Dissipative area-preserving one-dimensional Fermi accelerator model

The influence of dissipation on the simplified Fermi-Ulam accelerator model (SFUM) is investigated. The model is described in terms of a two-dimensional nonlinear mapping obtained from differential equations. It is shown that a dissipative SFUM possesses regions of phase space characterized by the property of area preservation.

Figure 1

(Color online) (a) The velocity $V\left(n\right)$ for $ϵ=1\times {10}^{-3}$, $\delta =1\times {10}^{-5}$ for $V_0=1$ and ${\varphi }_0=0$. The inset shows in more detail how $V$ varies with $n$ near to the KAM island of order $i=1$. (b) Details of a trajectory passing near to a KAM island, plotting $V\left(n\right)$ as a function of the phase $\varphi$.

Figure 2

(Color online) Stable regions in the velocity $V$ vs phase $\varphi$ phase space for the dissipative SFUM. The parameters used were $ϵ=1\times {10}^{-3}$ and $\delta =1\times {10}^{-5}$. The period-one stable fixed points are represented by the dots and classified by the label $i$ $(i=1,2,3\dots 10)$, as shown.

Figure 3

(Color online) (a) Behavior of the iteration number ${\overline{n}}_x$ as a function of the ensemble of initial conditions for the low energy region using $\delta =2\times {10}^{-7}$ in the simplified version of the dissipative model. (b) The iteration number for the relaxation transient as a function of the parameter $\delta$. The error bars represent the standard deviation for the ensemble of initial conditions used.