Chaotic transients of two particles in a Paul trap: Interpretation as a boundary crisis

Calculations based on coupled Mathieu-Coulomb equations indicate that the transition from transient to stationary chaos for two ions in a Paul trap near the edge of the stability region is due to a boundary crisis. Numerical simulations reproduce the long-lived chaotic transients observed in ion trap experiments, obeying the power-law dependence T(q)∝(${\mathit{q}}_{\mathit{c}}$-q${)}^{\mathrm{-}\gamma }$ where T is the average transient lifetime and q the dimensionless trap voltage. The unstable, periodic orbits which are fundamental to a heteroclinic boundary crisis were identified and the intersection of their invariant manifolds in the four-dimensional phase space was located, yielding a prediction for ${\mathit{q}}_{\mathit{c}}$, the transition point between transient and stationary chaos, that agrees well with the experimental value. This provides a theoretical understanding of a transition which previously has been a subject of controversy. Finally, a heuristic derivation is given for the critical exponent γ, based on the stability properties of the mediating periodic orbits. Thus solutions of the deterministic, time-dependent equations of motion can be used to accurately describe the duration of transient two-ion chaos near criticality, with only a single free scale factor.