# American Institute of Mathematical Sciences

2016, 21(4): 1119-1148. doi: 10.3934/dcdsb.2016.21.1119

## Impulsive SICNNs with chaotic postsynaptic currents

 1 Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, Turkey

Received  December 2014 Revised  December 2015 Published  March 2016

In the present study, we investigate the dynamics of shunting inhibitory cellular neural networks (SICNNs) with impulsive effects. We give a mathematical description of the chaos for the multidimensional dynamics of impulsive SICNNs, and prove its existence rigorously by taking advantage of the external inputs. The Li-Yorke definition of chaos is used in our theoretical discussions. In the considered model, the impacts satisfy the cell and shunting principles. This enriches the applications of SICNNs and makes the analysis of impulsive neural networks deeper. The technique is exceptionally useful for SICNNs with arbitrary number of cells. We make benefit of unidirectionally coupled SICNNs to exemplify our results. Moreover, the appearance of cyclic irregular behavior observed in neuroscience is numerically demonstrated for discontinuous dynamics of impulsive SICNNs.
Citation: Mehmet Onur Fen, Marat Akhmet. Impulsive SICNNs with chaotic postsynaptic currents. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1119-1148. doi: 10.3934/dcdsb.2016.21.1119
##### References:
 [1] M. Abeles, Neural Codes for Higher Brain Functions,, in Information Processing by the Brain (ed. H.J. Markowitsch), (1988). [2] S. Ahmad and I. M. Stamova, Global exponential stability for impulsive cellular neural networks with time-varying delays,, Nonlinear Anal.: Theory Methods Appl., 69 (2008), 786. doi: 10.1016/j.na.2008.02.067. [3] K. Aihara and G. Matsumoto, Chaotic oscillations and bifurcations in squid giant axons,, in Chaos (ed. A.V. Holden), (1986), 257. [4] K. Aihara, T. Takebe and M. Toyoda, Chaotic neural networks,, Phys. Lett. A, 144 (1990), 333. doi: 10.1016/0375-9601(90)90136-C. [5] M. U. Akhmet, Creating a chaos in a system with relay,, Int. J. Qual. Theory Differ. Equat. Appl., 3 (2009), 3. [6] M. U. Akhmet, Devaney's chaos of a relay system,, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 1486. doi: 10.1016/j.cnsns.2008.03.013. [7] M. U. Akhmet, Li-Yorke chaos in the system with impacts,, J. Math. Anal. Appl., 351 (2009), 804. doi: 10.1016/j.jmaa.2008.11.015. [8] M. U. Akhmet, Homoclinical structure of the chaotic attractor,, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 819. doi: 10.1016/j.cnsns.2009.05.042. [9] M. Akhmet, Principles of Discontinuous Dynamical Systems,, New York, (2010). doi: 10.1007/978-1-4419-6581-3. [10] M. Akhmet, Nonlinear Hybrid Continuous/Discrete-Time Models,, Atlantis Press, (2011). doi: 10.2991/978-94-91216-03-9. [11] M. U. Akhmet and M. O. Fen, Chaotic period-doubling and OGY control for the forced Duffing equation,, Commun. Nonlinear. Sci. Numer. Simulat., 17 (2012), 1929. doi: 10.1016/j.cnsns.2011.09.016. [12] M. U. Akhmet and M. O. Fen, Shunting inhibitory cellular neural networks with chaotic external inputs,, Chaos, 23 (2013). doi: 10.1063/1.4805022. [13] M. U. Akhmet and M. O. Fen, Replication of chaos,, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2626. doi: 10.1016/j.cnsns.2013.01.021. [14] M. Akhmet and E. Yílmaz, Neural Networks with Discontinuous/Impact Activations,, Springer, (2014). doi: 10.1007/978-1-4614-8566-7. [15] M. U. Akhmet and M. O. Fen, Entrainment by chaos,, J. Nonlinear Sci., 24 (2014), 411. doi: 10.1007/s00332-014-9194-9. [16] M. Akhmet and M. O. Fen, Chaotification of impulsive systems by perturbations,, Int. J. Bifurcation and Chaos, 24 (2014). doi: 10.1142/S0218127414500783. [17] M. Akhmet and M. O. Fen, Generation of cyclic/toroidal chaos by Hopfield neural networks,, Neurocomputing, 145 (2014), 230. doi: 10.1016/j.neucom.2014.05.038. [18] M. Akhmet and M. O. Fen, Replication of Chaos in Neural Networks, Economics and Physics,, Springer-Verlag, (2016). doi: 10.1007/978-3-662-47500-3. [19] E. Akin and S. Kolyada, Li-Yorke sensitivity,, Nonlinearity, 16 (2003), 1421. doi: 10.1088/0951-7715/16/4/313. [20] P. Balasubramaniam, M. Kalpana and R. Rakkiyappan, Stationary oscillation of interval fuzzy cellular neural networks with mixed delays under impulsive perturbations,, Neural Comput. & Applic., 22 (2013), 1645. doi: 10.1007/s00521-012-0816-6. [21] P. Balasubramaniam and P. Muthukumar, Synchronization of chaotic systems using feedback controller: An application to Diffie-Hellman key exchange protocol and ElGamal public key cryptosystem,, J. Egyptian Math. Soc., 22 (2014), 365. doi: 10.1016/j.joems.2013.10.003. [22] R. Barrio, M. A. Martinez, S. Serrano and A. Shilnikov, Macro- and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons,, Chaos, 24 (2014). doi: 10.1063/1.4882171. [23] A. Bouzerdoum and R. B. Pinter, 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, (1990), 149. doi: 10.1109/IJCNN.1990.137560. [24] A. Bouzerdoum, B. Nabet and R. B. Pinter, Analysis and analog implementation of directionally sensitive shunting inhibitory cellular neural networks,, in Visual Information Processing: From Neurons to Chips, 1473 (1991), 29. [25] A. Bouzerdoum and R. B. Pinter, Nonlinear lateral inhibition applied