Generation of Periodic Orbits in the Restricted Three-Body Problem with a Variational Autoencoder

This work presents the application of Variational Autoencoders (VAEs) to the generation and analysis of periodic orbits in the Circular Restricted Three-Body Problem (CR3BP). A VAE architecture based on Convolutional Neural Networks (CNNs) was trained on a dataset of time series representing periodic trajectories. The encoder provided a compact, low-dimensional representation that captured the underlying geometric and dynamical features of the trajectories. By sampling the latent space, the model was able to generate approximations of new quasi-periodic trajectories with prescribed characteristics. A continuation technique was subsequently implemented both in the physical and latent domains. Continuation in physical space enabled the convergence toward periodic orbits by starting from the approximations produced by the VAE, whereas continuation in latent space facilitated the systematic generation of trajectories belonging to selected orbital families. The results demonstrate the potential use of generative models to design trajectories as well as automatically discover periodic orbits in complex dynamical systems.