参考文献

［1］Hopfield J J. Neural networks and physical systems with emergent collective computational abilities [J]. Proceedings of the National Academy of Sciences, 1982, 79 (8): 2554-2558.

［2］Chua L O, Yang L. Cellular neural networks: theory and applications [J]. IEEE Transactions on Circuits and Systems I, 1988, 35 (10): 1257-1290.

［3］邢红杰，哈明虎.前馈神经网络及其应用［M］.北京：科学出版社，2013.

［4］Cohen M A, Grossberg S. Absolute stability of global pattern formation and parallel memory storage by competitive neural networks [J]. IEEE Transactions on Systems, Man, and Cybernetics, 1983, 13 (1): 815-826.

［5］Chua L O. Stability of a class of nonreciprocal cellular neural networks [J]. IEEE Transactions on Circuits and systems, 1990, 37 (12): 1520-1527.

［6］周立群.几类细胞神经网络的稳定性研究［D］.哈尔滨：哈尔滨工业大学，2007.

［7］廖晓昕.细胞神经网络的数学理论（Ⅰ）（Ⅱ）［J］.中国科学（A），1994，24（9）：902-910，24（10）：1037-1046.

［8］黄立宏，李雪梅.细胞神经网络的动力学行为［M］.北京：科学出版社，2007.

［9］钟守铭，刘碧森，王晓梅，等.神经网络稳定性理论［M］.北京：科学出版社，2008.

［10］Espejo S, Carmona R, Castro R D, et al. A VLSI-oriented continuous-time CNN model [J]. International Journal of Circuit Theory and Applications, 1996, 24 (3): 341-356.

［11］Kosko B. Bidirectional associative memories [J]. IEEE Transactions on Systems, Man and Cybernetics, 1988, 18 (10): 49-60.

［12］Kosko B. Unsupervised learning in noise [J]. IEEE Transactions on Neural Networks, 1991, 1 (1): 44-57.

［13］Kosko B. Neural networks and fuzzy systems-a dynamical system approach to machine intelligence [M]. Prentice-Hall: Englewood Cliffs, 1992.

［14］Kosko B. Structual stability of unsupervised learning in feedback neural networks [J]. IEEE Transactions on Automatic Control, 1991, 36 (5): 785-790.

［15］Bouzerdoum A, Pinter R B. Shunting inhibitory cellular networks: derivation and stability analysis [J]. IEEE Transactions on Circuits and Systems, 1993, 40 (3), 215-221.

［16］Roska T, Chua L O. Cellular neural methods with nonlinear and delay-type template elements [C]∥Proceedings of the IEEE International Workshop on Cellular Neural Networks and Their Applications, 1990, 12-25.

［17］Zhong S M. Exponential stability and periodicity of cellular neural networks with time delay [J]. Mathematical and Computer Modelling, 2007, 45 (9-10): 1231-1240.

［18］Chen W, Zheng W. A new method for complete stability analysis of cellular neural networks with time delay [J]. IEEE Transactions on Neural Networks, 2010, 21 (7): 1126-1137.

［19］Chen W H. A new method for complete stability analysis of cellular neural networks with time delay [J]. IEEE Transactions on Neural Networks, 2010, 21 (7): 1126-1139.

［20］Balasubramaniam P, Syed M. Stochastic stability of uncertain fuzzy recurrent neural networks with Markovian jumping parameters [J]. Journal of Applied Mathematics and Computing, 2011, 88 (5): 892-904.

［21］Han W, Liu Y, Wang L S. Global exponential stability of delayed fuzzy cellular neural networks with Markovian jumping parameters [J]. Neural Computing & Applications, 2012, 21 (1): 67-72.

［22］Cheng CY, Lin KH, Shih CW, et al. Multistability for delayed neural networks via sequential contracting [J]. IEEE Transactions on Neural Networks and Learning Systems, 2015, 26 (12): 3109-3122.

［23］Zhang H, Wang G. New criteria of global exponential stability for a class of generalized neural networks with time-varying delays [J]. Neurocomputing, 2007, 7 (13-15): 2486-2486-2494.

［24］Song Q, Wang Z. A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays [J]. Physics Letters A, 2007, 368 (1-2): 134-145.

