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

摘要

We consider a four-dimensional Hamiltonian system representing the reduced-order (two-mode) 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.