to motion detection in the fly visual system,, Nonlinear Vision (ed. R.B. Pinter, (1992), 423. [26] A. Bouzerdoum and R. B. Pinter, Shunting inhibitory cellular neural networks: Derivation and stability analysis,, IEEE Trans. Circuits Systems-I: Fund. Theory and Appl., 40 (1993), 215. doi: 10.1109/81.222804. [27] M. Cai and W. Xiong, Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz and bounded activation functions,, Phys. Lett. A, 362 (2007), 417. doi: 10.1016/j.physleta.2006.10.076. [28] J. Cao, Global stability conditions for delayed CNNs,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 48 (2001), 1330. doi: 10.1109/81.964422. [29] J. Cao and J. Lu, Adaptive synchronization of neural networks with or without time-varying delay,, Chaos, 16 (2006). doi: 10.1063/1.2178448. [30] J. Cao, D. W. C. Ho and X. Huang, LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay,, Nonlinear Anal., 66 (2007), 1558. doi: 10.1016/j.na.2006.02.009. [31] R. Caponetto, M. Lavorgna and L. Occhipinti, Cellular neural networks in secure transmission applications,, in Proceedings of CNNA96: Fourth IEEE International Workshop on Cellular Neural Networks and Their Applications, (1996), 411. doi: 10.1109/CNNA.1996.566609. [32] G. A. Carpenter and S. Grossberg, The ART of adaptive pattern recognition by a self-organizing neural network,, Computer, 21 (1988), 77. doi: 10.1109/2.33. [33] A. Chen and J. Cao, Almost periodic solution of shunting inhibitory CNNs with delays,, Phys. Lett. A, 298 (2002), 161. doi: 10.1016/S0375-9601(02)00469-3. [34] L. Chen and H. Zhao, Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients,, Chaos Soliton. Fract., 35 (2008), 351. doi: 10.1016/j.chaos.2006.05.057. [35] H. N. Cheung, A. Bouzerdoum and W. Newland, Properties of shunting inhibitory cellular neural networks for colour image enhancement,, in Proceedings of 6th International Conference on Neural Information Processing, (1999), 1219. doi: 10.1109/ICONIP.1999.844715. [36] L. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits Syst., 35 (1988), 1257. doi: 10.1109/31.7600. [37] L. O. Chua and L. Yang, Cellular neural networks: Applications,, IEEE Trans. Circuits Syst., 35 (1988), 1273. doi: 10.1109/31.7601. [38] H.-S. Ding, J. Liang and T.-J. Xiao, Existence of almost periodic solutions for SICNNs with time-varying delays,, Phys. Lett. A, 372 (2008), 5411. doi: 10.1016/j.physleta.2008.06.042. [39] M. J. Feigenbaum, Universal behavior in nonlinear systems,, Los Alamos Sci., 1 (1980), 4. [40] W. J. Freeman, Tutorial on neurobiology: From single neurons to brain chaos,, Int. J. Bifurcation and Chaos, 2 (1992), 451. doi: 10.1142/S0218127492000653. [41] K. Fukushima, Analysis of the process of visual pattern recognition by the neocognitron,, Neural Netw., 2 (1989), 413. doi: 10.1016/0893-6080(89)90041-5. [42] W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511815706. [43] J. Guckenheimer and R. A. Oliva, Chaos in the Hodgkin-Huxley Model,, SIAM J. Appl. Dyn. Syst., 1 (2002), 105. doi: 10.1137/S1111111101394040. [44] Z. Gui and W. Ge, Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses,, Chaos, 16 (2006). doi: 10.1063/1.2225418. [45] J. Hale and H. Koçak, Dynamics and Bifurcations,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-4426-4. [46] W. Hu, C. Li and S. Wu, Stochastic robust stability for neutral-type impulsive interval neural networks with distributed time-varying delays,, Neural Comput. Appl., 21 (2012), 1947. doi: 10.1007/s00521-011-0598-2. [47] X. Huang and J. Cao, Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay,, Phys. Lett. A, 314 (2003), 222. doi: 10.1016/S0375-9601(03)00918-6. [48] E. M. Izhikevich, Weakly connected quasi-periodic oscillators, FM interactions, and multiplexing in the brain,, SIAM J. Appl. Math., 59 (1999), 2193. doi: 10.1137/S0036139997330623. [49] S. Jankowski, A. Londei, C. Mazur and A. Lozowski, Synchronization phenomena in 2D chaotic CNN, in Proceedings of CNNA-94 Third IEEE International Workshop on Cellular Neural Networks and Their Applications,, Rome, (1994), 339. [50] M. E. Jernigan and G. F. McLean, Lateral inhibition and image processing,, in Nonlinear Vision (ed. R.B. Pinter, (1992), 451. [51] M. Kalpana and P. Balasubramaniam, Stochastic asymptotical synchronization of chaotic Markovian jumping fuzzy cellular neural networks with mixed delays and Wiener process based on sampled-data control,, Chin. Phys. B, 22 (2013). doi: 10.1088/1674-1056/22/7/078401. [52] E. Kaslik and S. Balint, Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture,, Neural Netw., 22 (2009), 1411. doi: 10.1016/j.neunet.2009.03.009. [53] Q. Ke and B. J. Oommen, Logistic neural networks: Their chaotic and pattern recognition properties,, Neurocomputing, 125 (2014), 184. doi: 10.1016/j.neucom.2012.10.039. [54] A. Khadra, X. Liu and X. Shen, Application of impulsive synchronization to communication security,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 50 (2003), 341. doi: 10.1109/TCSI.2003.808839. [55] P. Kloeden and Z. Li, Li-Yorke chaos in higher dimensions: A review,, J. Differ. Equ. Appl., 12 (2006), 247. doi: 10.1080/10236190600574069. [56] J. Kuroiwa, N. Masutani, S. Nara and K. Aihara, Chaotic wandering and its sensitivity to external input