Two-ion chaos

We review the evidence that the order-chaos transition of two ions in a Paul trap near the edge of the stability region is due to boundary crisis. A crucial element is that the transition is from stationary to transient chaos. Near the crisis the average lifetime T of chaotic transients scales with the dimensionless trap voltage q as T(q) propto (q_c-q)^-γ. The unstable, periodic orbits fundamental to a heteroclinic boundary crisis are identified and the intersection of their invariant manifolds in the four-dimensional phase space is located, yielding a prediction for q_c, the transition point between transient and stationary chaos, that agrees well with experiment. This provides a theoretical understanding of an order-chaos transition which previously has been a subject of controversy. With one additional assumption, the critical exponent γ can be calculated as well, yielding a completely deterministic description of the transient lifetime near criticality.