Floquet engineering on the dispersion relation with shaken optical lattice provides a versatile playground for studying dynamics and phases in exotic materials. We have built such a platform with ultracold fermionic Li-6 that opens possibilities to studying topological transitions of the Fermi surface or non-BCS pairing with tunable parameters. We demonstrate an interesting dynamic in the deep BEC regime under the presence of a shaken lattice and potential, where bifurcations are observed from an instable initial condition, and patterns form in positional space. We understand the bifurcated clusters' trajectories with a semi-classical model and GPE simulations, the latter also revealing the role of the interaction in the clusters' solitonic behaviors.