Quasi-periodically forced nonlinear Helmholtz oscillators

Abstract

In this paper we study a model that describes the dynamics of the tidal elevation within an almost-enclosed short basin that is connected to a tidal sea by a narrow strait. This model has the form of a forced nonlinear Helmholtz oscillator. The forcing is prescribed by the tide at sea and has a quasi-periodic character of a special nature, since the difference between the frequencies of the lunar and solar components of the forcing tide are very small. We focus on the interactions between the nonlinearity in the oscillator caused by the geometry of the basin, and the external forcing tide. The behavior of small amplitude solutions of the weakly (and quasi-periodically) forced Helmholtz oscillator is studied by a combination of the averaging method and the Melnikov method. We construct three different Melnikov functions. These Melnikov functions enable us to distinguish between six structurally different types of chaotic behavior.