in a chaotic neural network,, in Proceedings of the 9th International Conference on Neural Information Processing (ed. L. Wang, 1 (2002), 353. doi: 10.1109/ICONIP.2002.1202192. [57] J. Lei and Z. Lei, The chaotic cipher based on CNNs and its application in network,, in Proceedings of International Symposium on Intelligence Information Processing and Trusted Computing, (2011), 184. doi: 10.1109/IPTC.2011.54. [58] K. Li, X. Zhang and Z. Li, Global exponential stability of impulsive cellular neural networks with time-varying and distributed delay,, Chaos Soliton. Fract., 41 (2009), 1427. doi: 10.1016/j.chaos.2008.06.003. [59] P. Li, Z. Li, W. A. Halang and G. Chen, Li-Yorke chaos in a spatiotemporal chaotic system,, Chaos Soliton. Fract., 33 (2007), 335. doi: 10.1016/j.chaos.2006.01.077. [60] Y. Li, C. Liu and L. Zhu, Global exponential stability of periodic solution for shunting inhibitory CNNs with delays,, Phys. Lett. A, 337 (2005), 46. doi: 10.1016/j.physleta.2005.01.008. [61] Y. Li and J. Shu, Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales,, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 3326. doi: 10.1016/j.cnsns.2010.11.004. [62] T. Y. Li and J. A. Yorke, Period three implies chaos,, The American Mathematical Monthly, 82 (1975), 985. doi: 10.2307/2318254. [63] W. Lin and J. Ruan, Chaotic dynamics of an integrate-and-fire circuit with periodic pulse-train input,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 50 (2003), 686. doi: 10.1109/TCSI.2003.811015. [64] B. Liu and L. Huang, Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with time-varying delays,, Chaos Soliton. Fract., 31 (2007), 211. doi: 10.1016/j.chaos.2005.09.052. [65] Q. Liu and S. Zhang, Adaptive lag synchronization of chaotic Cohen-Grossberg neural networks with discrete delays,, Chaos, 22 (2012). doi: 10.1063/1.4745212. [66] W. Liu and L. Wang, Variable thresholds in the chaotic cellular neural network,, in Proceedings of International Joint Conference on Neural Networks, (2007), 711. doi: 10.1109/IJCNN.2007.4371044. [67] J. Lu, D. W. C. Ho, J. Cao and J. Kurths, Exponential synchronization of linearly coupled neural networks with impulsive disturbances,, IEEE Trans. Neural Netw., 22 (2011), 329. [68] W. Lu and T. Chen, Synchronization of coupled connected neural networks with delays,, IEEE Trans. Circuits Syst.-I Regul. Pap., 51 (2004), 2491. doi: 10.1109/TCSI.2004.838308. [69] F. R. Marotto, Snap-back repellers imply chaos in $R^n$,, J. Math. Anal. Appl., 63 (1978), 199. doi: 10.1016/0022-247X(78)90115-4. [70] B. L. McNaughton, C. A. Barnes and P. Andersen, Synaptic efficacy and EPSP summation in granule cells of rat fascia dentata studied in vitro,, J. Neurophysiol, 46 (1981), 952. [71] P. Muthukumar and P. Balasubramaniam, Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography,, Nonlinear Dynamics, 74 (2013), 1169. doi: 10.1007/s11071-013-1032-3. [72] P. Muthukumar, P. Balasubramaniam and K. Ratnavelu, Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES),, Nonlinear Dynamics, 77 (2014), 1547. doi: 10.1007/s11071-014-1398-x. [73] S. Nara and P. Davis, Chaotic wandering and search in a cycle-memory neural network,, Prog. Theor. Phys., 88 (1992), 845. doi: 10.1143/ptp/88.5.845. [74] S. Nara, P. Davis, M. Kawachi and H. Totsuji, Chaotic memory dynamics in a recurrent neural network with cycle memories embedded by pseudoinverse method,, Int. J. Bifurcation and Chaos, 5 (1995), 1205. [75] M. Ohta and K. Yamashita, A chaotic neural network for reducing the peak-to-average power ratio of multicarrier modulation,, in Proceedings of International Joint Conference on Neural Networks, 2 (2003), 864. doi: 10.1109/IJCNN.2003.1223803. [76] E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos,, Phys. Rev. Lett., 64 (1990), 1196. doi: 10.1103/PhysRevLett.64.1196. [77] C. Ou, Almost periodic solutions for shunting inhibitory cellular neural networks,, Nonlinear Anal.: Real World Appl., 10 (2009), 2652. doi: 10.1016/j.nonrwa.2008.07.004. [78] L. Pan and J. Cao, Anti-periodic solution for delayed cellular neural networks with impulsive effects,, Nonlinear Anal.: Real World Appl., 12 (2011), 3014. doi: 10.1016/j.nonrwa.2011.05.002. [79] G. Peng and L. Huang, Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays,, Nonlinear Anal.: Real World Appl., 10 (2009), 2434. doi: 10.1016/j.nonrwa.2008.05.001. [80] R. B. Pinter, R. M. Olberg and E. Warrant, Luminance adaptation of preferred object size in identified dragonfly movement detectors,, in Proceedings of IEEE International Conference on Systems, 2 (1989), 682. doi: 10.1109/ICSMC.1989.71382. [81] Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, Commun. Math. Phys., 74 (1980), 189. doi: 10.1007/BF01197757. [82] A. Potapov and M. K. Ali, Robust chaos in neural networks,, Phys. Lett. A, 277 (2000), 310. doi: 10.1016/S0375-9601(00)00726-X. [83] K. Pyragas, Continuous control of chaos by self-controlling feedback,, Phys. Rev. A, 170 (1992), 421. [84] D. J. Rijlaarsdam and V. M. Mladenov, Synchronization of chaotic cellular neural networks based on Rössler cells,, in Proceedings of 8th Seminar on Neural Network Applications in Electrical Engineering, (2006), 41. [85] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations,, World