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Dynamical analysis of a novel 4-neurons based Hopfield neural network: emergences of antimonotonicity and coexistence of multiple stable states

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Abstract

In this contribution, we investigate the dynamics of a novel model of 4-neurons based Hopfield neural networks. Our analyses highlight complex phenomena such as chaotic and periodic behaviors which have been classified by Panahi et al. (Chaos Solitons Fractals 105:150–156, 2017) as some brain behaviors. More interestingly, it has been revealed several sets of synaptic weights matrix for which the proposed HNNs displays multiple coexisting stable states including two, four and six disjoined orbits. Basins of attraction of coexisting stable states have been computed showing different regions in which each solution can be captured. Beside the presence of coexisting bifurcations, the model displays remerging Feigenbaum trees bifurcations also known as antimonotonicity for some judicious sets of synaptic weights. PSpice investigations are finally used to confirm results of the theoretical investigations.

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References

  1. Hopfield J (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088–3092

    Article  MATH  Google Scholar 

  2. Qiu H, Chen X, Liu W, Zhou G, Wang Y, Lai J (2012) A fast l1-solver and its applications to robust face recognition. J Ind Manag Optim 8:163–178

    MathSciNet  MATH  Google Scholar 

  3. Wang YJ, Zhou GL, Caccetta L, Liu WQ (2011) An alternative Lagrange-dual based algorithm for sparse signal reconstruction. IEEE Trans Signal Process 59:1895–1901

    Article  Google Scholar 

  4. Yang XS, Yuan Q (2005) Chaos and transient chaos in simple Hopfield neural networks. Neurocomputing 69:232–241

    Article  Google Scholar 

  5. Panahi S, Aram Z, Jafari S, Ma M, Sprott JC (2017) Modeling of epilepsy based on chaotic artificial neural network. Chaos Solitons Fractals 105:150–156

    Article  MathSciNet  Google Scholar 

  6. Sprott JC, Wildenberg JC, Vano JA (2005) A simple spatiotemporal chaotic Lotka–Volterra model. Chaos Solitons Fractals 26:1035–1043

    Article  MathSciNet  MATH  Google Scholar 

  7. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544

    Article  Google Scholar 

  8. Izhikevich EM (2004) Which model to use for cortical spiking neurons. IEEE Trans Neuron Netw 15:1063–1070

    Article  Google Scholar 

  9. Izhikevich EM (2007) Systems in neuroscience. MIT Press, Cambridge

    Google Scholar 

  10. Morris C, Leca H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213

    Article  Google Scholar 

  11. Güçlü U, van Gerven AJ (2017) Modeling the dynamics of human brain activity with recurrent neural networks. Front Comput Neurosci 11(7):1–14

    Google Scholar 

  12. Li Q, Yang X (2005) Complex dynamics in a simple Hopfield-type neural network. In: Wang J, Liao X, Yi Z (eds) Advances in neural networks—ISNN 2005. ISNN. Lecture Notes in Computer Science, vol 3496. Springer, Berlin, pp 357–362

  13. Zheng P, Tang W, Zang J (2010) Some novel double-scroll chaotic attractors in Hopfield networks. Neurocomputing 73:2280–2285

    Article  Google Scholar 

  14. Li Q, Tang S, Zeng H, Zhou T (2014) On hyperchaos in a small memristive neural network. Nonlinear Dyn 78:1087–1099

    Article  MATH  Google Scholar 

  15. Danca MF, Kuznets L (2017) Hidden chaotic sets in a Hopfield neural system. Chaos Solitons Fractals 103:144–150

    Article  MathSciNet  MATH  Google Scholar 

  16. Bao B, Qian H, Xu Q, Chen M, Wang J, Yu Y (2017) Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Front Comput Neurosci 11(81):1–14

    Google Scholar 

  17. Bao B, Qian H, Wang J, Xu Q, Chen M, Wu H, Yu Y (2017) Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network. Nonlinear Dyn. https://doi.org/10.1007/s11071-017-3808-3

    MathSciNet  Google Scholar 

  18. Njitacke ZT, Kengne J, Fotsin HB (2018) A plethora of behaviors in a memristor based Hopfield neural networks (HNNs). Int J Dyn Control. https://doi.org/10.1007/s40435-018-0435-x

    Google Scholar 

  19. Njitacke ZT, Kengne J (2018) Complex dynamics of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight: coexistence of multiple attractors and remerging Feigenbaum trees. Int J Electron Commun (AEÜ) 93:242–252

    Article  Google Scholar 

  20. Kengne J (2016) On the dynamics of Chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors. Nonlinear Dyn. https://doi.org/10.1007/s11071-016-3047-z

    Google Scholar 

  21. Njitacke ZT, Kengne J, Fotsin HB, Negou AN, Tchiotsop D (2016) Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit. Chaos Solitons Fractals 91:180–197

    Article  MATH  Google Scholar 

  22. Kengne J, Njitacke ZT, Negou AN, Fouodji MT, Fotsin HB (2015) Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. Int J Bifurc Chaos 25(4):1550052

    Article  MATH  Google Scholar 

  23. Kengne J, Njitacke ZT, Fotins HB (2015) Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. https://doi.org/10.1007/s11071-015-2364-y

    Google Scholar 

  24. Njitacke ZT, Kengne J, Kamdjeu Kengne L (2017) Antimonotonicity, chaos and multiple coexisting attractors in a simple hybrid diode-based jerk circuit. Chaos Solitons Fractals 105:77–91

    Article  MathSciNet  Google Scholar 

  25. Kengne J, Njitacke ZT, Fotins HB (2016) Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing–Holmes type chaotic oscillators. Commun Nonlinear Sci Numer Simul 36:29–44

