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The problem of lateral vibrations of footbridges due to the synchronization of the pedestrians' motion with that of the supporting structure is analyzed by means of a 3D discrete time model. The map is linear in the mechanical part, and nonlinear in the synchronization part. A very simple and predictive formula is obtained for the critical number of pedestrians, which also takes into account the imperfect resonance between the pedestrians' natural frequencies and the bridge frequency. It is shown that the underlying mechanism triggering the sudden appearance of swaying bridge motion is a perturbation of a pitchfork bifurcation. The results presented in this paper are not related to a specific real case and are based on a quite reasonable hypothesis, and therefore it is expected that they have general validity.

The control of chaos (suppression and enhancement) of a damped pendulum subjected to two perpendicular periodic excitations of its pivot (one chaos inducing and the other chaos controlling) is investigated. Analytical (Melnikov analysis) and numerical (Lyapunov exponents) results show that the initial phase difference between the two excitations plays a fundamental role in the control scenario. We demonstrate the effectiveness of the method in suppressing spatiotemporal chaos of chains of identical chaotic coupled pendula where homogeneous regularization is obtained under localized control on a minimal number of pendula. Additionally, we demonstrate the robustness of the control scenario against changes in the coupling function. In particular, synchronization-induced homogeneous regularization of chaotic chains can be highly enhanced by considering time-varying couplings instead of stationary couplings.