An Investigation of Chaotic Diffusion in a Family of Hamiltonian Mappings Whose Angles Diverge in the Limit of Vanishingly Action

Abstract

The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, I, and angle, θ and controlled by two control parameters: (i) ǫ, controlling the nonlinearity of the system, particularly a transition from integrable for ǫ = 0 to non-integrable for ǫ = 0 and; (ii) γ denoting the power of the action in the equation defining the angle. For ǫ = 0 the phase space is mixed and chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures.

Chaotic diffusion for particles moving in a time dependent potential well

“…The islands present in the positive side of I in Ref. [17] cancel the influence of the islands observed in the negative side of I and such effect does not cause differences in the saturation between the theoretical point of view obtained from the solution of the diffusion equation and the numerical results, obtained directly from the dynamical equations. By using the fraction of the chaotic sea to correct the density of points in the phase space, the curves of e rms (n) obtained from the diffusion equation saturate closer to the ones obtained from numerical simulations, as seen in Fig.…”

“…It is important to mention that such a difference was not observed in Ref. [17] because the phase space considered there is symmetric with respect to the action axis in the positive and negative sides. The islands present in the positive side of I in Ref.…”

“…Diffusive processes are observed in a wide variety of systems ranging from pollen diffusing from plant to plant [1], in medicine to the investigation of drugs delivered through the blood current [2], how diseases propagate in the air [3][4][5], population movements in European prehistory [6], pollution in atmosphere [7] and in the oceans [8], percolation [9] with the investigation of the chemical compounds being transported to the water table, memes in social media [10], the influence of the clime in pests spreading [11] and many other applications from physical side [12][13][14] turning the topic of interest to many [15][16][17].…”

“…As we discuss along the text, such a difference is mostly connected to the shape of the phase space where islands are present surrounded by chaotic sea. As an assumption for the solution of the diffusion equation, such islands are not taken into account [17] and their existence change the saturation of the curves for the diffusion at long enough time. We estimate the fraction that the islands occupy in the phase space and use it to made a first order correction in the saturation of the curves as an attempt to consider the same density of points in the phase space generated by the set of equations of the mapping and that used in the solution of the diffusion equation.…”

Characterization of a continuous phase transition in a chaotic system

“…We also notice a scaling present in the curves even if the initial action is not small enough, as is the case of the continuous curves. The latter were obtained by the analytical solution of the diffusion equation under specific boundary conditions [11].…”