OS-net: Orbitally Stable Neural Networks

In particular, periodic orbits play a significant role in chaos theory. In [6], chaotic systems are defined as systems that are sensitive to initial conditions, are topologically transitive (meaning that any region of the phase space can be reached from any other region), and have dense periodic orbits. Notably, chaotic systems are constituted of infinitely many Unstable Periodic Orbits (UPOs) which essentially form a structured framework, or a "skeleton", for chaotic attractors. A periodic orbit is (orbitally) unstable if trajectories that start near the orbit do not remain close to it. Finding and stabilizing UPOs is an interesting and relevant research field with numerous applications such as the design of lasers [18], the control of seizure activities [22] or the design of control systems for satellites [30]. An important tool when studying the stability of periodic orbits of a given system is the Poincaré or return map which allows one to study the dynamics of this system in a lower dimensional subspace.