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Most ultra-luminous X-ray sources (ULXs) are now thought to be powered bystellar-mass compact objects accreting at super-Eddington rates. While thediscovery of evolutionary cycles have marked a breakthrough in ourunderstanding of the accretion flow changes in the sub-Eddington regime inGalactic Black Hole Binaries, their evidence in the super-Eddington regimeremained elusive. However, recent circumstantial evidence had hinted thepresence of a recurrent evolutionary cycle in two archetypal ULXs: Holmberg IIX-1 and NGC 5204 X-1. Here we build on our previous work and exploit thelong-term high-cadence monitoring of Swift-XRT in order to provide evidence ofthe evolutionary cycle in these two sources and investigate the main physicalparameters inducing their spectral transitions. We study the long-termevolution of both sources using hardness-intensity diagrams (HID) and by meansof Lomb-Scargle periodograms and Gaussian processes modelling to look forperiodic variability. We show that both sources follow a recurrent evolutionarypattern in the HID that can be characterized by the hard ultraluminous (HUL)and soft ultraluminous (SUL) spectral regimes, and a third state withcharacteristics similar to the supersoft ultraluminous (SSUL) state. Thetransitions between the soft states seem aperiodic, as revealed by timinganalysis of the light curve of Holmberg II X-1, albeit further investigation iswarranted. The light curve of NGC 5204 X-1 shows a periodicity of $\sim$ 200days, possibly associated with the duration of the evolutionary cycle. Wesupport a scenario in which the spectral changes from HUL to SUL are due to aperiodic increase of the mass-transfer rate and subsequent narrowing of theopening angle of the supercritical funnel. The narrower funnel, combined withstochastic variability imprinted by the wind, might explain the SUL--SSULspectral changes.

Discovery of a recurrent spectral evolutionary cycle in the ultra-luminous X-ray sources Holmberg II X-1 and NGC 5204 X-1