In this chapter we present in detail the first resultsonglobal attractionto stationary orbits obtained for a 1D Klein--Gordon equationcoupled to a nonlinear oscillator. The proofs rely on the concept of the omega-limit trajectory and a nonlinear analog of the Kato theorem on the absence of embedded eigenvalues, and on new theory of multipliers in the space of quasimeasures and a novel application of the Titchmarsh convolution theorem. Besides the formal proof, we give an informal explanation of the nonlinear radiative mechanism, which causes theglobal attraction: nonlinear energy transfer from lower to higher harmonics and subsequent dispersive radiation of energy to infinity. In conclusion, we specifythe general conjecture on global attractors, which summarizes all results obtained thus far.