  In this paper we investigate the distribution of energy between weakly
coupled linear and nonlinear oscillators in a two-degree-of-freedom (2D)
system. Two classes of problems are studied analytically and numerically: (1) a
periodic force with constant frequency is applied to the nonlinear (Duffing)
oscillator with slowly time-decreasing linear stiffness; (2) the
time-independent nonlinear oscillator is excited by a force with slowly
increasing frequency. In both cases, stiffness of the attached linear
oscillator and linear coupling remain constant, and the system is initially
engaged in resonance. This paper demonstrates that in the systems of the first
type autoresonance (AR) occurs in both oscillators while in systems of the
second type AR occurs only in the excited nonlinear oscillator but the coupled
linear oscillator exhibits small bounded oscillations. Considering slow
detuning, we obtain explicit asymptotic approximations for the amplitudes and
the phases of oscillations close to exact (numerical) results.
