  The oscillation coupling and different nonlinear effects are observed in a
single trapped $^{40}Ca^{+}$ ion confined in our home-built surface-electrode
trap (SET). The coupling and the nonlinearity are originated from the
high-order multipole potentials due to different layouts and the fabrication
asymmetry of the SET. We solve a complicated Duffing equation with coupled
oscillation terms by the multiple scale method, which fits the experimental
values very well. Our investigation in the SET helps for exploring nonlinearity
using currently available techniques and for suppressing instability of qubits
in quantum information processing with trapped ions.