Scientific, (1995). doi: 10.1142/9789812798664. [86] E. Sander and J. A. Yorke, Period-doubling cascades galore,, Ergod. Th. & Dynam. Sys., 31 (2011), 1249. doi: 10.1017/S0143385710000994. [87] J. Shao, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays,, Phys. Lett. A, 372 (2008), 5011. doi: 10.1016/j.physleta.2008.05.064. [88] Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces,, Chaos Soliton. Fract., 22 (2004), 555. doi: 10.1016/j.chaos.2004.02.015. [89] Y. Shi and G. Chen, Discrete chaos in Banach spaces,, Sci. China Ser. A Math., 48 (2005), 222. doi: 10.1360/03ys0183. [90] Y. Shi, P. Zhu and K. Qin, Projective synchronization of different chaotic neural networks with mixed time delays based on an integral sliding mode controller,, Neurocomputing, 123 (2014), 443. doi: 10.1016/j.neucom.2013.07.044. [91] M. Shibasaki and M. Adachi, Response to external input of chaotic neural networks based on Newman-Watts model,, in Proceedings of WCCI 2012 IEEE World Congress on Computational Intelligence (ed. J. Liu, (2012), 1. doi: 10.1109/IJCNN.2012.6252394. [92] C. A. Skarda and W. J. Freeman, How brains make chaos in order to make sense of the world?,, Behav. Brain Sci., 10 (1987), 161. doi: 10.1017/S0140525X00047336. [93] X. Song, X. Xin and W. Huang, Exponential stability of delayed and impulsive cellular neural networks with partially Lipschitz continuous activation functions,, Neural Netw., 29-30 (2012), 29. doi: 10.1016/j.neunet.2012.01.006. [94] I. M. Stamova and R. Ilarionov, On global exponential stability for impulsive cellular neural networks with time-varying delays,, Comput. Math. Appl., 59 (2010), 3508. doi: 10.1016/j.camwa.2010.03.043. [95] J. Sun, Stationary oscillation for chaotic shunting inhibitory cellular neural networks with impulses,, Chaos, 17 (2007). doi: 10.1063/1.2816944. [96] J. A. K. Suykens, M. E. Yalcin and J. Vandewalle, Coupled chaotic simulated annealing processes,, in Proceedings of the 2003 IEEE International Symposium on Circuits and Systems, 3 (2003), 582. doi: 10.1109/ISCAS.2003.1205086. [97] I. Tsuda, Chaotic itinerancy as a dynamical basis of hermeneutics in brain and mind,, World Futures, 32 (1991), 167. doi: 10.1080/02604027.1991.9972257. [98] X. Wang, Period-doublings to chaos in a simple neural network: An analytical proof,, Complex Systems, 5 (1991), 425. [99] Q. Wang and X. Liu, Exponential stability of impulsive cellular neural networks with time delay via Lyapunov functionals,, Appl. Math. Comput., 194 (2007), 186. doi: 10.1016/j.amc.2007.04.112. [100] Z. Wang, H. Zhang and B. Jiang, LMI-based approach for global asymptotic stability analysis of recurrent neural networks with various delays and structures,, IEEE Trans. Neural Netw., 22 (2011), 1032. [101] B. Wu, Y. Liu and J. Lu, New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays,, Math. Comput. Model., 55 (2012), 837. doi: 10.1016/j.mcm.2011.09.009. [102] Y. Xia, J. Cao and Z. Huang, Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses,, Chaos Soliton. Fract., 34 (2007), 1599. doi: 10.1016/j.chaos.2006.05.003. [103] L. Yan, H. He and P. Xiong, Algebraic condition of control for multiple time-delayed chaotic cellular neural networks,, in Proceedings of Fourth International Workshop on Advanced Computational Intelligence, (2011), 596. doi: 10.1109/IWACI.2011.6160078. [104] T. Yang and L. O. Chua, Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication,, Int. J. Bifurcation and Chaos, 7 (1997), 645. doi: 10.1142/S0218127497000443. [105] T. Yang, Impulsive Systems and Control: Theory and Applications,, Nova Science, (2001). [106] Y. Yang and J. Cao, Stability and periodicity in delayed cellular neural networks with impulsive effects,, Nonlinear Anal.: Real World Appl., 8 (2007), 362. doi: 10.1016/j.nonrwa.2005.11.004. [107] Z. Yang and D. Xu, Stability analysis of delay neural networks with impulsive effects,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 563. [108] Y. Yao and W. J. Freeman, Model of biological pattern recognition with spatially chaotic dynamics,, Neural Netw., 3 (1990), 153. doi: 10.1016/0893-6080(90)90086-Z. [109] Z. Yifeng and H. Zhengya, A secure communication scheme based on cellular neural network,, in Proceedings of IEEE International Conference on Intelligent Processing Systems, 1 (1997), 521. doi: 10.1109/ICIPS.1997.672837. [110] W. Yu, J. Cao and W. Lu, Synchronization control of switched linearly coupled neural networks with delay,, Neurocomputing, 73 (2010), 858. doi: 10.1016/j.neucom.2009.10.009. [111] W. Zhao and H. Zhang, On almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients and time-varying delays,, Nonlinear Anal.: Real World Appl., 9 (2008), 2326. doi: 10.1016/j.nonrwa.2007.05.015. [112] Q. Zhou, B. Xiao, Y. Yu and L. Peng, Existence and exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays,, Chaos Soliton. Fract., 34 (2007), 860. doi: 10.1016/j.chaos.2006.03.092. [113] F. Zou and J. A. Nossek, A chaotic attractor with cellular neural networks,, IEEE Trans. Circuits Syst., 38 (1991), 811. doi: 10.1109/31.135755. [114] F. Zou and J. A. Nossek, Bifurcation and chaos in cellular neural networks,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 40 (1993), 166. doi: 10.1109/81.222797.

