Phase control of escapes in the fractional damped Helmholtz oscillator

We analyze the nonlinear Helmholtz oscillator in the presence of fractional damping, a characteristic feature in several physical situations. In our specific scenario, as well as in the non-fractional case, for large enough excitation amplitudes, all initial conditions are escaping from the potential well. To address this, we incorporate the phase control technique into a parametric term, a feature commonly encountered in real-world situations. In the non-fractional case it has been shown that, a phase difference of {\phi_{OPT}} \simeq {\pi}, is the optimal value to avoid the escapes of the particles from the potential well. Here, our investigation focuses on understanding when particles escape, considering both the phase difference {\phi} and the fractional parameter {\alpha} as control parameters. Our findings unveil the robustness of phase control, as evidenced by the consistent oscillation of the optimal {\phi} value around its non-fractional counterpart when varying the fractional parameter. Additionally, our results underscore the pivotal role of the fractional parameter in governing the proportion of bounded particles, even when utilizing the optimal phase.