  We examine the possibility that Universe expansion be made of some $\Lambda
CDM$ expansions repeating periodically, separated by some inflation and
radiation dominated phases. This so-called $\Lambda CDM$ periodic cosmology is
motivated by the possibility that inflation and the present phase of
accelerated expansion be due to the same dark energy. Then, in a phase space
showing the variation of matter density parameter $\Omega_m$ with respect to
this of the radiation $\Omega_r$, the curve $\Omega_m(\Omega_r)$ looks like a
closed trajectory that Universe could run through forever. In this case, the
end of the expansion acceleration of the $\Lambda CDM$ phase is the beginning
of a new inflation phase. We show that such a scenario implies the coupling of
matter and/or radiation to dark energy. We consider the simplest of these
$\Lambda CDM$ periodic models i.e. a vacuum energy coupled to radiation. From
matter domination phase to today, it behaves like a $\Lambda CDM$ model, then
followed by an inflation phase. But a sudden and fast decay of the dark energy
into radiation periodically ends the expansion acceleration. This leads to a
radiation dominated Universe preceding a new $\Lambda CDM$ type expansion. The
model is constrained with Markov Chain Monte Carlo simulations using
supernovae, Hubble expansion, BAO and CMB data and fits the data as well as the
$\Lambda CDM$ one.
