Ion traps are devices used to contain charged particles, and for highly selective mass spectroscopy. For a nonlinear trap subjected to secondary end-cap excitation and including a buffer gas, the equations of motion are coupled, damped, nonlinear, quasiperiodic Mathieu equations. The attractors for this system are non-trivially "knotted" in the 4-D phase space. The investigator and his colleagues have developed tools to characterize them topologically and geometrically, and have identified new bifurcations associated with their changes of type. Important outstanding problems include: how topological equivalence of attractors through torus braids and through surfaces is related; whether chaos is associated with transverse intersections of stable and unstable manifolds of saddle-type knotted tori. A sequence of the newly identified bifurcations leads to chaotic dynamics through a route observed, but not yet characterized. In some cases, two bifurcations bound extremely sharp resonances which are useful in applications.
The new knowledge being developed about this system can be used to suggest new operating regimes for ion traps used in discrimination of species in mass spectroscopy. The sharp resonances already observed suggest the use of an ion trap as a tunable electromechanical filter. Two similar ion trap systems may be used for the purpose of masking and unmasking signals for secure communication. Finally, the results suggest that a trapped charged particle or ion can be used as an accelerometer and gyroscope. These technological innovations or improvements flow directly from a detailed understanding of the underlying mathematics.
This award is co-funded by the Applied Mathematics and Topology programs of the Division of Mathematical Sciences and the Analytical and Surface Chemistry program of the Chemistry Division under the umbrella of the NSF-wide Mathematical Sciences Priority Area.
