A short review of phase transition in a chaotic system

The subject approached here is a dynamical phase transition observed in Hamiltonian systems, which is a transition from integrability to non-integrability. Using the dynamics defined by a discrete mapping on the variables action I and angle $$\theta$$, we perform a description of the behaviour of the chaotic diffusion to particles in the chaotic sea using two methods. One is a phenomenological description obtaining the critical exponents via numerical simulation, and the other is an analytical result obtained by the solution of the diffusion equation. The scaling invariance is observed in the chaotic sea leading to an universal chaotic diffusion. This is a clear signature that the system is passing through a phase transition. We investigate a set of four questions that characterize a phase transition: (1) identify the broken symmetry; (2) define the order parameter; (3) identify what are the elementary excitations and; (4) detect the topological defects which impact on the transport of the particles