Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation |
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Authors: | Randolph S Zounes Richard H Rand |
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Institution: | (1) Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, U.S.A;(2) Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14853, U.S.A |
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Abstract: | In this paper, we investigate the interaction of subharmonicresonances in the nonlinear quasiperiodic Mathieu equation,x + + (cos 1 t + cos 2 t)] x + x3 = 0.We assume that 1 and that the coefficient of the nonlinearterm, , is positive but not necessarily small.We utilize Lie transform perturbation theory with elliptic functions –rather than the usual trigonometric functions – to study subharmonic resonances associated with orbits in 2m:1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on ( , , 1, 2) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov's overlap criterion to obtain an overview of the O( ) global behavior of equation (1) as a function of and 2 with 1, , and fixed. |
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Keywords: | nonlinear quasiperiodic Mathieu equation elliptic functions Lie transforms resonance |
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