［25］Zhang B, Xu S, Zou Y. Improved delay-dependent exponential stability criteria for discrete-time recurrent neural networks with time-varying delays [J]. Neurocomputing, 2008, 72 (1-3): 321-330.

［26］Ma K, Yu L, Zhang W. Global exponential stability of cellular neural networks with time-varying discrete and distributed delays [J]. Neurocomputing, 2009, 72 (10-12): 2705-2709.

［27］Balasubramaniam P, Syedali M, Arik S. Global asymptotic stability of stochastic fuzzy cellular neural networks with multi time-varying delays [J]. Expert Systems with Applications, 2010, 37 (12): 7737-7744.

［28］Wang Y, Lin P, Wang L. Exponential stability of reaction-diffusion high-order Markovian jump Hopfield neural works with time-varying delays [J]. Nonlinear Analysis: Real World Applications, 2012, 13 (3): 1353-1361.

［29］Liu P, Zeng Z G, Wang J. Multistability analysis of a general class of recurrent neural networks with non-monotonic activation functions and time-varying delays [J]. Neural Networks, 2016, 79: 117-127.

［30］Zhang F, Zeng Z. Multistability and instability analysis of recurrent neural networks with time-varying delays [J]. Neural Networks, 2018, 97: 116-126.

［31］Cao J, Yuan K, Li H. Global Asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays [J]. IEEE Transactions on Neural Networks, 2006, 17 (6): 1646-1651.

［32］Liu Y, Wang Z, Liu X. Global exponential stability of generalized recurrent neural networks with discrete and distributed delays [J]. Neural Networks, 2006, 19 (5): 667-675.

［33］Li T, Fei S. Exponential state estimation for recurrent neural networks with distributed delays [J]. Neurocomputing, 2007, 71 (1-3): 428-438.

［34］Huang C. Almost sure exponential stability of stochastic cellular neural networks with unbounded distributed delays [J]. Neurcomputing, 2009, 72 (13-15): 3352-3356.

［35］Tan M. Global asymptotic stability of fuzzy cellular neural networks with unbounded distributed delays [J]. Neural Processing Letters, 2010, 31 (2): 147-157.

［36］Li T, Song A. Fei S, et al. Delay-derivative-dependent stability for delayed neural networks with unbound distributed delay [J]. IEEE Transactions on Neural Networks, 2010, 21 (8): 1365-1371.

［37］Kaslik E, Sivasundaram S. Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis [J]. Neural Networks, 2011, 24 (4): 370-377.

［38］Li Y, Zhao K, Ye Y. Stability of reaction-diffusion recurrent neural networks with distributed delays and Neumann boundary conditions on time scales [J]. Neural Processing Letters, 2012, 36 (3): 217-234.

［39］Yang W, Wang Y, Zeng Z, et al. Multistability of discrete-time delayed Cohen-Grossberg neural networks with second-order synaptic connectivity [J]. Neurocomputing, 2015, 164: 252-261.

［40］Wang L M, Shen Y, Yin Q, et al. Adaptive synchronization of memristor-based neural networks with time-varying delays [J]. IEEE Transactions on Neural Networks and Learning Systems, 2015, 26: 2033-2042.

［41］Thirunavukkarasu R, Gnaneswaran N. Dissipativity analysis of stochastic memristor-based recurrent neural networks with discrete and distributed time-varying delays [J]. Network: Computation in Neural Systems, 2016, 27 (4): 237-267.

［42］Li D S, Shi L, Gaans O, et al. Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays [J]. Journal of Differential Equations, 2018, 264 (6): 3864-3898.

［43］Huang H, Huang T, Chen X. A mode-dependent approach to state estimation of recurrent neural networks with Markovian jumping parameters and mixed delays [J]. Neural Networks, 2013, 46: 50-61.

［44］Şayli M, Yilmaz E. Anti-periodic solutions for state-dependent impulsive recurrent neural networks with time-varying and continuously distributed delays [J]. Annals of Operation Research, 2017, 258 (1): 159-185.

［45］Liu P, Zeng ZG, Wang J. Multistability of recurrent neural networks with nonmonotonic activation functions and mixed time delays [J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2016, 46: 512-523.

