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Hopfield neuronal network of fractional order: A note on its numerical integration

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  • Danca, Marius-F.

Abstract

In this paper, the commensurate fractional-order variant of an Hopfield neuronal network is analyzed. The system is integrated with the ABM method for fractional-order equations. Beside the standard stability analysis of equilibria, the divergence of fractional order is proposed to determine the instability of the equilibria. The bifurcation diagrams versus the fractional order, and versus one parameter, reveal a strange phenomenon suggesting that the bifurcation branches generated by initial conditions outside neighborhoods of unstable equilibria are spurious sets although they look similar with those generated by initial conditions close to the equilibria. These spurious sets look “delayed” in the considered bifurcation scenario. Once the integration step-size is reduced, the spurious branches maintain their shapes but tend to the branches obtained from initial condition within neighborhoods of equilibria. While the spurious branches move once the integration step size reduces, the branches generated by the initial conditions near the equilibria maintain their positions in the considered bifurcation space. This phenomenon does not depend on the integration-time interval, and repeats in the parameter bifurcation space.

Suggested Citation

  • Danca, Marius-F., 2021. "Hopfield neuronal network of fractional order: A note on its numerical integration," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    • Handle: RePEc:eee:chsofr:v:151:y:2021:i:c:s0960077921005737
    DOI: 10.1016/j.chaos.2021.111219
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    References listed on IDEAS

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    1. Danca, Marius-F. & Lampart, Marek, 2021. "Hidden and self-excited attractors in a heterogeneous Cournot oligopoly model," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    2. Xuerong Shi & Zuolei Wang, 2020. "Stability Analysis of Fraction-Order Hopfield Neuron Network and Noise-Induced Coherence Resonance," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-12, June.
    3. Jafari, Sajad & Sprott, J.C., 2013. "Simple chaotic flows with a line equilibrium," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 79-84.
    4. Xia Huang & Zhen Wang & Yuxia Li, 2013. "Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-9, November.
    5. Huang, Chuangxia & Huang, Lihong & Feng, Jianfeng & Nai, Mingyong & He, Yigang, 2007. "Hopf bifurcation analysis for a two-neuron network with four delays," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 795-812.
    6. Xu, Changjin & Li, Peiluan, 2017. "Global exponential convergence of neutral-type Hopfield neural networks with multi-proportional delays and leakage delays," Chaos, Solitons & Fractals, Elsevier, vol. 96(C), pages 139-144.
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    Cited by:

    1. Marius-F. Danca & Michal Fečkan & Nikolay Kuznetsov & Guanrong Chen, 2021. "Coupled Discrete Fractional-Order Logistic Maps," Mathematics, MDPI, vol. 9(18), pages 1-14, September.
    2. Danca, Marius-F., 2022. "Fractional order logistic map: Numerical approach," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).

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