Abstract
Using a quantum gas setup consisting of a Bose-Einstein condensate strongly coupled to a high-finesse optical cavity by a transverse pump laser, we experimentally observe an instability of a dissipative continuous time crystal (CTC) towards a time crystalline state exhibiting two prominent oscillation frequencies. Applying a mean-field approximation model and a Floquet analysis, we theoretically confirm that this transition is a manifestation in a many-body system of a torus bifurcation between a limit cycle (LC) and a limit torus (LT). We theoretically illustrate the LC and LT attractors using the minimal model and experimentally reconstruct them using Takens' embedding theorem applied to the non-destructively measured intracavity photon dynamics.