Abstract
Weakly nonlinear versions of the Mathieu equation, relevant among other thingsto Paul trap mass spectrometers, are studied in the neighborhoodof parameter values where the unperturbed solution is periodic,but where the unperturbed (or linear) Mathieu equation is not solvable in closedform using elementary functions. At these parameter valuesthe method of averaging is consideredapplicable 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 averagingcalculation using harmonic balance, approximate and simpleslow flows can be obtained. Comparisons with numerically obtainedPoincaré sections show that these `approximate' slow flows are quite accurate (though not asymptotically so). These slow flows provideuseful insights into the dynamics near these resonances.Such simple descriptions were not available before.
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Abraham, G.T., Chatterjee, A. Approximate Asymptotics for a Nonlinear Mathieu Equation Using Harmonic Balance Based Averaging. Nonlinear Dynamics 31, 347–365 (2003). https://doi.org/10.1023/A:1023293901305
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DOI: https://doi.org/10.1023/A:1023293901305