The Effect of Caputo Fractional Variable Difference Operator on a Discrete-Time Hopfield Neural Network with Non-Commensurate Order

Abstract

In this work, we recall some definitions on fractional calculus with discrete-time. Then, we introduce a discrete-time Hopfield neural network (D.T.H.N.N) with non-commensurate fractional variable-order (V.O) for three neurons. After that, phase-plot portraits, bifurcation and Lyapunov exponents diagrams are employed to verify that the proposed discrete time Hopfield neural network with non-commensurate fractional variable order has chaotic behavior. Furthermore, we use the 0-1 test and C0 complexity algorithm to confirm and prove the results obtained about the presence of chaos. Finally, simulations are carried out in Matlab to illustrate the results.

Keywords

• Hopfield neural network
• fractional variable order
• chaos
• bifurcation
• Lyapunov exponents
• <i>C</i>₀ complexity