NONLINEAR INVARIANT MANIFOLD AND HOMOCLINIC ORBITS OF A TWO-MODE MODEL OF BUCKLED BEAM VIA

This work deals with the detection of homoclinic orbits of systems having a large number of degrees of freedom (d.o.f.). Homoclinic orbits are particular solutions of the governing equations connecting, in an infinite time, a given unstable equilibrium point. In spite of the fact that they are special and isolated solutions, they play a fundamental role in the organization of the global dynamics of complex systems, as repeatedly verified in practice [1]. In the past, these solutions have been studied for systems with few (usually one) mechanical d.o.f.. However, real mechanical structures have a large number of d.o.f., or they are continuous, i.e. they have infinite d.o.f., and this calls for a more refined analysis. Due to the special and isolated character of these solutions, their detection is not easy, not even when numerical methods are employed. Even more difficult is their analytical detection, which is the object of this work, although in some cases the conservativeness of the system provides some help. Thus, we consider the four dimensional Hamiltonian system x& & +∂V(x,y)/∂x=0, y& & +∂V(x,y)/∂y=0, V(x,y)=–(1/2)x2+(1/2)y2+(c1x2+εxy+c2y2)2 (1