Oscillation of Non-Linear Systems Close to Equilibrium Position in the Presence of Coarse-Graining in Time and Space - Jumarie, G. (Nonlinear Analysis: Modelling and Control, 2009)
One considers the motion of nonlinear systems close to their equilibrium positions in the presence of coarse-graining in time on the one hand, and coarse-graining in time on the other hand. By considering a coarse-grained time as a time in which the increment is not dt but rather (dt)c > dt, one is led to introduce a modeling in terms of fractional derivative with respect to time; and likewise for coarse-graining with respect to the space variable x. After a few prerequisites on fractional calculus via modified Riemann-Liouville derivative, one examines in a detailed way the solutions of fractional linear differential equations in this framework, and then one uses this result in the linearization of nonlinear systems close to their equilibrium positions.
Keywords: coarse-grained time; coarse-grained space; fractal time; fractional analysis; fractional Taylor’s series; stability; linearization; Lyapunov function