caa a22 4500 445360402 CHVBK 20180317142913.0 cr unu---uuuuu 170323e20110601xx s 000 0 eng 10.1007/s11071-010-9870-8 doi (NATIONALLICENCE)springer-10.1007/s11071-010-9870-8 A scaling law near the primary resonance ofthequasiperiodic Mathieu equation [Elektronische Daten] [L. Romero, J. Torczynski, A. Kraynik] We analyze the quasiperiodic damped Mathieu equation $$\ddot{x}+ \gamma\dot{x}+ x \bigl( 1 + \delta+ \epsilon q(t) \bigr )=0 ,$$ with $$q(t) = \frac{1}{2} \bigl( (1+ \kappa) \cos(\omega_1 t ) +( 1- \kappa) \cos(\omega_2 t) \bigr),$$ where ω 1=2+α/2 and ω 2=2−α/2. In this equation, κ determines the relative weight of the two frequencies, and α determines how close the two frequencies are to each other. After applying the method of averaging to remove the fastest time scale in the problem, we analyze the averaged equations assuming that the parameter κ is small. That is, the two frequencies have nearly the same amplitude. We find a simple scaling law for ε ⋆=ε/α as a function of the other parameters in the system. The scaling law shows that it gets more difficult to excite an instability (α must be smaller) the smaller κ becomes. Springer Science+Business Media B.V., 2010 Quasiperiodic nationallicence Mathieu equation nationallicence Stability nationallicence Romero L. Computational Mathematics and Algorithms Department, Sandia National Laboratories, MS 1320, P.O. Box 5800, 87185-1320, Albuquerque, NM, USA aut Torczynski J. Microscale Science and Technology Department, Sandia National Laboratories, MS 0346, P.O. Box 5800, 87185-0346, Albuquerque, NM, USA aut Kraynik A. Thermal and Fluid Processes Department, Sandia National Laboratories, MS 0836, P.O. Box 5800, 87185-0836, Albuquerque, NM, USA aut Nonlinear Dynamics Springer Netherlands 64/4(2011-06-01), 395-408 0924-090X 64:4<395 2011 64 11071 https://doi.org/10.1007/s11071-010-9870-8 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11071-010-9870-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Romero L. Computational Mathematics and Algorithms Department, Sandia National Laboratories, MS 1320, P.O. Box 5800, 87185-1320, Albuquerque, NM, USA aut NATIONALLICENCE 700 1- Torczynski J. Microscale Science and Technology Department, Sandia National Laboratories, MS 0346, P.O. Box 5800, 87185-0346, Albuquerque, NM, USA aut NATIONALLICENCE 700 1- Kraynik A. Thermal and Fluid Processes Department, Sandia National Laboratories, MS 0836, P.O. Box 5800, 87185-0836, Albuquerque, NM, USA aut NATIONALLICENCE 773 0- Nonlinear Dynamics Springer Netherlands 64/4(2011-06-01), 395-408 0924-090X 64:4<395 2011 64 11071 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer