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Homothety and Quasi Self Similarity in Asymptotically Flat and AdS Spacetimes

For a spherically symmetric self-gravitating scalar field we study self similar and quasi-self similar solutions in asymptotically flat and AdS spacetimes in various dimensions. Our main approach relies on reducing the Einstein-Klein-Gordon system of equations under the self similarity assumption to a dynamical system. Applying standard techniques in the analysis of dynamical system we study the corresponding phase diagrams, gaining some insight into the short and long-term evolution of the system. In the asymptotically flat case the fixed points of the dynamical system correspond to conformally flat spacetimes. We observed a new transition from a fixed point to a black hole and we compute the corresponding exponent analytically. In the case of AdS spacetime, we observed the quasi self similar scalar field scattering to infinity after interfering near the center corresponding to the dynamical system with no fixed points.