A dynamical phase transition for a family of Hamiltonian mappings: A phenomenological investigation to obtain the critical exponents

• Scaling formalism to describe a transition from integrable to non-integrable;
• Homogeneous function used to obtain critical exponents;
• A break of symmetry of the probability function explains an additional scaling.

A dynamical phase transition from integrability to non-integrability for a family of 2-D Hamiltonian mappings whose angle, θ, diverges in the limit of vanishingly action, I, is characterised. The mappings are described by two parameters: (i) ϵ  , controlling the transition from integrable (ϵ=0ϵ=0) to non-integrable (ϵ≠0ϵ≠0); and (ii) γ, denoting the power of the action in the equation which defines the angle. We prove the average action is scaling invariant with respect to either ϵ or n and obtain a scaling law for the three critical exponents.