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##### References:
 [1] M. Abeles, Neural Codes for Higher Brain Functions,, in Information Processing by the Brain (ed. H.J. Markowitsch), (1988). [2] S. Ahmad and I. M. Stamova, Global exponential stability for impulsive cellular neural networks with time-varying delays,, Nonlinear Anal.: Theory Methods Appl., 69 (2008), 786. doi: 10.1016/j.na.2008.02.067. [3] K. Aihara and G. Matsumoto, Chaotic oscillations and bifurcations in squid giant axons,, in Chaos (ed. A.V. Holden), (1986), 257. [4] K. Aihara, T. Takebe and M. Toyoda, Chaotic neural networks,, Phys. Lett. A, 144 (1990), 333. doi: 10.1016/0375-9601(90)90136-C. [5] M. U. Akhmet, Creating a chaos in a system with relay,, Int. J. Qual. Theory Differ. Equat. Appl., 3 (2009), 3. [6] M. U. Akhmet, Devaney's chaos of a relay system,, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 1486. doi: 10.1016/j.cnsns.2008.03.013. [7] M. U. Akhmet, Li-Yorke chaos in the system with impacts,, J. Math. Anal. Appl., 351 (2009), 804. doi: 10.1016/j.jmaa.2008.11.015. [8] M. U. Akhmet, Homoclinical structure of the chaotic attractor,, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 819. doi: 10.1016/j.cnsns.2009.05.042. [9] M. Akhmet, Principles of Discontinuous Dynamical Systems,, New York, (2010). doi: 10.1007/978-1-4419-6581-3. [10] M. Akhmet, Nonlinear Hybrid Continuous/Discrete-Time Models,, Atlantis Press, (2011). doi: 10.2991/978-94-91216-03-9. [11] M. U. Akhmet and M. O. Fen, Chaotic period-doubling and OGY control for the forced Duffing equation,, Commun. Nonlinear. Sci. Numer. Simulat., 17 (2012), 1929. doi: 10.1016/j.cnsns.2011.09.016. [12] M. U. Akhmet and M. O. Fen, Shunting inhibitory cellular neural networks with chaotic external inputs,, Chaos, 23 (2013). doi: 10.1063/1.4805022. [13] M. U. Akhmet and M. O. Fen, Replication of chaos,, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2626. doi: 10.1016/j.cnsns.2013.01.021. [14] M. Akhmet and E. Yílmaz, Neural Networks with Discontinuous/Impact Activations,, Springer, (2014). doi: 10.1007/978-1-4614-8566-7. [15] M. U. Akhmet and M. O. Fen, Entrainment by chaos,, J. Nonlinear Sci., 24 (2014), 411. doi: 10.1007/s00332-014-9194-9. [16] M. Akhmet and M. O. Fen, Chaotification of impulsive systems by perturbations,, Int. J. Bifurcation and Chaos, 24 (2014). doi: 10.1142/S0218127414500783. [17] M. Akhmet and M. O. Fen, Generation of cyclic/toroidal chaos by Hopfield neural networks,, Neurocomputing, 145 (2014), 230. doi: 10.1016/j.neucom.2014.05.038. [18] M. Akhmet and M. O. Fen, Replication of Chaos in Neural Networks, Economics and Physics,, Springer-Verlag, (2016). doi: 10.1007/978-3-662-47500-3. [19] E. Akin and S. Kolyada, Li-Yorke sensitivity,, Nonlinearity, 16 (2003), 1421. doi: 10.1088/0951-7715/16/4/313. [20] P. Balasubramaniam, M. Kalpana and R. Rakkiyappan, Stationary oscillation of interval fuzzy cellular neural networks with mixed delays under impulsive perturbations,, Neural Comput. & Applic., 22 (2013), 1645. doi: 10.1007/s00521-012-0816-6. [21] P. Balasubramaniam and P. Muthukumar, Synchronization of chaotic systems using feedback controller: An application to Diffie-Hellman key exchange protocol and ElGamal public key cryptosystem,, J. Egyptian Math. Soc., 22 (2014), 365. doi: 10.1016/j.joems.2013.10.003. [22] R. Barrio, M. A. Martinez, S. Serrano and A. Shilnikov, Macro- and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons,, Chaos, 24 (2014). doi: 10.1063/1.4882171. [23] A. Bouzerdoum and R. B. Pinter, 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, (1990), 149. doi: 10.1109/IJCNN.1990.137560. [24] A. Bouzerdoum, B. Nabet and R. B. Pinter, Analysis and analog implementation of directionally sensitive shunting inhibitory cellular neural networks,, in Visual Information Processing: From Neurons to Chips, 1473 (1991), 29. [25] A. Bouzerdoum and R. B. Pinter, Nonlinear lateral inhibition applied to motion detection in the fly visual system,, Nonlinear Vision (ed. R.B. Pinter, (1992), 423. [26] A. Bouzerdoum and R. B. Pinter, Shunting inhibitory cellular neural networks: Derivation and stability analysis,, IEEE Trans. Circuits Systems-I: Fund. Theory and Appl., 40 (1993), 215. doi: 10.1109/81.222804. [27] M. Cai and W. Xiong, Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz and bounded activation functions,, Phys. Lett. A, 362 (2007), 417. doi: 10.1016/j.physleta.2006.10.076. [28] J. Cao, Global stability conditions for delayed CNNs,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 48 (2001), 1330. doi: 10.1109/81.964422. [29] J. Cao and J. Lu, Adaptive synchronization of neural networks with or without time-varying delay,, Chaos, 16 (2006). doi: 10.1063/1.2178448. [30] J. Cao, D. W. C. Ho and X. Huang, LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay,, Nonlinear Anal., 66 (2007), 1558. doi: 10.1016/j.na.2006.02.009. [31] R. Caponetto, M. Lavorgna and