On the existence of chaos in a class of two-degree-of-freedom, damped, strongly parametrically forced mechanical systems with brokenO(2) symmetry

Abstract

In this paper we study some aspects of the global dynamics associated with a normal form that arises in the study of a class of two-degree-of-freedom, damped, parametrically forced mechanical systems. In our analysis the amplitude of the forcing is an ϕ(1) quantity, hence of the same order as the nonlinearity. The normal form is relevant to the study of modal interactions in parametrically excited surface waves in nearly square tanks, parametrically excited, nearly square plates, and parametrically excited beams with nearly square cross sections. These geometrical constraints result in a normal form with brokenO(2) symmetry and the two interacting modes have nearly equal frequencies. Our main result is a method for determining the parameter values for which a “Silnikov type” homoclinic orbit exists. Such a homoclinic orbit gives rise to a well-described type of chaos. In this problem chaos arises as a result of a balance between symmetry breaking and dissipative terms in the normal form. We use a new global perturbation technique developed by Kovacic and Wiggins that is a combination of higher dimensional Melnikov methods and geometrical singular perturbation methods.