    Article  MathSciNet  Google Scholar 

  26. Kengne J, Njitacke ZT, Kamdoum VT, Negou AN (2015) Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos Interdiscip J Nonlinear Sci 25:103126

    Article  MathSciNet  MATH  Google Scholar 

  27. Hilborn RC (1994) Chaos and nonlinear dynamics—an introduction for scientists and engineers. Oxford University Press, Oxford

    MATH  Google Scholar 

  28. Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics: analytical, computational and experimental methods. Wiley, New York

    Book  MATH  Google Scholar 

  29. Nik HS, Effati S, Saberi-Nadjafi J (2015) Ultimate bound sets of a hyperchaotic system and its application in chaos synchronization. Complexity 20:30–44

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen M, Xu Q, Lin Y, Bao BC (2017) Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit. Nonlinear Dyn 87:789–802

    Article  Google Scholar 

  31. Singh JP, Roy BK (2017) Hidden attractors in a new complex generalized Lorenz hyperchaotic system, its synchronization using adaptive contraction theory, circuit validation and application. Nonlinear Dyn. https://doi.org/10.1007/s11071-018-4062-z

    Google Scholar 

  32. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D Nonlinear Phenom 16(3):285–317

    Article  MathSciNet  MATH  Google Scholar 

  33. Parlitz U, Lauterborn W (1987) Period-doubling cascades and devil’s staircases of the driven van der Pol oscillator. Phys Rev A 36:1428

    Article  Google Scholar 

  34. Kocarev L, Halle K, Eckert K, Chua L (1993) Experimental observation of antimonotonicity in Chua’s circuit. Int J Bifurc Chaos 3:1051–1055

    Article  MATH  Google Scholar 

  35. Kengne J, Folifack Signing VR, Chedjou JC, Leutcho GD (2017) Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors. Int J Dyn Control. https://doi.org/10.1007/s40435-017-0318-6.15

    Google Scholar 

  36. Kengne J, Jafari S, Njitacke ZT, Yousefi Azar Khanian M, Cheukem A (2017) Dynamic analysis and electronic circuit implementation of a novel 3D autonomous system without linear terms. Commun Nonlinear Sci Numer Simul. https://doi.org/10.1016/j.cnsns.2017.04.017

    Google Scholar 

  37. Dawson SP, Grebogi C, Yorke JA, Kan I, Koçak H (1992) Antimonotonicity: inevitable reversals of period-doubling cascades. Phys Lett A 162:249–254

    Article  MathSciNet  Google Scholar 

  38. Bier M, Boutis TC (1984) Remerging Feigenbaum trees in dynamical systems. Phys Lett A 104:239–244

    Article  MathSciNet  Google Scholar 

  39. Dawson SP (1993) Geometric mechanism for antimonotonicity in scalar maps with two critical points. Phys Rev E 48:1676–1680

    Article  MathSciNet  Google Scholar 

  40. Pham VT, Jafari S, Vaidyanathan S, Volos C, Wang X (2015) A novel memristive neural network with hidden attractors and its circuitry implementation. Sci China Technol Sci. https://doi.org/10.1007/s11431-015-5981-2

    Google Scholar 

  41. Njitacke ZT, Kengne J, Negou AN (2017) Dynamical analysis and electronic circuit realization of an equilibrium free 3D chaotic system with a large number of coexisting attractors. Optik 130:356–364

    Article  Google Scholar 

  42. Filali RL, Benrejeb M, Borne P (2014) Observer-based secure communication design using discrete-time hyperchaotic systems. Comm Nonlinear Sci Numer Simul 19:1424

    Article  MathSciNet  Google Scholar 

  43. Volos C, Kyprianidis IM, Stouboulos IN (2013) Image encryption process based on chaotic synchronization phenomena. Sig Process 93:1328–1340

    Article  Google Scholar 

  44. Nguimdo RM, Tchitnga R, Woafo P (2013) Dynamics of coupled simplest chaotic two-component electronic circuits and its potential application to random bit generation. Chaos 23:043122

    Article  MathSciNet  Google Scholar 

  45. Fortuna L, Frasca M, Rizzo A (2003) Chaotic pulse position modulation to improve the efficiency of sonar sensors. IEEE Trans Instrum Meas 52(6):1809

    Article  Google Scholar 

  46. Patel MS, Patel U, Sen A, Sethia GC, Hens C, Dana SK, Feudel U, Showalter K, Ngonghala CN, Amritkar RE (2014) Experimental observation of extreme multistability in an electronic system of two coupled Rossler oscillators. Phys Rev E 89:022918

    Article  Google Scholar 

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Authors and Affiliations

  1. Unité de Recherche d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

    Z. T. Njitacke, J. Kengne & T. Fonzin Fozin

  2. Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (LAMACETS), Department of Physics, University of Dschang, P. O. Box 67, Dschang, Cameroon

    Z. T. Njitacke, T. Fonzin Fozin & H. B. Fotsin

  3. Department of Chemistry, Faculty of Science, The University of Maroua, P. O. Box 46, Maroua, Cameroon

    B. P. Leutcha

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  1. Z. T. Njitacke
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  2. J. Kengne
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  3. T. Fonzin Fozin
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  4. B. P. Leutcha
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  5. H. B. Fotsin
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Correspondence to Z. T. Njitacke.

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Njitacke, Z.T., Kengne, J., Fozin, T.F. et al. Dynamical analysis of a novel 4-neurons based Hopfield neural network: emergences of antimonotonicity and coexistence of multiple stable states. Int. J. Dynam. Control 7, 823–841 (2019). https://doi.org/10.1007/s40435-019-00509-w

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