L. Occhipinti, Cellular neural networks in secure transmission applications,, in Proceedings of CNNA96: Fourth IEEE International Workshop on Cellular Neural Networks and Their Applications, (1996), 411. doi: 10.1109/CNNA.1996.566609. [32] G. A. Carpenter and S. Grossberg, The ART of adaptive pattern recognition by a self-organizing neural network,, Computer, 21 (1988), 77. doi: 10.1109/2.33. [33] A. Chen and J. Cao, Almost periodic solution of shunting inhibitory CNNs with delays,, Phys. Lett. A, 298 (2002), 161. doi: 10.1016/S0375-9601(02)00469-3. [34] L. Chen and H. Zhao, Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients,, Chaos Soliton. Fract., 35 (2008), 351. doi: 10.1016/j.chaos.2006.05.057. [35] H. N. Cheung, A. Bouzerdoum and W. Newland, Properties of shunting inhibitory cellular neural networks for colour image enhancement,, in Proceedings of 6th International Conference on Neural Information Processing, (1999), 1219. doi: 10.1109/ICONIP.1999.844715. [36] L. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits Syst., 35 (1988), 1257. doi: 10.1109/31.7600. [37] L. O. Chua and L. Yang, Cellular neural networks: Applications,, IEEE Trans. Circuits Syst., 35 (1988), 1273. doi: 10.1109/31.7601. [38] H.-S. Ding, J. Liang and T.-J. Xiao, Existence of almost periodic solutions for SICNNs with time-varying delays,, Phys. Lett. A, 372 (2008), 5411. doi: 10.1016/j.physleta.2008.06.042. [39] M. J. Feigenbaum, Universal behavior in nonlinear systems,, Los Alamos Sci., 1 (1980), 4. [40] W. J. Freeman, Tutorial on neurobiology: From single neurons to brain chaos,, Int. J. Bifurcation and Chaos, 2 (1992), 451. doi: 10.1142/S0218127492000653. [41] K. Fukushima, Analysis of the process of visual pattern recognition by the neocognitron,, Neural Netw., 2 (1989), 413. doi: 10.1016/0893-6080(89)90041-5. [42] W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511815706. [43] J. Guckenheimer and R. A. Oliva, Chaos in the Hodgkin-Huxley Model,, SIAM J. Appl. Dyn. Syst., 1 (2002), 105. doi: 10.1137/S1111111101394040. [44] Z. Gui and W. Ge, Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses,, Chaos, 16 (2006). doi: 10.1063/1.2225418. [45] J. Hale and H. Koçak, Dynamics and Bifurcations,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-4426-4. [46] W. Hu, C. Li and S. Wu, Stochastic robust stability for neutral-type impulsive interval neural networks with distributed time-varying delays,, Neural Comput. Appl., 21 (2012), 1947. doi: 10.1007/s00521-011-0598-2. [47] X. Huang and J. Cao, Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay,, Phys. Lett. A, 314 (2003), 222. doi: 10.1016/S0375-9601(03)00918-6. [48] E. M. Izhikevich, Weakly connected quasi-periodic oscillators, FM interactions, and multiplexing in the brain,, SIAM J. Appl. Math., 59 (1999), 2193. doi: 10.1137/S0036139997330623. [49] S. Jankowski, A. Londei, C. Mazur and A. Lozowski, Synchronization phenomena in 2D chaotic CNN, in Proceedings of CNNA-94 Third IEEE International Workshop on Cellular Neural Networks and Their Applications,, Rome, (1994), 339. [50] M. E. Jernigan and G. F. McLean, Lateral inhibition and image processing,, in Nonlinear Vision (ed. R.B. Pinter, (1992), 451. [51] M. Kalpana and P. Balasubramaniam, Stochastic asymptotical synchronization of chaotic Markovian jumping fuzzy cellular neural networks with mixed delays and Wiener process based on sampled-data control,, Chin. Phys. B, 22 (2013). doi: 10.1088/1674-1056/22/7/078401. [52] E. Kaslik and S. Balint, Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture,, Neural Netw., 22 (2009), 1411. doi: 10.1016/j.neunet.2009.03.009. [53] Q. Ke and B. J. Oommen, Logistic neural networks: Their chaotic and pattern recognition properties,, Neurocomputing, 125 (2014), 184. doi: 10.1016/j.neucom.2012.10.039. [54] A. Khadra, X. Liu and X. Shen, Application of impulsive synchronization to communication security,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 50 (2003), 341. doi: 10.1109/TCSI.2003.808839. [55] P. Kloeden and Z. Li, Li-Yorke chaos in higher dimensions: A review,, J. Differ. Equ. Appl., 12 (2006), 247. doi: 10.1080/10236190600574069. [56] J. Kuroiwa, N. Masutani, S. Nara and K. Aihara, Chaotic wandering and its sensitivity to external input in a chaotic neural network,, in Proceedings of the 9th International Conference on Neural Information Processing (ed. L. Wang, 1 (2002), 353. doi: 10.1109/ICONIP.2002.1202192. [57] J. Lei and Z. Lei, The chaotic cipher based on CNNs and its application in network,, in Proceedings of International Symposium on Intelligence Information Processing and Trusted Computing, (2011), 184. doi: 10.1109/IPTC.2011.54. [58] K. Li, X. Zhang and Z. Li, Global exponential stability of impulsive cellular neural networks with time-varying and distributed delay,, Chaos Soliton. Fract., 41 (2009), 1427. doi: 10.1016/j.chaos.2008.06.003. [59] P. Li, Z. Li, W. A. Halang and G. Chen, Li-Yorke