Projective synchronization for distinct fractional-order neural networks consist of inconsistent orders via sliding mode control

The present study delves into the projective synchronization for distinct fractional-order neural networks consist of inconsistent orders, employing the principles of sliding mode control and incorporating fractional operators within the controller. Firstly, incorporating the fractional-order derivative into the controller facilitates the derivation of a synchronous error system, and a well-suited integral-type sliding switching surface is subsequently constructed, along with the development of an appropriate sliding mode controller, thereby ensuring the existence of sliding movement derived from the principle of sliding mode control. Then through the application of the Lyapunov direct method, it is shown that the sliding mode controller steers the evolution of error system towards the designated sliding switching surface and maintains therein indefinitely, meanwhile, novel criteria for achieving projective synchronization for distinct fractional-order neural networks consist of inconsistent orders is established. Lastly, to exemplify the efficacy and practical applications of the proposed approach, simulations with two types of distinct fractional-order neural networks are carried out utilizing a continuous frequency distribution model.