Chaotic properties of a time-modulated barrier

Some chaotic properties of a classical particle interacting with a time-modulated barrier are studied. The dynamics of this problem is obtained by use of a two-dimensional nonlinear area-preserving map. The chaotic low energy region is characterized in terms of Lyapunov exponents. The time that the particle stays trapped in the well is such that the distributions of successive reflections, and of the corresponding successive reflection times, obey power laws with the same exponent. Using time series analysis, we show that the chaotic sea exhibits an interesting scaling property over a large range of control parameters. Our results indicate that the particle experiences unlimited energy growth when the barrier behaves randomly.

Figure 1

A sketch of the potential $V(x,t)$ for the periodic case; the zeroes of $x$ and $V$ are in the bottom left-hand corner.

Figure 2

Iteration of different initial conditions for the parameters $M=4.7$, $r=0.5$, $b∕l=0.2$, and $L∕l=1$. In the low energy regime it is easy to see $\mathrm{KAM}$ islands surrounded by a chaotic sea that is limited by one spanning curve. For intermediate energies it is possible to observe a small chaotic region that is enclosed by two different spanning curves. The high energy domain shows many different spanning curves.

Figure 3

Distributions of successive reflection (a) numbers and (b) times. The parameters used here are $M=4.7$, $b∕l=L∕l=1$ and $r=0.5$. The power law fits yield $P_m\propto m^{{\gamma }_m}$ and $P_t\propto t^{{\gamma }_t}$, where ${\gamma }_m=-3.03\left(2\right)$ and ${\gamma }_t=-3.00\left(2\right)$.

Figure 4

Convergence of the Lyapunov exponent in the low energy domain. The parameters used are $M=4.7$, $r=0.5$, $b∕l=0.2$, and $L∕l=1$ and the Lyapunov exponent value is $\overline{\lambda }=1.675\pm 0.003$.

Figure 5

Log linear plot of the Lyapunov exponent $\lambda$ as a function of $M$. The parameters used are $r=0.25$, $r=0.5$, $r=0.75$, $L∕l=1$ and (a) $b∕l=1$, i.e., the symmetrical case; and (b) $b∕l=0.5$, i.e., the asymmetrical case.

Figure 6

Log linear plot of the variation of the Lyapunov exponent $\lambda$ with $r$. The parameters used were $M=4.7$ and $b∕l=L∕l=1$.

Figure 7

(a) Iteration of three different initial conditions that lie, respectively, (i) below, (ii) inside, and (iii) above the first invariant spanning curve. (b) Iteration of only one initial condition. The parameters used were $M=4.7$, $b∕l=L∕l=1$ and (a) $r=0.0075$ and (b) $r=0.0076$.

Figure 8

Normalized distribution of successive reflection energies for parameters $M=4.7$, $r=0.5$, $b∕l=L∕l=1$.

Figure 9

The time evolution for one initial condition located in the chaotic sea for parameters $M=4.7$; $r=0.5$; $b∕l=L∕l=1$.

Figure 10

Roughness behavior for an ensemble of $\mathrm{10\hspace{0.17em}000}$ different initial phases $\varphi$ and the same initial energy $e=1+r$. Both initial conditions give rise to chaotic orbits. The inset shows the procedure used to obtain the roughness in the saturation regime after applying the transformation $N\to 1∕N$.

Figure 11

The dependencies on $M$ of (a) the saturation roughness ${\omega }_{sat}$ and (b) the crossover iteration number $N_x$.

Figure 12

(a) Roughness evolutions for different $M$. (b) Collapse of the curves onto the same saturation value. (c) Their collapse onto both the same saturation value and the same characteristic crossover iteration number.

Figure 13

Average total energy as a function of (a) $n$ and (b) $t$. The parameters used here are $M=4.7$, $r=0.5$, $b∕l=L∕l=1$. Note that $b∕l=L∕l=1$ is the symmetrical case. The power law gives us that $e\propto n^{{\delta }_n}$ and $e\propto t^{{\delta }_t}$, where ${\delta }_n=0.496\pm 0.001$ and ${\delta }_t=0.649\pm 0.001$.