Fractal differential equations and fractal-time dynamical systems

Differential equations and maps axe the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus called F-alpha-calculus, is a natural calculus on subsets F subset of R of dimension alpha, 0 < alpha <= 1. It involves integral and derivative of order alpha, called F-alpha-integral and F-alpha-derivative respectively. The F-alpha-integral is suitable for integrating functions with fractal support of dimension a, while the F-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions of F-alpha-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems. We discuss construction and solutions of some fractal differential equations of the form D-F,t(alpha) x = h(x, t), where h is a vector field and D-F,t(alpha) is a fractal differential operator of order alpha in time t. We also consider some equations of the form DF,t alpha W (x, t) = L[W (x, T)], where L is an ordinary differential operator in the real variable x, and (t, x) is an element of F x R-n where F is a Cantor-like set of dimension alpha. Further, we discuss a method of finding solutions to F-alpha-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a couple of examples.