Hopf and Double Hopf Bifurcations in a Delayed Lateral Vibration Model of Footbridges Induced by Pedestrians

In this paper, we investigate the dynamical behaviors of a delayed lateral vibration model of footbridges. The model is proposed based on the facts that pedestrians will reduce their walking speeds or stop walking when the response of the footbridge becomes sufficiently large, and that the angular velocity of bridge vibration cannot be changed instantaneously when pedestrians begin to walk on the bridge. By analyzing the distribution of roots of the associated characteristic equation, we prove that there are only two types of bifurcations in this model: Hopf bifurcation and double Hopf bifurcation. We also give the condition on the stability of the trivial solution. By using the center manifold theorem and bifurcation theory of delayed differential equations, we obtain the dynamical behavior in these bifurcations. Specifically, we analyze the stability of periodic solutions and invariant tori bifurcating from the trivial solution. The conditions for these bifurcations and the stability are provided in explicit forms in terms of the model’s parameters, making them convenient to verify. Finally, we prove that this model exhibits quasi-periodic vibrations by Kolmogorov–Arnold–Moser (KAM) theorem, in addition to periodic vibrations. Our results show that the delayed lateral vibration model exhibits abundant periodic and quasi-periodic vibrations, but the corresponding model without the time delay only has periodic vibration, which means that the time delay can cause quasi-periodic vibration.