Soliton molecules in optical resonators have attracted remarkable attention in nonlinear dynamics

driven by their compelling analogies with matter molecules. So far

while extensive research has been conducted on their generation

pulsations

and dissociation behaviors

the investigation of their quasi-periodic dynamics has been relatively limited. Here

we present a systematic exploration of the quasi-periodic dynamics of soliton molecules using advanced balanced optical cross-correlation techniques. The incommensurable quasi-period bifurcations constituted of cascaded Hopf bifurcations are found

providing an unambiguous pathway toward chaotic soliton molecules. The chaotic intramolecular dynamics are analyzed by time series

radio frequency spectra

phase portraits

and Lyapunov exponent analysis. In addition

we reveal an intrinsic frequency entrainment phenomenon experimentally. Such frequency entrainment provides a novel perspective on synchronization in optical resonators

encompassing the competition and interaction of oscillations across multiple temporal scales. Our experimental findings offer clear proof that the gain dynamics serve as the origin of the binding forces between solitons within the molecule

which are well supported by the numerical simulations. By advancing the understanding of sub-femtosecond resolved quasi-period dynamics of optical soliton molecules

this study contributes to the broader field of complex nonlinear dynamics

paving the way for future explorations into the intricate behaviors of solitons within optical resonators and relevant fields.