Detecting Stable-Unstable Nonlinear Invariant Manifold and Homoclinic Orbits in Mechanical Systems

We consider a four dimensional Hamiltonian system representing the reduced-order (two-mode) dynamics of a buckled beam. The system has a saddle-center equilibrium point, and we pay attention to the existence and detection of the stable-unstable nonlinear manifold and of homoclinic solutions, which are the sources of complex and chaotic dynamics observed in the system response. The system has also a coupling nonlinear parameter, which depends on the boundary conditions, and is zero, e.g., for the hinged-hinged beam and different from zero, e.g., for the fixed-fixed beam. The invariant manifold in the latter case is detected assuming that it can be represented as a graph over the plane spanned by the unstable (principal) variable and its velocity. We show by a series solution that the manifold exists but has a limited extension, not sufficient for the deployment of the homoclinic orbit. Thus, the homoclinic orbit is addressed directly, irrespective of its belonging to the invariant manifold. By means of the perturbation method it is shown that it exists only on some curves of the governing parameters space, which branch from a fundamental path. This shows that the homoclinic orbit is not generic. These results have been confirmed by numerical simulations and by a different analytical technique.