  Generative Adversarial Networks (GANs) are powerful Machine Learning models
capable of generating fully synthetic samples of a desired phenomenon with a
high resolution. Despite their success, the training process of a GAN is highly
unstable and typically it is necessary to implement several accessory
heuristics to the networks to reach an acceptable convergence of the model. In
this paper, we introduce a novel method to analyze the convergence and
stability in the training of Generative Adversarial Networks. For this purpose,
we propose to decompose the objective function of the adversary min-max game
defining a periodic GAN into its Fourier series. By studying the dynamics of
the truncated Fourier series for the continuous Alternating Gradient Descend
algorithm, we are able to approximate the real flow and to identify the main
features of the convergence of the GAN. This approach is confirmed empirically
by studying the training flow in a $2$-parametric GAN aiming to generate an
unknown exponential distribution. As byproduct, we show that convergent orbits
in GANs are small perturbations of periodic orbits so the Nash equillibria are
spiral attractors. This theoretically justifies the slow and unstable training
observed in GANs.
