of $S_Q$ is regarded as a measure for spreading, where $Q$ is defined by Eq. (6). The black line is $S_Q$ for an ensemble of 1-cycle (normal panel) and 10-cycles (log-log panel, base 10) trajectories that were launched in the chaotic sea. In the log-scale panel, the lower and upper dashed lines indicate $\propto t$ and $\propto t^2$ dependence. The red and blue lines are $S_Q$ if the control parameter $\mathsf{R}$ were frozen. The selected values are at $\theta =0$ (red) and $\theta =\pi$ (blue), for which the spreading is relatively fast or slow, respectively. According to the traditional paradigm of quasistatic processes, the black line should be roughly between the red and the blue lines, which is clearly not the case. The red and blue dotted lines (lower data lines in both panels) are $S_Q$ for a signal that is composed of a randomized sequence of the same pulses, i.e., they provide the variance that would be expected if the actual signal did not have phase-space correlations. We conclude that correlations are enhanced due to the time dependence of $\theta \left(t\right)$.

Figure 5

Poincaré sequences of pulses. The rows of each image displays the color-coded sequences ${\overline{F}}_j$ of the length-$T$ trajectories for which the spreading has been calculated in the upper panel of Fig. 4. The three panels are for $\theta =\pi$, and for $\theta =0$, and for the actual $\theta \left(t\right)$ of the Kepler motion. The trajectories in each panel are ordered by the average value of ${\overline{F}}_j$. The red stretches for $\theta =0$ indicate stickiness in red regions that will be identified in Fig. 6. The additional blue stretches in the lower panel indicate excess dwell time in the chaotic sea.

Figure 6

Poincaré mapping of pulses. The upper panel displays the Poincaré section for the $\theta =0$ Hamiltonian. Those are the same chaotic trajectories as in Fig. 1 (the quasi-integrable regions are left empty), but the points are color coded by the values of ${\overline{F}}_j$. The black line indicates the border of the energy surface (the forbidden region is outside). The lower panel shows a chaotic trajectory $\left(x\right(t),y(t\left)\right)$. The color encodes the time. The black stretch indicates motion in the “red” sticky region of the upper panel. It is characterized by an “$\infty$” shaped loops.

Figure 7

Driven system dynamics. In the upper panel a trajectory of the Kepler-driven Hamiltonian is presented using the same section as that of Fig. 6. Gray is used for all the points of the trajectory, while red and blue are used for the correlated stretches of Fig. 5. Parameters are the same as in Fig. 1. The Hamiltonian is time dependent. Accordingly, the cloud of trajectories spreads in energy and can migrate between different regions. This is demonstrated in the lower panel, where we use the same presentation method as in Fig. 1. The native chaotic region, which corresponds to the red areas in Fig. 1, is in black. The evolving cloud of the Kepler-driven system is displayed using blue, red and gray dots. Red indicates sticking in periphery regions, while blue indicates sticking in the native chaotic sea. The nonsticking gray points expand into “swamp” regions that are located outside of the native chaotic sea. See text for further details.

Figure 8

Transition probability matrix. The matrix element $P_{n,m}$ is the probability to make a transition form bin $m$ to bin $n$. The images of the matrix are for the frozen $\theta =0$ dynamics and for the Kepler-driven dynamics. In both cases one observes high probability to stay in the red region (last bin). In the Kepler-driven case there is also enhance probability to stay in the blue region (first bin).

Figure 9

Survival probability $\mathcal{P}\left(\tau \right)$. The probability to find a pulse that has at least $\tau$ consecutive pulses at the same region. The red and blue refer respectively to survival in the red and in the blue regions. The $+$ symbols are for the time-independent $\theta =0$ Hamiltonian, and the $*$ symbols are for the Kepler-driven time-dependent Hamiltonian. The solid lines are the analytic results based on the minimal stochastic model (the thicker lines are for the Kepler-driven Hamiltonian). The dashed lines illustrate the naive exponential decay for the time-independent $\theta =0$ Hamiltonian.

Figure 10

Stochastic modeling of spreading. The solid lines are $\text{Var}\left(Q\right)$ for the $F_j$ digitized sequences. See Appendix pp4 for technical details regarding the “digitization.” Note that the “time” ($\tau )$ is the number of Poincaré steps. The red lines are for the $\theta =0$ Hamiltonian, and the black lines are for the Kepler-driven Hamiltonian. The lower dashed lines are the prediction of the effective stochastic model. The dotted lines are for a randomized $F_j$ sequence, namely the expected result if the pulses were uncorrelated. The lower and upper dashed lines indicate $\propto t$ and $\propto t^2$ dependence (the former cannot be resolved from the dotted lines).

Figure 11

The billiard paradigm. This caricature clarifies the mechanism of dissipation in the quasistatic limit. The control parameters are $\mathit{R}=(X,W)$, where $X$ is the position of the piston and $W$ is the height of a dividing barrier. Consider the “blue” quasistatic cycle: The barrier is turned on ($W=\infty$); the piston is pushed in; the barrier is turned off ($W=0$); and the piston is pushed out. Such cycle, unlike the “red” modulation, raises the average energy of the system ($E\mapsto \alpha E$ with $\alpha >1$). Note that the blue and red $E\left(t\right)$ plots reflect the average over an ensemble of trajectories.

Figure 12

Cyclic driving. An example for a generic driving cycle in $(\mathsf{R},\mathsf{K})$ is illustrated by the blue line, where $\mathsf{R}\left(\theta \right)\propto {[1+\varepsilon cos\left(\theta \right)+{\varepsilon }^{\prime }sin\left(2\theta \right)]}^{-1}$ and $\mathsf{K}$ is determined by Eq. (2), with $\varepsilon =0.2$ and ${\varepsilon }^{\prime }=0.1$. For Kepler driving we set ${\varepsilon }^{\prime }=0$ and get the black line, which is a single-parameter modulation.

Figure 13

Directionality dependence. The probability distribution of the $F_j$ over the bins is uniform by definition. It can be regarded as the sum of “forward” pulses distribution and “backward” pulses distribution. Here we plot the deviation of the cumulative probability distribution of the backward pulses from a uniform distribution. The horizontal axis is the indexed $F$ value (the smallest value is indexed as 1 and the largest value is indexed as $38516$). We see that the small values (blue pulses) become slightly more frequent, as opposed to the large values (red pulses) that become slightly less frequent.