Abstract
In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks.
Highlights
Fractional calculus has a history of 300 years
Studying the chaotic fractional-order Hopfield neural network (FHNN) with time-delay is a very significant issue that can narrow the gap between biological neuronal systems and artificial neural networks
Motivated by the discussions above, in this paper, we propose a new class of time-delayed FHNN and analyze the rich dynamics, design its generalized projective synchronization (GPS) scheme
Summary
Fractional calculus has a history of 300 years. In the early days, because of its computational complexity and lack of intuitive physical and geometric explanations, it did not attract the interest of researchers [1,2,3]. It is natural to consider that incorporating the two memory terms (fractional calculus and the Hopfield neural network) will potentially be a huge step toward the enhancement of both memory characteristics and the efficiency of information processing [8]. Studying the chaotic fractional-order Hopfield neural network (FHNN) with time-delay is a very significant issue that can narrow the gap between biological neuronal systems and artificial neural networks. Only a few of them studied the chaotic characteristics and generalized projective synchronization (GPS) schemes of time-delayed FHNN. Motivated by the discussions above, in this paper, we propose a new class of time-delayed FHNN and analyze the rich dynamics, design its GPS scheme.
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