［46］王林山.时滞递归神经网络［M］.北京：科学出版社，2008.

［47］王晓红，付主木.时滞型神经网络动力学分析及在电力系统中的应用［M］.北京：科学出版社，2015.

［48］甘勤涛，徐睿.时滞神经网络的稳定性与同步性控制［M］.北京：科学出版社，2016.

［49］郭英新.非线性随机时滞神经网络稳定性分析与脉冲镇定［M］.北京：科学出版社，2017.

［50］李秀玲，王慧敏.具时滞的神经网络模型的分支问题研究［M］.北京：科学出版社，2017.

［51］谭满春.比例时滞线性系统的反馈镇定［J］.信息与控制，2006，35（6）：690-694.

［52］Lai Y C, Szu Y C. Achieving proportional delay and Loss differentiation in a wireless network with a multi-state link [C]∥Lecture Notes in Computer Science, 2008, 5200: 811-820.

［53］高昂，穆德俊，胡延苏，等.Web QoS中的预测控制与比例延迟保证［J］.计算机科学，2010，37（1）：57-59.

［54］Zhou L Q. On the global dissipativity of a class of cellular neural networks with multi-pantograph delays [J]. Advances in Artificial Neural Systems, 2011, DOI: 10. 1155/2011/ 941426: 1-7.

［55］张迎迎，周立群.一类具多比例延时的细胞神经网络的指数稳定性［J］.电子学报，2012，40（6）：1159-1163.

［56］周立群.多比例时滞细胞神经网络的全局一致渐近稳定性［J］.电子科技大学学报，2013，42（4）：625-629.

［57］Zhou L Q, Zhang Y Y. Global exponential stability of cellular neural networks with multi-proportional delays [J]. International Journal of Biomathematics, 2015, 8 (6): 1550071: 1-17.

［58］Zhou L Q. Dissipativity of a class of cellular neural networks with proportional delays [J]. Nonlinear Dynamics, 2013, 73 (3): 1895-1903.

［59］Zhou L Q. Global asymptotic stability of cellular neural networks with proportional delays [J]. Nonlinear Dynamics, 2014, 77 (1): 41-47.

［60］Zhou L Q. Delay-dependent exponential stability of cellular neural networks with multi-proportional delays [J]. Neural Processing Letters, 2013, 38 (3): 321-346.

［61］Zhou L Q, Zhang Y Y. Global exponential periodicity and stability of recurrent neural networks with multi-proportional delays [J]. ISA Transactions, 2016, 60 (1): 89-95.

［62］Zhou L Q, Chen X B, Yang Y. Asymptotic stability of cellular neural networks with multiple proportional delays [J]. Applied Mathematics and Computation, 2014, 229: 457-466.

［63］周立群.具比例时滞杂交双向联想记忆神经网络的全局指数稳定性［J］.电子学报，2014，42（1）：96-101.

［64］Zhou L Q. Novel global exponential stability criteria for hybrid BAM neural networks with proportional delays [J]. Neurocomputing, 2015, 161: 99-106.

［65］周立群.具比例时滞高阶广义细胞神经网络的全局指数周期性［J］.系统科学与数学，2015，35（9）：1-13.

［66］Zhou L Q. Delay-dependent exponential synchronization of recurrent neural networks with multiple proportional delays [J]. Neural Processing Letters, 2015, 42 (3): 619-632.

［67］Zhou L Q, Zhao Z Y. Exponential stability of a class of competitive neural networks with multi-proportional delays [J]. Neural Processing Letters, 2016, 44 (3): 651-663.

［68］Zhou L Q, Zhang Y Y. Global exponential stability of a class of impulsive recurrent neural networks with proportional delays via fixed point theory [J]. Journal of the Franklin Institute, 2016, 353 (2): 561-575.

［69］Zhou L Q. Delay-dependent exponential stability of recurrent neural networks with Markovian jumping parameters and proportional delay [J]. Neural Computing & Applications, 2017, 28 (S1): 765-773.

［70］Zhou L Q, Liu X T. Mean-square exponential input-to-state stability of stochastic recurrent neural networks with multi-proportional delays [J]. Neurocomputing, 2017, 219 (1): 396-403.

