Universal instabilities of radio-frequency traps

Using standard tools of nonlinear dynamics we analyze recently discovered instabilities of radio-frequency charged-particle traps. In the cw-driven cylindrical Kingdon trap the instabilities occur at the two values ${\ensuremath{\eta}}_{3}^{*}=3.613 046 7\dots{}$ and ${\ensuremath{\eta}}_{4}^{*}=4.431 124 4\dots{}$ of the trap's control parameter $\ensuremath{\eta}$. Analytical estimates based on the theory of Mathieu functions predict ${\ensuremath{\eta}}_{3}^{*}=\ensuremath{\pi}\sqrt{(363\ensuremath{-}32{\ensuremath{\pi}}^{2})∕(66\ensuremath{\pi}\sqrt{6}\ensuremath{-}48{\ensuremath{\pi}}^{2})}=3.692 392 2\dots{}$ and ${\ensuremath{\eta}}_{4}^{*}=(\sqrt{\ensuremath{\pi}}∕2){[(363\ensuremath{-}32{\ensuremath{\pi}}^{2})∕(\sqrt{1089+48{\ensuremath{\pi}}^{2}}\ensuremath{-}12\ensuremath{\pi})]}^{1∕2}=4.496 546 6\dots{}$. The kicked Kingdon trap, an analytically solvable model, predicts ${\ensuremath{\eta}}_{3}^{*}=\frac{1}{3}\sqrt{105}=3.415 650 2\dots{}$ and ${\ensuremath{\eta}}_{4}^{*}=\sqrt{17}=4.123 105 6\dots{}$. We show that similar instabilities occur in the two-particle Paul trap and the cw-driven spherical Kingdon trap.