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.

We explain in detail the strategy of Buslaev--Perelman--Sulem (BPS): symplectic projection of a trajectory on a solitary manifold andmodulation equations for the projection and time decay for a transversal component using the Poincar\'e normal form and Fermi Golden Rule for the transversal dynamics. We present an extensive list of results onasymptotic stability of stationary orbitsand solitons that rely on the BPS strategy and its generalizations byS. Cuccagna, Y. Martel, F. Merle, T. Mizumachi, K. Nakanishi, I. Rodnianski,W. Schlag,I. M. Sigal, A. Soffer,R. L. Pego,T. P. Tsai,M. I. Weinstein, H. T. Yau, and others. We also mention the results on stability and instability ofself-similar, spherically symmetric solutions and rotating Kerr solutions ofequations of the General Theory of Relativity by T. Harada, C. E. Kenig, H. Maeda,F. Merle, W. Schlag, and others. Moreover, we illustrate the BPS strategyin the simplest modelof a 1D Schrödinger equation coupled to a nonlinear oscillator, giving complete proofs with all details.