chaos in a spatiotemporal chaotic system,, Chaos Soliton. Fract., 33 (2007), 335. doi: 10.1016/j.chaos.2006.01.077. [60] Y. Li, C. Liu and L. Zhu, Global exponential stability of periodic solution for shunting inhibitory CNNs with delays,, Phys. Lett. A, 337 (2005), 46. doi: 10.1016/j.physleta.2005.01.008. [61] Y. Li and J. Shu, Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales,, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 3326. doi: 10.1016/j.cnsns.2010.11.004. [62] T. Y. Li and J. A. Yorke, Period three implies chaos,, The American Mathematical Monthly, 82 (1975), 985. doi: 10.2307/2318254. [63] W. Lin and J. Ruan, Chaotic dynamics of an integrate-and-fire circuit with periodic pulse-train input,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 50 (2003), 686. doi: 10.1109/TCSI.2003.811015. [64] B. Liu and L. Huang, Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with time-varying delays,, Chaos Soliton. Fract., 31 (2007), 211. doi: 10.1016/j.chaos.2005.09.052. [65] Q. Liu and S. Zhang, Adaptive lag synchronization of chaotic Cohen-Grossberg neural networks with discrete delays,, Chaos, 22 (2012). doi: 10.1063/1.4745212. [66] W. Liu and L. Wang, Variable thresholds in the chaotic cellular neural network,, in Proceedings of International Joint Conference on Neural Networks, (2007), 711. doi: 10.1109/IJCNN.2007.4371044. [67] J. Lu, D. W. C. Ho, J. Cao and J. Kurths, Exponential synchronization of linearly coupled neural networks with impulsive disturbances,, IEEE Trans. Neural Netw., 22 (2011), 329. [68] W. Lu and T. Chen, Synchronization of coupled connected neural networks with delays,, IEEE Trans. Circuits Syst.-I Regul. Pap., 51 (2004), 2491. doi: 10.1109/TCSI.2004.838308. [69] F. R. Marotto, Snap-back repellers imply chaos in $R^n$,, J. Math. Anal. Appl., 63 (1978), 199. doi: 10.1016/0022-247X(78)90115-4. [70] B. L. McNaughton, C. A. Barnes and P. Andersen, Synaptic efficacy and EPSP summation in granule cells of rat fascia dentata studied in vitro,, J. Neurophysiol, 46 (1981), 952. [71] P. Muthukumar and P. Balasubramaniam, Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography,, Nonlinear Dynamics, 74 (2013), 1169. doi: 10.1007/s11071-013-1032-3. [72] P. Muthukumar, P. Balasubramaniam and K. Ratnavelu, Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES),, Nonlinear Dynamics, 77 (2014), 1547. doi: 10.1007/s11071-014-1398-x. [73] S. Nara and P. Davis, Chaotic wandering and search in a cycle-memory neural network,, Prog. Theor. Phys., 88 (1992), 845. doi: 10.1143/ptp/88.5.845. [74] S. Nara, P. Davis, M. Kawachi and H. Totsuji, Chaotic memory dynamics in a recurrent neural network with cycle memories embedded by pseudoinverse method,, Int. J. Bifurcation and Chaos, 5 (1995), 1205. [75] M. Ohta and K. Yamashita, A chaotic neural network for reducing the peak-to-average power ratio of multicarrier modulation,, in Proceedings of International Joint Conference on Neural Networks, 2 (2003), 864. doi: 10.1109/IJCNN.2003.1223803. [76] E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos,, Phys. Rev. Lett., 64 (1990), 1196. doi: 10.1103/PhysRevLett.64.1196. [77] C. Ou, Almost periodic solutions for shunting inhibitory cellular neural networks,, Nonlinear Anal.: Real World Appl., 10 (2009), 2652. doi: 10.1016/j.nonrwa.2008.07.004. [78] L. Pan and J. Cao, Anti-periodic solution for delayed cellular neural networks with impulsive effects,, Nonlinear Anal.: Real World Appl., 12 (2011), 3014. doi: 10.1016/j.nonrwa.2011.05.002. [79] G. Peng and L. Huang, Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays,, Nonlinear Anal.: Real World Appl., 10 (2009), 2434. doi: 10.1016/j.nonrwa.2008.05.001. [80] R. B. Pinter, R. M. Olberg and E. Warrant, Luminance adaptation of preferred object size in identified dragonfly movement detectors,, in Proceedings of IEEE International Conference on Systems, 2 (1989), 682. doi: 10.1109/ICSMC.1989.71382. [81] Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, Commun. Math. Phys., 74 (1980), 189. doi: 10.1007/BF01197757. [82] A. Potapov and M. K. Ali, Robust chaos in neural networks,, Phys. Lett. A, 277 (2000), 310. doi: 10.1016/S0375-9601(00)00726-X. [83] K. Pyragas, Continuous control of chaos by self-controlling feedback,, Phys. Rev. A, 170 (1992), 421. [84] D. J. Rijlaarsdam and V. M. Mladenov, Synchronization of chaotic cellular neural networks based on Rössler cells,, in Proceedings of 8th Seminar on Neural Network Applications in Electrical Engineering, (2006), 41. [85] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations,, World Scientific, (1995). doi: 10.1142/9789812798664. [86] E. Sander and J. A. Yorke, Period-doubling cascades galore,, Ergod. Th. & Dynam. Sys., 31 (2011), 1249. doi: 10.1017/S0143385710000994. [87] J. Shao, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays,, Phys. Lett. A, 372 (2008), 5011. doi: 10.1016/j.physleta.2008.05.064. [88] Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces,, Chaos