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.

The New Four-Dimensional Fractional Chaotic Map with Constant and Variable-Order: Chaos, Control and Synchronization

“…In [21], the authors explained the dynamics of the discrete fractional neural network with incommensurate order. Very recently, some papers have been published on the discrete fractional variable-order system and their dynamics [22][23][24][25]. For example, in [22], the dynamic and discrete systems of variable fractional-order in the sense of the Lozi structure map is studied.…”

“…For example, in [22], the dynamic and discrete systems of variable fractional-order in the sense of the Lozi structure map is studied. In [23], the C 0 complexity and the 0-1 test were employed to determine the chaotic attractors of the discrete non-commensurate variable-order Hopfild neural network. However, in [24], the Lyapunov exponents and the approximate entropy were exploited to discuss the chaotic behavior of the fractional Tinkerbell system.…”