An instability result in the theory of suspension bridges

We consider a second order system of two ODEs which arises as a single mode Galerkin projection of the so-called fish-bone (Berchio and Gazzola, 2015) model of suspension bridges. The two unknowns represent flexural and torsional modes of vibration of the deck of the bridge. The elastic response of the cables is supposed to be asymptotically linear under traction, and asymptotically constant when compressed (a generalization of the slackening regime). We establish a condition depending on a set of 3 parameters under which the flexural motions are unstable provided the energy is sufficiently large.