Scaling investigation of Fermi acceleration on a dissipative bouncer model

The phenomenon of Fermi acceleration is addressed for the problem of a classical and dissipative bouncer model, using a scaling description. The dynamics of the model, in both the complete and simplified versions, is obtained by use of a two-dimensional nonlinear mapping. The dissipation is introduced using a restitution coefficient on the periodically moving wall. Using scaling arguments, we describe the behavior of the average chaotic velocities on the model both as a function of the number of collisions with the moving wall and as a function of the time. We consider variations of the two control parameters; therefore critical exponents are obtained. We show that the formalism can be used to describe the occurrence of a transition from limited to unlimited energy growth as the restitution coefficient approaches unity. The formalism can be used to characterize the same transition in two-dimensional time-varying billiard problems.

Figure 1

Chaotic orbits for a full version of the dissipative bouncer model. The control parameters used in the construction of the figures were (a) $\alpha =0.99$ and $ϵ=10$; (b) $\alpha =0.999$ and $ϵ=10$; (c) $\alpha =0.99$ and $ϵ=100$; (d) $\alpha =0.999$ and $ϵ=1000$.

Figure 2

(Color online) Convergence for the positive Lyapunov exponent. The control parameters used were (a) $\alpha =0.99$ and $ϵ=10$; (b) $\alpha =0.999$ and $ϵ=10$; (c) $\alpha =0.999$ and $ϵ=100$; (d) $\alpha =0.999$ and $ϵ=1000$.

Figure 3

(Color online) Plot of the averaged positive Lyapunov exponent as a function of (a) $(1-\alpha )$ and (b) $ϵ$.

Figure 4

(Color online) (a) Plot of $\omega$ as a function of $n$. (b) The same plot shown in (a) after the transformation $n\to n{ϵ}^2$. The control parameters used in the construction of these curves are shown in the figure.

Figure 5

(Color online) Behavior of (a) ${\omega }_{\mathrm{sat}}$ vs $(1-\alpha )$, (b) $n_x$ vs $(1-\alpha )$, (c) ${\omega }_{\mathrm{sat}}$ vs $ϵ$, and (d) $n_x{ϵ}^2$ vs $ϵ$.

Figure 6

(Color online) Universal plot for different curves of $\omega$.

Figure 7

(Color online) Behavior of $\overline{\lambda }$ vs $ϵ$ and $\overline{\lambda }$ vs $(1-\alpha )$.

Figure 8

Frequency histograms, for the variable velocity, of chaotic attractors for (a) simplified and (b) complete versions of the model. The control parameters used were $\alpha =0.999$ and $ϵ=10$.

Figure 9

(Color online) (a) Frequency histograms, for the variable velocity, regarding chaotic attractors for the complete version of the model. (b) Their collapse onto a single and universal curve.

Figure 10

(Color online) Behavior of $V_p$ as a function of the control parameters.

Figure 11

(Color online) Log-log plot of $\omega$ as a function of $t{\epsilon }^2$ for different values of ${\alpha }_r$ and $\epsilon$.

Figure 12

(Color online) (a) Saturation values of deviation of the velocity as function of $\epsilon$. A power law fit gives ${\gamma }_1=0.99\left(1\right)$. (b) shows the least-squares fit to the numerical values of ${\omega }_{\mathrm{sat}}$ as a function of the parameter ${\alpha }_r$. The procedure furnishes ${\gamma }_2=-0.510\left(1\right)$. (c) and (d) show, respectively, the crossover value $t_z{\epsilon }^2$ as a function of $\epsilon$ and ${\alpha }_r$. The corresponding fits furnish $z_1=2.98\left(2\right)$ and $z_2=-1.49\left(1\right)$.

Figure 13

(Color online) Collapse of the $\omega$ curves into a single and universal curve.

Figure 14

(Color online) Average collision number curves as functions of $t{\epsilon }^2$ for different values of $\epsilon$ and ${\alpha }_r$.

Figure 15

(Color online) (a) A least-squares fit to the numerical data in an $N∕{\left(t{\epsilon }^2\right)}^{{\xi }_1}$ vs $\epsilon$ plot furnishes ${\eta }_1=-2.12\pm 0.01$. (b) The best fit to the data in an $N∕{\left(t{\epsilon }^2\right)}^{{\xi }_2}$ vs $\epsilon$ plot furnishes ${\eta }_2=-2.97\pm 0.01$. (c) The best fit to the numerical data in an $N∕{\left(t{\epsilon }^2\right)}^{{\xi }_2}$ vs ${\alpha }_r$ plot furnishes $\zeta =0.492\pm 0.005$.

Figure 16

(Color online) After appropriate variable transformations the average collision number collapses into a universal curve.