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Hopfield-Type Neural Networks with Poincaré Chaos

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Smart Applications with Advanced Machine Learning and Human-Centred Problem Design (ICAIAME 2021)

Part of the book series: Engineering Cyber-Physical Systems and Critical Infrastructures ((ECPSCI,volume 1))

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Abstract

The paper focuses on the problem of unpredictable oscillations for the Hopfield-type neural networks. Since the unpredictable dynamics is associated with Poincaré chaos, the importance of the motions is indisputable for problems of artificial intelligence and deep learning. The presence of chaos in each coordinate of the state space is productive in applied problems. This is why, we consider the phenomenon of the unpredictability for each coordinate of the network. The theoretical results have been illustrated with numerical analysis.

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References

  1. Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci U S A 79:2554–2558

    Article  MathSciNet  MATH  Google Scholar 

  2. Dong Q, Matsui K, Huang X (2002) Existence and stability of periodic solutions for Hopfield neural network equations with periodic input. Nonlinear Anal 49:471–479

    Article  MathSciNet  MATH  Google Scholar 

  3. Akhmet M, Karacaören M (2018) A Hopfield neural network with multi-compartmental activation. Neural Comput Appl 29:815–822

    Article  Google Scholar 

  4. Akhmet MU, Arugaslan D, Yilmaz E (2010) Stability analysis of recurrent neural networks with piecewise constant argument of generalized type. Neural Netw 23:805–811

    Article  MATH  Google Scholar 

  5. Jin D, Peng J (2009) A new approach for estimating the attraction domain for Hopfield-type neural networks. Neural Comput 21:101–120

    Article  MathSciNet  MATH  Google Scholar 

  6. Aihara K, Takabe T, Toyoda M (1990) Chaotic neural networks. Phys Lett A 6:333–340

    Article  MathSciNet  Google Scholar 

  7. Das A, Roy AB, Das P (2000) Chaos in a three dimensional neural network. Appl Math Model 24:511–522

    Article  MATH  Google Scholar 

  8. Yuan Q, Li QD, Yang X-S (2009) Horseshoe chaos in a class of simple Hopfield neural networks. Chaos Solitons Fractals 39:1522–1529

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu K, Maidana JP, Castro S et al (2018) Synchronization transition in neuronal networks composed of chaotic or non-chaotic oscillators. Sci Rep 8:8370

    Article  Google Scholar 

  10. Sussillo D, Abbott LF (2009) Generating coherent patterns of activity from chaotic neural networks. Neuron 63:544–557

    Article  Google Scholar 

  11. Liu Q, Zhang S (2012) Adaptive lag synchronization of chaotic Cohen-Grossberg neural networks with discrete delays. Chaos 22:033123

    Article  MathSciNet  MATH  Google Scholar 

  12. Ke Q, Oommen J (2014) Logistic neural networks: their chaotic and pattern recognition propertie. Neurocomputing 125:184–194

    Article  Google Scholar 

  13. He G, Chen L, Aihara K (2008) Associative memory with a controlled chaotic neural network. Neurocomputing 71:2794–2805

    Article  Google Scholar 

  14. Akhmet M, Fen MO (2017) Poincaré chaos and unpredictable functions. Commun Nonlinear Sci Numer Simul 48:85–94

    Article  MathSciNet  MATH  Google Scholar 

  15. Akhmet M, Fen MO (2016) Unpredictable points and chaos. Commun Nonlinear Sci Numer Simul 40:1–5

    Article  MathSciNet  MATH  Google Scholar 

  16. Akhmet M, Fen MO (2018) Non-autonomous equations with unpredictable solutions. Commun Nonlinear Sci Numer Simul 59:657–670

    Article  MathSciNet  MATH  Google Scholar 

  17. Akhmet M, Fen MO, Tleubergenova M, Zhamanshin A (2019) Unpredictable solutions of linear differential and discrete equations. Turk J Math 43:2377–2389

    Article  MathSciNet  MATH  Google Scholar 

  18. Akhmet M, Tleubergenova M, Zhamanshin A (2020) Quasilinear differential equations with strongly unpredictable solutions. Carpathion J Math 36:341–349

    Article  MathSciNet  MATH  Google Scholar 

  19. Akhmet M, Tleubergenova M, Zhamanshin A (2019) Poincaré chaos for a hyperbolic quasilinear system. Miskolc Math Notes 20:33–44

    Article  MathSciNet  MATH  Google Scholar 

  20. Akhmet MU, Fen MO, Alejaily EM (2020) Dynamics with Chaos and fractals. Springer, Switzerland

    Book  MATH  Google Scholar 

  21. Akhmet MU (2021) Domain structured dynamics: unpredictability, chaos randomness, fractals, differential equations and neural networks. IOP Publishing, UK

    Book  Google Scholar 

Download references

Acknowledgements

MA is supported by 2247-A National Leading Researchers Program of e8ÜBİTAK, Turkey, N 120C138. MT, RS and ZN are supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (grants No.AP08856170 and No. AP09258737).

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Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University, Ankara, Turkey

    Marat Akhmet

  2. Department of Mathematics, Suleyman Demirel University, Isparta, 32260, Turkey

    Duygu Aruğaslan Çinçin

  3. Department of Mathematics, K. Zhubanov Aktobe Regional University, Aktobe, 030000, Kazakhstan

    Madina Tleubergenova, Roza Seilova & Zakhira Nugayeva

  4. Institute of Information and Computational Technologies CS MES RK, Almaty, 050000, Kazakhstan

    Madina Tleubergenova, Roza Seilova & Zakhira Nugayeva

Authors
  1. Marat Akhmet
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  2. Duygu Aruğaslan Çinçin
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  3. Madina Tleubergenova
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  4. Roza Seilova
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  5. Zakhira Nugayeva
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Corresponding author

Correspondence to Marat Akhmet .

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Akhmet, M., Çinçin, D.A., Tleubergenova, M., Seilova, R., Nugayeva, Z. (2023). Hopfield-Type Neural Networks with Poincaré Chaos. In: Smart Applications with Advanced Machine Learning and Human-Centred Problem Design. ICAIAME 2021. Engineering Cyber-Physical Systems and Critical Infrastructures, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-031-09753-9_42

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