  It is well-known that nonlinearity may lead to localization effects and
coupling of internally resonant modes. However, research focused primarily on
conservative systems commonly assumes that the near-resonant forced response
closely follows the autonomous dynamics. Our results for even a simple system
of two coupled oscillators with a cubic spring clearly contradict this common
belief. We demonstrate analytically and numerically global effects of a weak
local damping source in a harmonically forced nonlinear system under condition
of 1:3 internal resonance: The global motion becomes asynchronous, i.e., mode
complexity is introduced with a non-trivial phase difference between the modal
oscillations. In particular, we show that a maximum mode complexity with a
phase difference of $90^\circ$ is attained in a multi-harmonic sense. This
corresponds to a transition from generalized standing to traveling waves in the
system's modal space. We further demonstrate that the localization is crucially
affected by the system's damping. Finally, we propose an extension of the
definition of mode complexity and mode localization to nonlinear quasi-periodic
motions, and illustrate their application to a quasi-periodic regime in the
forced response.
