  Within the framework of the kinetic energy driven superconducting mechanism,
the dynamical spin response of cuprate superconductors is studied from
low-energy to high-energy. The spin self-energy is evaluated explicitly in
terms of the collective charge carrier modes in the particle-hole and
particle-particle channels, and employed to calculate the dynamical spin
structure factor. Our results show the existence of damped but well-defined
dispersive spin excitations in the whole doping phase diagram. In particular,
the low-energy spin excitations in the superconducting-state have an
hour-glass-shaped dispersion, with commensurate resonance that appears in the
superconducting-state only, while the low-energy incommensurate spin
fluctuations can persist into the normal-state. The high-energy spin
excitations in the superconducting-state on the other hand retain roughly
constant energy as a function of doping, with spectral weights and dispersion
relations comparable to those in the corresponding normal-state. The theory
also shows that the unusual magnetic correlations in cuprate superconductors
can be ascribed purely to the spin self-energy effects which arise directly
from the charge carrier-spin interaction in the kinetic energy of the system.
