Abstract
This paper is concerned with the problem of bifurcation for a ring fractional Hopfield neural network with leakage time delay and communication time delay. The stability and the Hopf bifurcations of such a network without and with time delays are investigated by analyzing the associated characteristic equations. Specifically, some criteria for the occurrence of Hopf bifurcations at the trivial steady state are established. It is shown that the dynamical property of the network is not only crucially dependent on the communication time delay, but also significantly influenced by the leakage time delay. Furthermore, the effects of the order on the Hopf bifurcation are numerically demonstrated. Finally, four numerical examples are provided to illustrate the feasibility of the theoretical results.
Highlights
The studies for various Hopfield neural networks (HNNs) have been continuously active over the past three decades because of their successful applications in numerous areas, for instance, optimizations, signal processing, image processing, solving nonlinear algebraic equations, pattern recognitions, associative memories [1,2,3,4,5]
One major and often encountered difficulty in the analysis of neural network dynamics is the ubiquity of time delays that can result in instability, oscillation, periodic solution, anti-periodic solution, almost periodic solution, quasi-periodic solution, and even give rise to multistability and chaotic motion
Theorem 4.2 For system (3.2), the following results hold: (1) If (C1) and (C5) are satisfied, the zero equilibrium point is globally asymptotically stable for τ ∈ [0, +∞). (2) If (C1), (C5)–(C6) hold, (i) the zero equilibrium point is locally asymptotically stable for τ ∈ [0, τ0∗); (2019) 2019:179. This theorem demonstrates that the stability and the Hopf bifurcation of the neural network are crucially dependent on the communication delays, and heavily influenced by the leakage delay
Summary
The studies for various Hopfield neural networks (HNNs) have been continuously active over the past three decades because of their successful applications in numerous areas, for instance, optimizations, signal processing, image processing, solving nonlinear algebraic equations, pattern recognitions, associative memories [1,2,3,4,5]. Theorem 4.2 For system (3.2), the following results hold: (1) If (C1) and (C5) are satisfied, the zero equilibrium point is globally asymptotically stable for τ ∈ [0, +∞). (2) If (C1), (C5)–(C6) hold, (i) the zero equilibrium point is locally asymptotically stable for τ ∈ [0, τ0∗); Figure 1 Time responses of system (5.1) with φ = 0.92, τ = 0.47 < τ0 = 0.5421 This theorem demonstrates that the stability and the Hopf bifurcation of the neural network are crucially dependent on the communication delays, and heavily influenced by the leakage delay. It is essential for considering the effects of communication and leakage delays in designing or controlling neural networks
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