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Open AccessArticle

Resonance Enhancement by Suitably Chosen Frequency Detuning

by Denys Dutykh 1,2,† and Elena Tobisch 3,*,†
1
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
2
LAMA UMR 5127, CNRS, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le Bourget-du-Lac CEDEX, France
3
Institute of Analysis, Johannes Kepler University, 4040 Linz, Austria
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(3), 450; https://doi.org/10.3390/math8030450
Received: 21 February 2020 / Revised: 12 March 2020 / Accepted: 13 March 2020 / Published: 19 March 2020
(This article belongs to the Special Issue Turbulence Modeling)
The theory of exact resonances (kinematics and dynamics) is well developed while even the very concept of detuned resonance is ambiguous and only studies of their kinematic characteristics (that is, those not depending on time) are available in the literature. In this paper, we report novel effects enforced by the resonance detuning on solutions of the dynamical system describing interactions of three spherical planetary waves. We establish that the energy variation range can significantly exceed the range of the exact resonance for suitably chosen values of the detuning. The asymmetry of system’s solutions with respect to the sign of the detuning parameter is demonstrated. Finally, a non-monotonic dependence of the energy oscillation period with respect to detuning magnitude is discovered. These results have direct implications in physics of atmosphere, e.g., for prediction of weather extremes in the Northern Hemisphere midlatitudes (Proc. Nat. Acad. Sci. USA 2016, 133(25), 6862–6867). Moreover, similar study can be conducted for a generic three-wave system taken in the Hamiltonian form which makes our results applicable for an arbitrary Hamiltonian three-wave system met in climate prediction theory, geophysical fluid dynamics, plasma physics, etc. View Full-Text
Keywords: nonlinear resonance; frequency detuning; three-wave system; resonance enhancement nonlinear resonance; frequency detuning; three-wave system; resonance enhancement
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Dutykh, D.; Tobisch, E. Resonance Enhancement by Suitably Chosen Frequency Detuning. Mathematics 2020, 8, 450.

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