Mixed-coexistence of periodic orbits and chaotic attractors in an inertial neural system with a nonmonotonic activation function

Zigen Song; Jian Xu; Bin Zhen

doi:10.3934/mbe.2019320

Mathematical Biosciences and Engineering, 2019, 16(6): 6406-6425. doi: 10.3934/mbe.2019320. Research article

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Mixed-coexistence of periodic orbits and chaotic attractors in an inertial neural system with a nonmonotonic activation function

Zigen Song, Jian Xu, Bin Zhen

1 College of Information Technology, Shanghai Ocean University, Shanghai, 201306, P.R. China
2 School of Aerospace and Mechanics Engineering, Tongji University, Shanghai 200092, P.R. China
3 School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 200093, P.R. China

Received: , Accepted: , Published:

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In this paper, we construct an inertial two-neuron system with a non-monotonic activation function. Theoretical analysis and numerical simulation are employed to illustrate the complex dynamics. It is found that the neural system exhibits the mixed coexistence with periodic orbits and chaotic attractors. To this end, the equilibria and their stability are analyzed. The system parameters are divided into some regions with the different number of equilibria by the static bifurcation curve. Then, employing some numerical simulations, including the phase portraits, Lyapunov exponents, bifurcation diagrams, and the sensitive dependence to initial values, we find that the system generates two coexisting single-scroll chaotic attractors via the period-doubling bifurcation. Further, the single-scroll chaos will evolve into the double-scroll chaotic attractor. Finally, to view the global evolutions of dynamical behavior, we employ the combined bifurcation diagrams including equilibrium points and periodic orbits. Many types of multistability are presented, such as the bistable periodic orbits, multistable periodic orbits, and multistable chaotic attractors with multi-periodic orbits. The phase portraits and attractor basins are shown to verify the coexisting attractors. Additionally, transient chaos in neural system is observed by phase portraits and time histories.

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Keywords: inertial neuron system; nonmonotonic activation function; multistability; attractor merging crisis; period-doubling bifurcation; transient chaos

Citation: Zigen Song, Jian Xu, Bin Zhen. Mixed-coexistence of periodic orbits and chaotic attractors in an inertial neural system with a nonmonotonic activation function. Mathematical Biosciences and Engineering, 2019, 16(6): 6406-6425. doi: 10.3934/mbe.2019320

References:

