Critical exponents and Scaling Properties for the Chaotic Dynamics of a Particle in a Time-Dependent Potential Barrier

Some scaling properties of the chaotic sea for a particle confined inside an infinitely deep potential box containing a time varying barrier are studied. The dynamics of the particle is described by a two-dimensional, nonlinear and area-preserving mapping for the variables energy of the particle and time. The phase space of the model exhibits a mixed structure with Kolmogorov– Arnold–Moser islands, chaotic seas and invariant spanning curves limiting the chaotic orbits. Average properties of the chaotic sea including the first momenta and the deviation of the second momenta are obtained as a function of: (i) number of iterations (n), and (ii) time (t). By the use of scaling arguments, critical exponents for the ensemble average of the first momenta are obtained and compared for both cases (i) and (ii). Scaling invariance of the average properties for the chaotic sea is obtained as a function of the control parameters that describe the model.