  We investigate cyclic and singularity-free evolutions in a universe governed
by Lagrange-multiplier modified gravity, either in scalar-field cosmology, as
well as in $f(R)$ one. In the scalar case, cyclicity can be induced by a
suitably reconstructed simple potential, and the matter content of the universe
can be successfully incorporated. In the case of $f(R)$-gravity, cyclicity can
be induced by a suitable reconstructed second function $f_2(R)$ of a very
simple form, however the matter evolution cannot be analytically handled.
Furthermore, we study the evolution of cosmological perturbations for the two
scenarios. For the scalar case the system possesses no wavelike modes due to a
dust-like sound speed, while for the $f(R)$ case there exist an oscillation
mode of perturbations which indicates a dynamical degree of freedom. Both
scenarios allow for stable parameter spaces of cosmological perturbations
through the bouncing point.
