Quasi-normal forms in the problem of vibrations of pedestrian bridges

From a discrete model describing the vibrations of a pedestrian bridge, a transition is made to a system of nonlinear integro-differential equations that continuously depend on time and space variables. Critical cases are singled out in the problem of stationary state stability. The local dynamics of the resulting model is investigated using the formalism of the method of normal forms. As a consequence of the infinite-dimensionality of the critical cases, it is shown that the role of the normal form is played by a special evolutionary boundary value problem. Families of simple stepwise time-periodic solutions of this boundary value problem are constructed.