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This paper considers the formulation of the variational model (VM) of autonomous circuits (oscillators) working in periodic steady-state conditions. The shooting method, which is largely used to compute the solution in the time domain when the VM is forced by a small-signal perturbation, is studied. The proposed analytical approach can be exploited to improve accuracy in the simulation of the effects of noise sources. In particular, we justify from an analytical standpoint the adoption of a suitable periodicity constraint in the shooting method. We exploit the properties of block circulant matrices that naturally arise in the description of the problem. We prove that the frequency of the small-signal perturbation must be different from that of the unperturbed oscillator to avoid inaccuracy of the shooting method due to the existence of singularities in the VM formulation, and derive a method that allows us to get closer to the singularity.