Fractal differential equations and fractal-time dynamical systems

Abstract

Differential equations and maps are 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 calledF ^α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF ^α-integral andF ^α-derivative respectively. TheF ^α-integral is suitable for integrating functions with fractal support of dimension α, while theF ^α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions ofF ^α-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

$$D_{F,t}^\alpha x = h(x,t),$$

whereh is a vector field andD ^α _F,t is a fractal differential operator of order α in timet. We also consider some equations of the form

$$D_{F,t}^\alpha W(x,t) = L[W(x,t)],$$

whereL is an ordinary differential operator in the real variablex, and(t,x) ∈F × Rⁿ whereF is a Cantor-like set of dimension α.

Further, we discuss a method of finding solutions toF ^α-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.