Rich Dynamics for a Model Arising in the Study of Suspension Bridges

In this paper, a nonlinear and nonlocal model for suspension bridge-type structures with piers is studied. The model encompasses a coupled dynamics involving longitudinal u(x, t) and torsional θ(x,t) oscillations. Focusing the dynamics on a single specific Fourier component for both the variables, a coupled system of ODEs is obtained. For this latter system, we prove the occurrence of rich and complex dynamics—including infinitely many periodic solutions (harmonic and subharmonic)—for the longitudinal time component, when the torsional one is small. This goal is achieved by applying a rigorous analytical approach, based on the theory of linked twist maps. With respect to previous works on suspension bridge models, one of the main contributions concerns the fact that the nonlinear function f appearing in the original model is transformed, via finite-dimensional projection, into a new effective nonlinearity Γf,λ whose dynamical effects may strikingly differ from those of f. This also depends on the magnitude of the coefficients related to the nonlocal terms.