In the present study we investigate the secondary instability of the in-phase synchronized vortex shedding from two side-by-side circular cylinders at low Reynolds numbers. Two distinct Floquet modes become unstable for different values of the Reynolds number and of the non-dimensional gap spacing, leading to the onset of the well-known flip-flop instability of the two cylinder wakes. In both cases the two-dimensional Floquet analysis reveals that at very low Reynolds numbers, a pair of complex-conjugate multipliers crosses the unit circle, showing the same frequency as the biased gap-flow flip-over. In the past literature this behaviour has been often ascribed to a bistability of the flow. On the contrary, the present DNS and stability results provide evidence that at low Reynolds numbers, the flip-flopping behaviour originates from a Neimark–Sacker bifurcation of the in-phase shedding cycle.