This paper investigates the modified function projective synchronization (MFPS) in drive-response dynamical networks (DRDNs) with different nodes, which means that systems in nodes are strictly different. An adaptive open-plus-closed-loop (AOPCL) control method is proposed, which is a practically realizable method and can overcome the model mismatched to achieve synchronization. It is well known that each of the close-loop and open-loop control method possesses some advantages and disadvantages. By combining their advantages, the open-plus-closed-loop (OPCL) control method was proposed by Jackson and Grosu. For arbitrary nonlinear dynamic systems, dx/dt = F(x,t), Jackson and Grosu proved that there exists solutions, x(t), in the neighborhood of any arbitrary goal dynamics g(t) that are entrained to g(t), through the use of an additive controlling action, K(g,x,t) = H(dg/dt,g) + C(g,t)(g(t) - x), which is the sum of the open-loop action, H(dg/dt,g), and a suitable linear closed-loop (feedback) action C(g,t). This method is a practically realizable method and robust to limited accuracy of data and effects of noise. The AOPCL control method preserve the merits of OPCL control method and its closed loop control part can be automatically adapted to suitable constants. Considering time-delays are always unavoidably in the practical situations, MFPS in DRDNs with time-varying coupling delayed is further investigated by the proposed method. Corresponding numerical simulations are performed to verify and illustrate the analytical results.
Analysis and modified function projective synchronization of integer and fractional-order autonomous Morse jerk oscillator
