Detecting stable–unstable nonlinear invariant manifold and homoclinic orbits in mechanical systems

Abstract

We consider a four-dimensional Hamiltonian system representing the reduced-order (twomode) dynamics of a buckled beam, where the nonlinearity comes from the axial deformation in moderate displacements, according to classical theories. 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 beam hinged-hinged ends and different from zero, e.g., for the beam fixed-fixed ends.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.