Analysis of double-resonance crossing in adiabatic trapping phenomena for quasi-integrable area-preserving maps with time-dependent exciters

Abstract

In this paper, we analyze the adiabatic crossing of a resonance for Hamiltonian systems when a double-resonance condition is satisfied by the linear frequency at an elliptic fixed point. We discuss in detail the phase-space structure on a class of Hamiltonians and area-preserving maps with an elliptic fixed point in the presence of a time-dependent exciter. Various regimes have been identified and carefully studied. This study extends results obtained recently for the trapping and transport phenomena for periodically perturbed Hamiltonian systems, and it could have relevant applications in the adiabatic beam splitting in accelerator physics.

Figures (12)

• Figure 1: Generic phase-space portrait divided into three regions ($\mathrm{I}$, $\mathrm{II}$, $\mathrm{III}$) by the separatrices $\ell_1(\lambda)$ and $\ell_2(\lambda)$.
• Figure 2: Phase-space portraits of the Poincaré map of Eq. \ref{['eq:doubleresmap']} sampled every $4$ iterations. We use three values of $\delta/\varepsilon^{2/3}$ that account for three possible resonance topologies (the other parameters values are: $\omega_\mathrm{r}/(2\pi)=1/4$, $\Delta=0$, $\varepsilon=10^{-4}$, $\kappa=0.1$, $\psi_0=0$).
• Figure 3: Separatrices (black lines) of the Poincaré map of Eq. \ref{['eq:doubleresmap']} sampled every $4$ iterations. The four islands and the core have been filled with the colors used in Figs. \ref{['fig:area_map_eps']}, \ref{['fig:areadiff']}, \ref{['fig:plot_distr_map']} and \ref{['fig:plot_eps_map']} to refer to each region. The naming convention of the various regions is also reported here (parameters values: $\omega_\mathrm{r}/(2\pi)=1/4$, $\Delta=0$, $\varepsilon=10^{-4}$, $\kappa=0.1$, $\psi_0=0$ and $\delta/\varepsilon^{2/3}=1$).
• Figure 4: Left: Plot of the area $A_i$ of each island and of the core region in the $4$th iterate of the map of Eq. \ref{['eq:doubleresmap']} as a function of $\delta/\varepsilon^{2/3}$. Note that the lines for the North and South islands are almost perfectly overlying. The slope quoted is the angular coefficient of the linear fit of each line performed in the interval $1\le \delta/\varepsilon^{2/3} \le 4$. Right: Ratio between the area of the main island (East) and the sum of all islands using the data of the left plot (parameters values: $\omega_\mathrm{r}/(2 \pi)=1/4$, $\Delta=0$, $\kappa=0.1$, $\psi_0=0$).
• Figure 5: Phase space portraits of the Poincaré map of Eq. \ref{['eq:doubleresmap']} sampled every $4$ iterations for three values of the initial phase $\psi_0$. The phase of exciter selects the island that becomes larger than the others (the parameters values are $\omega_\mathrm{r}/(2 \pi)=1/4$, $\Delta=0$, $\varepsilon=10^{-4}$, $\kappa=0.1$, $\delta/\varepsilon^{2/3}=0.75$).