Reduced-order synchronization of fractional order chaotic systems with fully unknown parameters using modified adaptive control

A novel reduced-order adaptive controller is extended and developed to synchronize two different fractional order chaotic systems with different dimensions. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third{order fractional order chaotic system can be synchronized with the projection of a fourth{order fractional order chaotic system even though their parameters are unknown. The techniques are successfully applied to fractional order hyperchaotic Chen (4th-order) and fractional order chaotic Liu (3rd-order) systems. Theoretical analysis and numerical simulations are shown to verify the results.