Inariant manifolds and chaotic vibrations in singularly perturbed nonlinear oscillators

Abstract This work concerns the forced nonlinear vibrations of a dissipative soft–stiff structural dynamical system consisting of a soft nonlinear oscillator coupled to a linear stiff oscillator. The equations of motion are cast into a set of singularly perturbed ordinary differential equations, with the ratio of linear frequency like quantities as the singular parameter. Then, using the theory of invariant manifolds, it is shown that, for sufficiently small coupling, the forced system possesses a 3-dimensional slow invariant manifold. The invariant manifold is a regular perturbation of a global invariant manifold for the conservative system. It is shown that the conservative system possesses a homoclinic orbit on the slow invariant manifold. Numerical simulations reveal that the forced system undergoes a period doubling cascade of bifurcations. The cascade of bifurcations gives rise to a weak strange attractor which undergoes a metamorphosis into a strong strange attractor as the forcing amplitude increases. Using Melnikov's method, it is shown that the strong strange attractor stems from transverse intersections of the invariant manifolds of a saddle-type periodic motion carried by the slow invariant manifold.