Soliton. Fract., 22 (2004), 555. doi: 10.1016/j.chaos.2004.02.015. [89] Y. Shi and G. Chen, Discrete chaos in Banach spaces,, Sci. China Ser. A Math., 48 (2005), 222. doi: 10.1360/03ys0183. [90] Y. Shi, P. Zhu and K. Qin, Projective synchronization of different chaotic neural networks with mixed time delays based on an integral sliding mode controller,, Neurocomputing, 123 (2014), 443. doi: 10.1016/j.neucom.2013.07.044. [91] M. Shibasaki and M. Adachi, Response to external input of chaotic neural networks based on Newman-Watts model,, in Proceedings of WCCI 2012 IEEE World Congress on Computational Intelligence (ed. J. Liu, (2012), 1. doi: 10.1109/IJCNN.2012.6252394. [92] C. A. Skarda and W. J. Freeman, How brains make chaos in order to make sense of the world?,, Behav. Brain Sci., 10 (1987), 161. doi: 10.1017/S0140525X00047336. [93] X. Song, X. Xin and W. Huang, Exponential stability of delayed and impulsive cellular neural networks with partially Lipschitz continuous activation functions,, Neural Netw., 29-30 (2012), 29. doi: 10.1016/j.neunet.2012.01.006. [94] I. M. Stamova and R. Ilarionov, On global exponential stability for impulsive cellular neural networks with time-varying delays,, Comput. Math. Appl., 59 (2010), 3508. doi: 10.1016/j.camwa.2010.03.043. [95] J. Sun, Stationary oscillation for chaotic shunting inhibitory cellular neural networks with impulses,, Chaos, 17 (2007). doi: 10.1063/1.2816944. [96] J. A. K. Suykens, M. E. Yalcin and J. Vandewalle, Coupled chaotic simulated annealing processes,, in Proceedings of the 2003 IEEE International Symposium on Circuits and Systems, 3 (2003), 582. doi: 10.1109/ISCAS.2003.1205086. [97] I. Tsuda, Chaotic itinerancy as a dynamical basis of hermeneutics in brain and mind,, World Futures, 32 (1991), 167. doi: 10.1080/02604027.1991.9972257. [98] X. Wang, Period-doublings to chaos in a simple neural network: An analytical proof,, Complex Systems, 5 (1991), 425. [99] Q. Wang and X. Liu, Exponential stability of impulsive cellular neural networks with time delay via Lyapunov functionals,, Appl. Math. Comput., 194 (2007), 186. doi: 10.1016/j.amc.2007.04.112. [100] Z. Wang, H. Zhang and B. Jiang, LMI-based approach for global asymptotic stability analysis of recurrent neural networks with various delays and structures,, IEEE Trans. Neural Netw., 22 (2011), 1032. [101] B. Wu, Y. Liu and J. Lu, New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays,, Math. Comput. Model., 55 (2012), 837. doi: 10.1016/j.mcm.2011.09.009. [102] Y. Xia, J. Cao and Z. Huang, Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses,, Chaos Soliton. Fract., 34 (2007), 1599. doi: 10.1016/j.chaos.2006.05.003. [103] L. Yan, H. He and P. Xiong, Algebraic condition of control for multiple time-delayed chaotic cellular neural networks,, in Proceedings of Fourth International Workshop on Advanced Computational Intelligence, (2011), 596. doi: 10.1109/IWACI.2011.6160078. [104] T. Yang and L. O. Chua, Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication,, Int. J. Bifurcation and Chaos, 7 (1997), 645. doi: 10.1142/S0218127497000443. [105] T. Yang, Impulsive Systems and Control: Theory and Applications,, Nova Science, (2001). [106] Y. Yang and J. Cao, Stability and periodicity in delayed cellular neural networks with impulsive effects,, Nonlinear Anal.: Real World Appl., 8 (2007), 362. doi: 10.1016/j.nonrwa.2005.11.004. [107] Z. Yang and D. Xu, Stability analysis of delay neural networks with impulsive effects,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 563. [108] Y. Yao and W. J. Freeman, Model of biological pattern recognition with spatially chaotic dynamics,, Neural Netw., 3 (1990), 153. doi: 10.1016/0893-6080(90)90086-Z. [109] Z. Yifeng and H. Zhengya, A secure communication scheme based on cellular neural network,, in Proceedings of IEEE International Conference on Intelligent Processing Systems, 1 (1997), 521. doi: 10.1109/ICIPS.1997.672837. [110] W. Yu, J. Cao and W. Lu, Synchronization control of switched linearly coupled neural networks with delay,, Neurocomputing, 73 (2010), 858. doi: 10.1016/j.neucom.2009.10.009. [111] W. Zhao and H. Zhang, On almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients and time-varying delays,, Nonlinear Anal.: Real World Appl., 9 (2008), 2326. doi: 10.1016/j.nonrwa.2007.05.015. [112] Q. Zhou, B. Xiao, Y. Yu and L. Peng, Existence and exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays,, Chaos Soliton. Fract., 34 (2007), 860. doi: 10.1016/j.chaos.2006.03.092. [113] F. Zou and J. A. Nossek, A chaotic attractor with cellular neural networks,, IEEE Trans. Circuits Syst., 38 (1991), 811. doi: 10.1109/31.135755. [114] F. Zou and J. A. Nossek, Bifurcation and chaos in cellular neural networks,, IEEE Trans. Circuits Syst.-I: Fund. Theory Appl., 40 (1993), 166. doi: 10.1109/81.222797.
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