Original research articleOn a function projective synchronization scheme for non-identical Fractional-order chaotic (hyperchaotic) systems with different dimensions and orders

Referring to fractional-order systems, this paper investigates the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions and different orders. By taking a master system of dimension n and a slave system of dimension m, the method enables each master system state to be synchronized with a linear combination of slave system states, where the scaling factor of the linear combination can be any arbitrary differentiable function. The approach presents some useful features: i) it enables commensurate and incommensurate non-identical fractional-order systems with different dimension n < m or n > m to be synchronized; ii) it can be applied to a wide class of chaotic (hyperchaotic) fractional-order systems for any differentiable scaling function; iii) it is rigorous, being based on two theorems, one for the case n < m and the other for the case n > m. Two different numerical examples are reported, involving chaotic/hyperchaotic fractional-order Lorenz systems (three-dimensional and four-dimensional master/slave, respectively) and hyperchaotic/chaotic fractional-order Chen systems (four-dimensional and three-dimensional master/slave, respectively). The examples clearly highlight the capability of the conceived approach in effectively achieving synchronized dynamics for any differentiable scaling function.