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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 called

where

where

*F*^{α}-calculus, is a natural calculus on subsets*F*⊂ R of dimension α,*0 < α ≤ 1.*It involves integral and derivative of order α, called*F*^{α}-integral and*F*^{α}-derivative respectively. The*F*^{α}-integral is suitable for integrating functions with fractal support of dimension α, while the*F*^{α}-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions of*F*^{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*h*is a vector field and*D*_{ F,t }^{α}is a fractal differential operator of order α in time*t.*We also consider some equations of the form*L*is an ordinary differential operator in the real variable*x*, and*(t,x)*∈*F*× R^{n}where*F*is a Cantor-like set of dimension α. Further, we discuss a method of finding solutions to*F*^{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. 相似文献**1**