We employed the laboratory (Laser-Produced Plasmas, LPP) and numerical (3D/PIC-code) simulations to study the resulting state of very strong compression of magnetopause (MP) by CME with effective ...energy E sub(0) greater than or equal to 10 super(34) ergs directed to the Earth. During probable formation of an Artificial Magnetosphere (AM, in a flow of CME' plasma around the Earth) with the MP stand-off at R sub( mp ) up to (2-3)R sub( E ), many catastrophic phenomena could occur in a space and ground networks due to very high curl electric fields induced by world-wide magnetic field's changes with a SC-rate >50nT/s. The laboratory models of AM (with R sub( mp )0.1-30cm) were formed around high-field, 1D and 3D magnetic obstacles, overflowing by LPP-blobs with E sub(0) up to kJ and magnetized ions. The shape and internal structure of a large-scale AM were studied at KI-1 facility of the Russian team using a set of B-dot magnetic probes, while the main goal of UT's small-AM experiment was to explore a possible shock's generation and relevant electron acceleration. Preliminary results of KI-1 experiments show that the both R sub( m )-size and SC (E sub(0)) of AM could be described by modified Chapman-Ferraro Scaling, while the whole SC-distribution (in front 'one-half' of equatorial plane)-by well-known 'Image Dipole' model of the Earth's magnetopause field.