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Neural Processing Letters

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23-04-2018

Homoclinical Structure of Retarded SICNNs with Rectangular Input Currents

Authors: Mehmet Onur Fen, Fatma Tokmak Fen

Published in: Neural Processing Letters | Issue 2/2019

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Abstract

The dynamics of retarded shunting inhibitory cellular neural networks (SICNNs) with rectangular input currents is investigated from the asymptotic point of view. It is rigorously proved that such networks possess homoclinic and heteroclinic outputs under certain conditions. Illustrative examples that support the theoretical results are provided. Moreover, the extension of the homoclinical structure is numerically demonstrated for unidirectionally coupled retarded SICNNs.

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Poincaré H (1899) Les Methodes Nouvelles de la Mecanique Celeste. Gauthier-Villars et fils, ParisCrossRefMATH

Gonchenko SV, Shil’nikov LP, Turaev DV (1996) Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos 6:15–31MathSciNetCrossRefMATH

Shil’nikov LP (1967) On a Poincaré-Birkhoff problem. Math USSR-Sbornik 3:353–371CrossRefMATH

Smale S (1965) Diffeomorphisms with many periodic points. In: Cairns SS (ed) Differential and combinatorial topology: a symposium in honor of marston morse. Princeton University Press, Princeton, pp 63–70

Bertozzi AL (1988) Heteroclinic orbits and chaotic dynamics in planar fluid flows. Siam J Math Anal 19:1271–1294MathSciNetCrossRefMATH

Chacón R, Bejarano JD (1995) Homoclinic and heteroclinic chaos in a triple-well oscillator. J Sound Vib 186:269–278MathSciNetCrossRefMATH

Rabinovich M, Volkovskii A, Lecanda P, Huerta R, Abarbanel HDI, Laurent G (2001) Dynamical encoding by networks of competing neuron groups: winnerless competition. Phys Rev Lett 87:068102CrossRef

Coullet P, Riera C, Tresser C (2004) A new approach to data storage using localized structures. Chaos 14:193–198MathSciNetCrossRefMATH

Bouzerdoum A, Pinter RB (1993) Shunting inhibitory cellular neural networks: derivation and stability analysis. IEEE Trans Circuits Syst I Fundam Theory Appl 40:215–221MathSciNetCrossRefMATH

10.

Bouzerdoum A, Pinter RB (1990) A shunting inhibitory motion detector that can account for the functional characteristics of fly motion-sensitive interneurons. In: Proceedings of IJCNN international joint conference on neural networks, San Diego, CA, USA, pp 149–153

11.

Carpenter GA, Grossberg S (1988) The ART of adaptive pattern recognition by a self-organizing neural network. Computer 21:77–88CrossRef

12.

Fukushima K (1989) Analysis of the process of visual pattern recognition by the neocognitron. Neural Netw 2:413–420CrossRef

13.

Jernigan ME, McLean GF (1992) Lateral inhibition and image processing. In: Nabet B, Printer RB (eds) Nonlinear vision. CRC Press, Boca Raton, pp 451–462

14.

Pinter RB, Olberg RM, Warrant E (1989) Luminance adaptation of preferred object size in identified dragonfly movement detectors. In: Proceedings of IEEE international conference on systems, man and cybernetics, pp 682–686

15.

Yao L (2017) Global exponential convergence of neutral type shunting inhibitory cellular neural networks with D operator. Neural Process Lett 45:401–409CrossRef

16.

Peng G, Huang L (2009) Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays. Nonlinear Anal Real World Appl 10:2434–2440MathSciNetCrossRefMATH

17.

Wang P, Li B, Li Y (2015) Square-mean almost periodic solutions for impulsive stochastic shunting inhibitory cellular neural networks with delays. Neurocomputing 167:76–82CrossRef

18.

Huang X, Cao J (2003) Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay. Phys Lett A 314:222–231MathSciNetCrossRefMATH

19.

Zhang A (2017) Pseudo almost periodic solutions for SICNNs with oscillating leakage coefficients and complex deviating arguments. Neural Process Lett 45:183–196CrossRef

20.

Zhang A (2017) Pseudo almost periodic solutions for neutral type SICNNs with D operator. J Exp Theor Artif Intell 29:795–807CrossRef

21.

Zhang A (2017) Almost periodic solutions for SICNNs with neutral type proportional delays and D operators. Neural Process Lett. https://doi.org/10.1007/s11063-017-9631-5

22.

Zhou Q, Shao J (2016) Weighted pseudo-anti-periodic SICNNs with mixed delays. Neural Comput Appl. https://doi.org/10.1007/s00521-016-2582-3

23.

Shao J (2008) Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Phys Lett A 372:5011–5016CrossRefMATH

24.

Liu X (2015) Exponential convergence of SICNNs with delay sand oscillating coefficients in leakage terms. Neurocomputing 168:500–504CrossRef

25.

Chen Z (2013) A shunting inhibitory cellular neural network with leakage delays and continuously distributed delays of neutral type. Neural Comput Appl 23:2429–2434CrossRef

26.

Akhmet M, Fen MO, Kirane M (2016) Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument. Neural Comput Appl 27:2483–2495CrossRef

27.

Zhao C, Wang Z (2015) Exponential convergence of a SICNN with leakage delays and continuously distributed delays of neutral type. Neural Process Lett 41:239–247CrossRef

28.

Ou C (2009) Almost periodic solutions for shunting inhibitory cellular neural networks. Nonlinear Anal Real World Appl 10:2652–2658MathSciNetCrossRefMATH

29.