1. A. N. Pisarchik and U. Feudel, Control of multistability, Phys. Rep., 540(2014), 167–218.
2. E. V. Felk, A. P. Kuznetsov and A. V. Savin, Multistability and transition to chaos in the degenerate Hamiltonian system with weak nonlinear dissipative perturbation, Physica A, 410(2014), 561–572.
3. Z. G. Song and J. Xu, Codimension-two bursting analysis in the delayed neural system with external stimulations, Nonlinear Dyn., 67(2012), 309–328.
4. J. L. Schwartz, N. Grimault, J. M. Hupé, et al., Multistability in perception: sensory modalities, an overview, Philos. Trans. R. Soc. B, 367(2012), 896–905.
5. P. A. Tass and C. Hauptmann, Therapeutic modulation of synaptic connectivity with desynchronizing brain stimulation, Int. J. Psychophysiol., 64(2007), 53–61.
6. F. Fröhlich and M. Bazhenov, Coexistence of tonic firing and bursting in cortical neurons, Phys. Rev. E, 74(2006), 031922.
7. H. A. Lechner, D. A. Baxter, J. W. Clark, et al., Bistability and its regulation by serotonin in the endogenously bursting neuron R15 in Aplysia, J. Neurophysiol., 75(1996), 957–962.
8. J. P. Newman and R. J. Butera, Mechanism, dynamics and biological existence of multistability in a large class of bursting neurons, Chaos, 20(2010), 023118.
9. J. Foss, A. Longtin, B. Mensour, et al., Multistability and delayed recurrent loops, Phys. Rev. Lett. 76(1996), 708–711.
10. C. Masoller, M. C. Torrent and J. García-Ojalvo, Dynamics of globally delay-coupled neurons displaying subthreshold oscillations, Phil. Trans. R. Soc. A, 367(2009), 3255–3266.
11. N. Buric and D. Rankovic, Bursting neurons with coupling delays, Phys. Lett. A, 363(2007), 282–289.
12. N. Buric, I. Grozdanovic and N. Vasovic, Excitable systems with internal and coupling delays, Chaos Solit. Fract., 36(2008), 853–861.
13. Z. G. Song, K. Yang, J. Xu, et al., Multiple pitchfork bifurcations and multiperiodicity coexistences in a delay-coupled neural oscillator system with inhibitory-to-inhibitory connection, Commun. Nonlinear Sci. Numer. Simulat., 29(2015), 327–345.
14. X. Mao, Bifurcation, synchronization, and multistability of two interacting networks with multiple time delays, Int. J. Bifurcat. Chaos, 26(2016), 1650156.
15. X. Mao and Z. Wang, Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays, Nonlinear Dyn., 82(2015), 1551–1567.
16. M. S. Baptista, R. M. Szmoski, R. F. Pereira, et al., Chaotic, informational and synchronous behaviour of multiplex networks, Sci. Rep., 6(2016), 22617.
17. G. He, L. Chen, K. Aihara, Associative memory with a controlled chaotic neural network, Neurocomputing, 71(2008), 2794–2805.
18. X. S Yang and Y. Huang, Complex dynamics in simple Hopfield neural networks, Chaos, 16(2006), 033114.
19. W. Z. Huang and Y. Huang, Chaos, bifurcation and robustness of a class of Hopfield neural networks, Int. J. Bifurcat. Chaos, 21(2011), 885–895.
20. X. S. Yang and Q. Yuan, Chaos and transient chaos in simple Hopfield neural networks, Neurocomputing, 69(2005), 232–241.
21. C. G. Li and G. R. Chen, Coexisting chaotic attractors in a single neuron model with adapting feedback synapse, Chaos Solit. Fract., 23(2005), 1599–1604.
22. C. Li and J. C. Sprott, Coexisting hidden attractors in a 4-D simplified Lorenz system, Int. J. Bifurcat. Chaos, 24(2014), 1450034.
23. J. Kengne, Z. T. Njitacke, A. Nguomkam Negou, et al., Coexistence of multiple attractors and crisis route to chaos in a novel chaotic Jerk circuit, Int. J. Bifurcat. Chaos, 26(2016), 1650081.
24. Z. T. Njitacke, J. kengne, H. B. Fotsin, et al., Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit, Chaos Solit. Fract., 91(2016), 180–197.
25. J. Kengne, Z. Njitacke Tabekoueng and H. B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing–Holmes type chaotic oscillators, Commun. Nonlinear Sci. Numer. Simulat., 36(2016), 29–44.
26. B. C. Bao, Q. D. Li, N. Wang, et al., Multistability in Chua's circuit with two stable node-foci, Chaos, 26(2016), 043111.
27. J. Kengne, Z. Njitacke Tabekoueng, V. Kamdoum Tamba, et al., Periodicity, chaos, and multiple attractors in a memristor-based Shinriki's circuit, Chaos, 25(2015), 103126.
28. A. P. Kuznetsov, S. P. Kuznetsov, E. Mosekilde, et al., Co-existing hidden attractors in a radio-physical oscillator, J. Phys. A: Math. Theor., 48(2015), 125101.
29. A. Massoudi, M. G. Mahjani and M. Jafarian, Multiple attractors in Koper–Gaspard model of electrochemical, J. Electroanalyt. Chem., 647(2010), 74–86.
30. C. Y. Cheng, Coexistence of multistability and chaos in a ring of discrete neural network with delays, Int. J. Bifurcat. Chaos, 20(2010), 1119–1136.
31. J. Li, F. Liu, Z. H. Guan, et al., A new chaotic Hopfield neural network and its synthesis via parameter switchings, Neurocomputing, 117(2013), 33–39.
32. W. C. Schieve, A. R. Bulsara and G. M. Davis, Single effective neuron, Phys. Rev. A, 43(1991), 2613–2623.
33. K. L. Badcock and R. M. Westervelt, Dynamics of simple electronic neural networks, Physical D, 28(1987), 305–316.
34. D. W. Wheeler and W. C. Schieve, Stability and chaos in an inertial two-neuron system, Physical D, 105(1997), 267–284.
35. Q. Liu, X. F. Liao, S. T. Guo, et al., Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation, Nonlinear Anal. Real World Appl., 10(2009), 2384–2395.
36. Q. Liu, X. F. Liao, Y. Liu, et al., Dynamics of an inertial two-neuron system with time delay, Nonlinear Dyn., 58(2009), 573–609.
37. Z. G. Song and J. Xu, Stability switches and Bogdanov–Takens bifurcation in an inertial two-neurons coupling system with multiple delays, Sci. China Tech. Sci., 57(2014), 893–904.
38. Z. G. Song, J. Xu and B. Zhen, Multitype activity coexistence in an inertial two-neuron system with multiple delays, Int. J. Bifurcat. Chaos, 25(2015), 1530040.
39. B. Crespi, Storage capacity of non-monotonic neurons, Neural Netw., 12(1999), 1377–1389.
40. S. Yao, L. Ding, Z. Song, et al., Two bifurcation routes to multiple chaotic coexistence in an inertial two-neural system with time delay, Nonlinear Dyn., 95(2019), 1549–1563.
41. Y. C. Lai and T. Tel, Transient Chaos: Complex Dynamics on Finite-Time Scales, Springer, New York, 2011.

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