On the dynamical investigation and synchronization of variable-order fractional neural networks: the Hopfield-like neural network model

Since the variable-order fractional systems show more complex characteristics and more degrees of freedom due to time-varying fractional derivatives, we introduce a variable-order fractional Hopfield-like neural network in this paper. First, the properties and dynamical behavior of the system are studied. The variable-order derivative’s effects on the system’s behavior are investigated through the Lyapunov exponents and bifurcation diagram; an emerging Feigenbaum tree of period-4 bubble is observed, which appears with the creation and annihilation of periodic orbits. A general basin of attraction for the fractional-order neural network is presented, demonstrating that its dynamical behaviors are extremely sensitive to initial conditions resulting in different periodic orbits and chaotic attractors’ coexistence. After that, an adaptive control scheme is proposed for the variable-order fractional system. Through Lyapunov theorem and Barbalat’s Lemma, the system’s convergence and stability under the proposed control scheme are proven. The main advantages of the proposed controller are its guaranteed stability, robustness against uncertainties, and simplicity. Finally, the synchronization results are presented. Numerical simulations show the excellent performance of the proposed controller for the variable-order fractional Hopfield-like neural network.