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

TRAPPIST-1: Dynamical analysis of the transit-timing variations and origin of the resonant chain