We study the dynamics of a two-degree-of-freedom nonlinear system consisting of a linear oscillator with an essentially nonlinear attachment. For the undamped system, we perform a numerical study based on non-smooth temporal transformations to determine its periodic solutions in a frequency-energy plot. It turns out that there is a sequence of periodic solutions bifurcating from the main backbone curve of the plot. We then study analytically the periodic orbits of the undamped system using the complexification/averaging technique in order to determine the frequency contents of the various branches of solutions, and to understand the types of oscillation performed by the system at the different regimes of the motion. The transient responses of the weakly damped system are then examined, and numerical wavelet transforms are used to study the time evolutions of their harmonic components. We show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics, as it causes complicated transitions between modes in the damped transient motion. In addition, there is the possibility of strong passive energy transfer from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions.