  We analyze solutions drawn from the recently published posterior distribution
of the TRAPPIST-1 system, which consists of seven Earth-size planets appearing
to be in a resonant chain around a red dwarf. We show that all the planets are
simultaneously in two-planet and three-planet resonances, apart from the
innermost pair for which the two-planet resonant angles circulate. By means of
a frequency analysis, we highlight that the transit-timing variation (TTV)
signals possess a series of common periods varying from days to decades, which
are also present in the variations of the dynamical variables of the system.
Shorter periods (e.g., the TTVs characteristic timescale of 1.3 yr) are
associated with two-planet mean-motion resonances, while longer periods arise
from three-planet resonances. By use of $N$-body simulations with migration
forces, we explore the origin of the resonant chain of TRAPPIST-1 and find that
for particular disc conditions, a chain of resonances - similar to the observed
one - can be formed which accurately reproduces the observed TTVs. Our analysis
suggests that while the 4-yr collected data of observations hold key
information on the two-planet resonant dynamics, further monitoring of
TRAPPIST-1 will soon provide signatures of three-body resonances, in particular
the 3.3 and 5.1 yr periodicities expected for the current best-fit solution.
dditional observations would help to assess whether the innermost pair of
planets is indeed resonant (its proximity to the 8:5 resonance being
challenging to explain), and therefore give additional constraints on formation
scenarios.
