Approximate Asymptotics for a Nonlinear Mathieu Equation Using Harmonic Balance Based Averaging

Abstract

Weakly nonlinear versions of the Mathieu equation, relevant among other things to Paul trap mass spectrometers, are studied in the neighborhood of parameter values where the unperturbed solution is periodic, but where the unperturbed (or linear) Mathieu equation is not solvable in closed form using elementary functions. At these parameter values the method of averaging is considered applicable in principle but not in practice, due to the impossibility of, e.g., evaluating certain integrals in closed form. However, on approximately carrying out the averaging calculation using harmonic balance, approximate and simple slow flows can be obtained. Comparisons with numerically obtained Poincare sections show that these ‘approximate’ slow flows are quite accurate (though not asymptotically so). These slow flows provide useful insights into the dynamics near these resonances. Such simple descriptions were not available before.