On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator

Abstract

Evolution in time of an arbitrary initial state for a parametrically driven quantum oscillator is an interesting problem since there exist regions in parameter space (defined by the amplitude and frequency of the driving) where the moments of the probability distribution can diverge in time. While the first moment satisfies a Mathieu equation, the higher-order moments follow Mathieu like equations of order greater than two. It is not very often that a physical problem gives rise to higher-order Mathieu equations. Hence, we give a detailed study of the different stability zones associated with the parametric quantum oscillator, using perturbative techniques traditionally associated with the Mathieu equation. We verify our results by numerical analysis, thus demonstrating that for the higher-order Mathieu equations, the traditional perturbation theory methods give a consistent account of the stability zones.

Numerical investigation of parametric resonance due to hydrodynamic coupling in a realistic wave energy converter

“…On the one hand, a large makes spar buoys unresponsive in roll/pitch, hence ideal for both classic offshore applications and wave energy converters operating in heave. On the other hand, since is so large, typically falls in the range of operational wave periods, generating conditions for roll/pitch parametric resonance to settle [ 4 ]. Several experimental studies have confirmed the appearance of parametric roll in container ships [ 35 ], spar platforms [ 29 ] and wave energy converters (WECs), either spar buoys [ 7 , 23 ] or self-reacting [ 5 , 30 , 39 ].…”

Instability Zones in the Dynamics of a Quantum Mechanical Quasiperiodic Parametric Oscillator

“…The dynamics of the variance as shown in Eq. (2.13) has been investigated in [4]- [5] for µ = 0 (periodic modulation). We note that from Eqs.…”

“…It is around the periodic responses that the instability zones exist. It was shown in Ref [5] from harmonic balance arguments that there is no instability zone around T = 3π ω and so in Sec III we discuss the case around T = 2π ω , i.e. twice the natural frequency.…”

“…While references [4] and [5] have dealt with the variance, it should be noted that reference [6] has dealt with a more physical quantity which is the energy expectation value and shown that the region of divergence of mean position and the average energy are the same. We show the connection between the variance and the average energy in Sec II.…”

Coupled Mathieu Equations:γ-Hamiltonian andμ-Symplectic

“…Most investigations in literature deal with the corresponding stability transition curves [3]. Some works analyze the stability of two coupled Mathieu equations [4][5][6]. In general, an asymptotic or a numerical analysis method is required for analyzing this class of systems.…”