Abstract
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.
Highlights
Numerous natural phenomena exhibit linear and nonlinear resonances
The goal is to approach the state of exact resonance, by reducing resonance detuning, in order to increase the efficiency of a process or device
In this paper we study for the first time the effects of frequency detuning in Equation (1) by means of the numerical simulation with corresponding dynamical system
Summary
Numerous natural phenomena exhibit linear and nonlinear resonances. In many technical cases occurrence of resonance must to be avoided, the widely known Tacoma Bridge dramatic collapse being an example for this. In mathematical definition frequencies ω j are variables, not functions, and the properties of corresponding dynamical systems are characterized by the ratios of frequencies This difference is very important and, in particular, shows that exact mathematical results available in this area cannot be directly used in solving a physical problem. The waves are said to interact resonantly (that is, to form an exact resonance) if resonant conditions given in Equations (1) and (2). The detuned three-wave system has the maximal range of the energy amplitudes variation when the elastic pendulum interacts resonantly with the external forcing. In this paper we study for the first time the effects of frequency detuning in Equation (1) by means of the numerical simulation with corresponding dynamical system.
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