INVARIANT-MANIFOLDS AND CHAOTIC VIBRATIONS IN SINGULARLY PERTURBED NONLINEAR OSCILLATORS

This work concerns the forced nonlinear vibrations of a dissipative so ft-stiff structural dynamical system consisting of a soft nonlinear os cillator coupled to a linear stiff oscillator. The equations of motion are cast into a set of singularly perturbed ordinary differential equ ations, with the ratio of linear frequency like quantities as the sing ular parameter. Then, using the theory of invariant manifolds, it is s hown that, for sufficiently small coupling, the forced system possesse s a 3-dimensional slow invariant manifold. The invariant manifold is a regular perturbation of a global invariant manifold for the conservat ive system. It is shown that the conservative system possesses a homoc linic orbit on the slow invariant manifold. Numerical simulations reve al that the forced system undergoes a period doubling cascade of bifur cations. The cascade of bifurcations gives rise to a weak strange attr actor which. undergoes a metamorphosis into a strong strange attractor as the forcing amplitude increases, Using Melnikov's method, it is sh own that the strong strange attractor stems from transverse intersecti ons of the invariant manifolds of a saddle-type periodic motion carrie d by the slow invariant manifold. (C) 1998 Elsevier Science Ltd. All r ights reserved.