Oscillation of Non-Linear Systems Close to Equilibrium Position in the Presenceof Coarse-Graining in Time and Space

摘要

One considers the motion of nonlinear systems close to their equilibriumpositions in the presence of coarse-graining in time on the one hand, and coarse-grainingin time on the other hand. By considering a coarse-grained time as a time in whichthe increment is not dt but rather (dt)c> dt, one is led to introduce a modelingin terms of fractional derivative with respect to time; and likewise for coarse-grainingwith respect to the space variable x. After a few prerequisites on fractional calculus viamodified Riemann-Liouville derivative, one examines in a detailed way the solutions offractional linear differential equations in this framework, and then one uses this result inthe linearization of nonlinear systems close to their equilibrium positions