Fermi-Ulam Accelerator Model under Scaling Analysis

The chaotic low energy region of the Fermi-Ulam simplified accelerator model is characterized by the use of scaling analysis. It is shown that the average velocity and the roughness (variance of the average velocity) obey scaling functions with the same characteristic exponents. The formalism is widely applicable, including to billiards and to other chaotic systems.

Figure 1

(a) Behavior of the roughness $\omega$ as a function of the iteration number $n$. (b) Behavior of $\omega$ as a function of $n{\epsilon }^2$. Both curves were derived from an ensemble average of 50 000 different initial conditions starting with $V_0\approx 0$.

Figure 2

(a) Plot of ${\omega }_{\mathrm{s}\mathrm{a}\mathrm{t}}$ against the control parameter $\epsilon$. (b) The crossover iteration number $n_x$ as a function of $\epsilon$.

Figure 3

(a) Roughness evolution for different values of the control parameter $\epsilon$. (b) Collapse of the curves from (a) onto a universal curve. Both (a) and (b) were obtained using $V_0\approx 0$.

Figure 4

Behavior of the average velocity $V$ as a function of $n$ for different values of $\epsilon$ and $V_0$. (a) The original time series. (b) Collapse of the data onto a universal curve.