  There is evidence that most chains of mean motion resonances of type
$k$:$k-1$ among exoplanets become unstable once the dissipative action from the
gas is removed from the system, particularly for large $N$ (the number of
planets) and $k$ (indicating how compact the chain is). We present a novel
dynamical mechanism that can explain the origin of these instabilities and thus
the dearth of resonant systems in the exoplanet sample. It relies on the
emergence of secondary resonances between a fraction of the synodic frequency
$2 \pi (1/P_1-1/P_2)$ and the libration frequencies in the mean motion
resonance. These secondary resonances excite the amplitudes of libration of the
mean motion resonances thus leading to an instability. We detail the emergence
of these secondary resonances by carrying out an explicit perturbative scheme
to second order in the planetary masses and isolating the harmonic terms that
are associated with them. Focusing on the case of three planets in the 3:2 --
3:2 mean motion resonance as an example, a simple but general analytical model
of one of these resonances is obtained which describes the initial phase of the
activation of one such secondary resonance. The dynamics of the excited system
is also briefly described. This scheme shows how one can obtain analytical
insight into the emergence of these resonances, and into the dynamics that they
trigger. Finally, a generalisation of this dynamical mechanism is obtained for
arbitrary $N$ and $k$. This leads to an explanation of previous numerical
experiments on the stability of resonant chains, showing why the critical
planetary mass allowed for stability decreases with increasing $N$ and $k$.
