Design of fractional-order hyperchaotic systems with maximum number of positive lyapunov exponents and their antisynchronisation using adaptive control

A systematic design procedure for generating fractional-order hyperchaotic systems (FOHSs) with a desired number of positive Lyapunov Exponents (LEs) remains an open problem. This paper puts forward a simple step-wise algorithm to tackle the above problem for a fractional-order system (FOS) to generate hyperchaos with effective dimension n(d) > 3 onwards; unlike the integer-order hyperchaotic systems where the net dimension must be n(d) > 4. The problem is a significant one as a lower-dimensional system with higher number of positive LEs has crucial potentiality in secure communication. Two proposals of controller design are put forward. The first one is to design an FOHS with maximum possible positive LEs from a stable system. The second is an adaptive control scheme in fractional dynamics to perform antisynchronisation between the generated FOHS with itself considering unknown parameters. Two representative examples are presented to validate the two control proposals. The superior effectiveness in generating FOHSs by following the proposed procedure is established in comparison with the existing hit-and-trial-methods.