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Identification of the unknown parameters and orders of fractional chaoticsystems is of vital significance in controlling and synchronization offractional-order chaotic systems. However there exist basic hypotheses intraditional estimation methods, that is, the parameters and fractional ordersare partially known or the known data series coincide with definite forms offractional chaotic differential equations except some uncertain parameters andfractional orders. What should I do when these hypotheses do not exist?  In this paper, a non-Lyapunov novel approach with a novel united mathematicalmodel is proposed to reconstruct fractional chaotic systems, through thefractional-order differential equations self-growing mechanism by some geneticoperations ideas independent of these hypotheses. And the cases of identifyingthe unknown parameters and fractional orders of fractional chaotic systems canbe thought as special cases of the proposed united mathematical reconstructionmethod in non-Lyapunov way. The problems of fractional-order chaosreconstruction are converted into a multiple modal non-negative specialfunctions' minimization through a proper translation, which takesfractional-order differential equations as its particular independent variablesinstead of the unknown parameters and fractional orders. And the objective isto find best form of fractional-order differential equations such that theobjective function is minimized. Simulations are done to reconstruct a seriesof hyper and normal fractional chaotic systems. The experiments' results showthat the proposed self-growing mechanism of fractional-order differentialequations with genetic operations is a successful methods for fractional-orderchaotic systems' reconstruction, with the advantages of high precision androbustness.

Reconstruction mechanism with self-growing equations for hyper, improper and proper fractional chaotic systems through a novel non-Lyapunov approach