  We consider f(R) modified gravity theories in the metric variation formalism
and attempt to reconstruct the function f(R) by demanding a background LCDM
cosmology. In particular we impose the following requirements: a. A background
cosmic history H(z) provided by the usual flat LCDM parametrization though the
radiation (w_eff=1/3), matter (w_eff=0) and deSitter (w_eff=-1) eras. b. Matter
and radiation dominate during the `matter' and `radiation' eras respectively
i.e. \Omega_m =1 when w_eff=0 and \Omega_r=1 when w_eff=1/3. We have found that
the cosmological dynamical system constrained to obey the LCDM cosmic history
has four critical points in each era which correspondingly lead to four forms
of f(R). One of them is the usual general relativistic form f(R)=R-2\Lambda.
The other three forms in each era, reproduce the LCDM cosmic history but they
do not satisfy requirement b. stated above. Only one of these forms (different
from general relativity) is found to be an attractor of the dynamical
cosmological evolution. It has (\Omega_DE=1, \Omega_r=0, \Omega_m=0) throughout
the evolution. Its phase space trajectory is numerically obtained.