Peng L, Wang W (2013) Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays in leakage terms. Neurocomputing 111:27–33CrossRef

30.

Gui Z, Ge W (2006) Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses. Chaos 16:033116MathSciNetCrossRefMATH

31.

Akhmet MU, Fen MO (2013) Shunting inhibitory cellular neural networks with chaotic external inputs. Chaos 23:023112MathSciNetCrossRefMATH

32.

Akhmet MU, Fen MO (2015) Attraction of Li-Yorke chaos by retarded SICNNs. Neurocomputing 147:330–342CrossRef

33.

Akhmet M, Fen MO, Kıvılcım A (2016) Li-Yorke chaos generation by SICNNs with chaotic/almost periodic postsynaptic currents. Neurocomputing 173:580–594CrossRef

34.

Fen MO, Akhmet M (2016) Impulsive SICNNs with chaotic postsynaptic currents. Discret Contin Dyn Syst Ser B 21:1119–1148MathSciNetCrossRefMATH

35.

Fen MO, Tokmak Fen F (2017) SICNNs with Li-Yorke chaotic outputs on a time scale. Neurocomputing 237:158–165CrossRef

36.

Marcus CM, Westervelt RM (1989) Stability of analog neural networks with delay. Phys Rev A 39:347–359MathSciNetCrossRef

37.

Cao J (2000) Global asymptotic stability of neural networks with transmission delays. Int J Syst Sci 31:1313–1316CrossRefMATH

38.

Roska T, Chua LO (1992) Cellular neural networks with non-linear and delay-type template elements and non-uniform grids. Int J Circuit Theory Appl 20:469–481CrossRefMATH

39.

Akhmet MU (2008) Hyperbolic sets of impact systems. Dyn Contin Discrete Impuls Syst Ser A Math. Anal. 15 (Suppl. S1):1-2. In: Proceedings of the \(5\)th international conference on impulsive and hybrid dynamical systems and applications, Beijing, Watan Press

40.

Akhmet MU (2010) Homoclinical structure of the chaotic attractor. Commun Nonlinear Sci Numer Simul 15:819–822MathSciNetCrossRefMATH

41.

Akhmet MU (2009) Devaney’s chaos of a relay system. Commun Nonlinear Sci Numer Simul 14:1486–1493MathSciNetCrossRefMATH

42.

Fen MO, Tokmak Fen F (2017) Homoclinic and heteroclinic motions in hybrid systems with impacts. Math Slovaca 67:1179–1188MathSciNetMATH

43.

Chen SS (2009) Delayed transiently chaotic neural networks and their application. Chaos 19:033125CrossRefMATH

44.

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

45.

Zou F, Katérle A, Nossek JA (1993) Homoclinic and heteroclinic orbits of the three-cell cellular neural networks. IEEE Trans Circuits Syst I Fundam Theory Appl 40:843–848CrossRefMATH

46.

Shil’nikov LP (1969) On a new type of bifurcation of multidimensional dynamical systems. Sov Math Dokl 10:1368–1371MATH

47.

Wang T, He X, Huang T (2016) Complex dynamical behavior of neural networks in circuit implementation. Neurocomputing 190:95–106CrossRef

48.

Izhikevich EM (1999) Weakly connected quasi-periodic oscillators, FM interactions, and multiplexing in the brain. SIAM J Appl Math 59:2193–2223MathSciNetCrossRefMATH

49.

Hoppensteadt FC, Izhikevich EM (1997) Weakly connected neural networks. Springer, New YorkCrossRefMATH

50.

Pasemann F, Hild M, Zahedi K (2003) SO(2)-Networks as neural oscillators. In: Mira J, Álvarez JR (eds) Computational methods in neural modeling. IWANN 2003. Lecture notes in computer science, vol 2686. Springer, Berlin, Heidelberg

51.

Tyagi BK, Mehotra RK (1976) Special type of emitter-coupled rectangular pulse generator. In: IEE-IERE Proceedings, India, pp 223–227

52.

Ping W, Jiali B, Hong W, Huiping W (2003) Multi-pulse generator for electroporation. In: Proceedings of the 25th annual international conference of the IEEE EMBS, Cancun, Mexico, September 17-21, pp 2970–2973

53.

Driver RD (1977) Ordinary and delay differential equations. Springer, New YorkCrossRefMATH

54.

Filippov AF (1988) Differential equations with discontinuous righthand sides: control systems. Springer, BerlinCrossRef

55.

Akhmet M (2011) Nonlinear hybrid continuous/discrete-time models. Atlantis Press, ParisCrossRefMATH

56.

Zhang Z, Quan Z (2015) Global exponential stability via inequality technique for inertial BAM neural networks with time delays. Neurocomputing 151:1316–1326CrossRef

57.

Hale J, Koçak H (1991) Dynamics and Bifurcations. Springer, New YorkCrossRefMATH

58.

Avrutin V, Schenke B, Gardini L (2015) Calculation of homoclinic and heteroclinic orbits in 1D maps. Commun Nonlinear Sci Numer Simul 22:1201–1214MathSciNetCrossRefMATH

Title: Homoclinical Structure of Retarded SICNNs with Rectangular Input Currents
Authors: Mehmet Onur Fen
Fatma Tokmak Fen
Publication date: 23-04-2018
Publisher: Springer US
Published in: Neural Processing Letters / Issue 2/2019
Print ISSN: 1370-4621
Electronic ISSN: 1573-773X
DOI: https://doi.org/10.1007/s11063-018-9832-6

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