［71］Su L J, Zhou L Q. Passivity of memristor-based recurrent neural networks with multi-proportional delays [J]. Neurocomputing, 2017, 266: 485-493.

［72］Zhou L Q. Delay-dependent and delay-independent passivity of a class of recurrent neural networks with impulse and multi-proportional delays [J]. Neurocomputing, 2018, 308: 235-244.

［73］Hien L V, Son D T. Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays [J]. Applied Mathematics and Computation, 2015, 14: 14-23.

［74］Hien L V, Son D T, Trinh H. On global dissipstivity of nonautonomous neural networks with multiple proportional delays [J]. IEEE Transactions on Neural Networks and Learning Systems, 2016, 29 (1): 225-231.

［75］Yu Y. Finite-time stability on a class of non-autonomous SICNNs with multi-proportional delays [J]. Asian Journal of Control, 2017, 19 (1): 87-94.

［76］Yu Y. Global exponential convergence for a class of HCNNs with neutral time-proportional delays [J]. Applied Mathematics and Computation, 2016, 285: 1-7.

［77］Liu B. Global exponential convergence of non-autonomous cellular neural networks with multi-proportional delays [J]. Neurcomputing, 2016, 191: 352-355.

［78］Xu C, Li P, Pang Y. Global exponential stability for interval general bidirectional associative memory (BAM)neural networks with proportional delays [J]. Mathematical Models and Methods in Applied Sciences, 2016, 39 (18): 5720-5731.

［79］Zheng C, Li N, Cao J. Matrix measure based stability criteria for high-order networks with proportional delay [J]. Neurcomputing, 2015, 149: 1149-1154.

［80］Feng N, Wu Y, Wang W, et al. Exponential cluster synchronization of neural networks with proportional delays [J]. Mathematica Problems in Engineering, 2015, DOI: org/10. 1155/2105. 52324.

［81］Song X, Zhao P, Xing Z, et al. Global asymptotic stability of CNNs with impulses and multi-proportional delays [J]. Mathematical Methods in the Applied Science, 2016, 39: 722-733.

［82］Wang W, Li L, Peng H, et al. Anti-synchronization control of memristive neural networks with multiple proportional delays [J]. Neural processing Letters, 2016, 43: 269-283.

［83］Liu J, Xu R. Passivity analysis and state estimation for a class of memristor-based neural networks with multiple proportional delays [J]. Advances in Difference Equations, 2017, 34: DOI 10. 1186/s13662-016-1069-y.

［84］Wang W. Finite-time synchronization for a class of fuzzy cellular neural networks with time-varying coefficients and proportional delays [J]. Fuzzy Sets and Systems, 2018, 338: 40-49.

［85］Cui N, Jiang H, Hu C, et al. Global asymptotic and robust stability of inertial neural networks with proportional delays [J]. Neurocomputing, 2018, 272: 326-333.

［86］郑祖麻.泛函微分方程理论［M］.合肥：安徽教育出版社，1994.

［87］廖晓昕.动力系统的稳定性理论及应用［M］.北京：国防工业出版社，2000.

［88］廖晓昕.稳定性的数学理论及应用［M］.2版.武汉：华中师范大学出版社，2004.

［89］Mao X R. Stochastic differential equations and their applications [M]. Cambridge, UK：Woodhead publishing, 1997: 144.

［90］Ockendon J R, Tayler A B. The dynamics of a current collection system for an electric locomotive [J]. Proceedings of the Royal Society of London (Series A), 1971, 332 (2): 447-468.

［91］王龙洪.几类具比例时滞的中立型微分方程解的定性性质研究［D］.衡阳：南华大学，2012.

［92］Fox L, Mayers D F, Ockendon J R, et al. On a functional differential equation [J]. Journal of the Institute of Mathematics and its Applications, 1971, 8: 271-307.

［93］Kato T, Maleod J B. The functional-differential equation y′ ( x )= ay ( λx )+ by ( x ) [J]. Bulletin of the American Mathematical Society, 1971, 77: 891-937.

［94］Carr J, Dyson J. The functional differential equation y′ ( x )= ay ( λx )+ by ( x ) [J]. Proceedings of the Royal Society of Edinburgh Section A, 1976, 74 (13): 165-174.

［95］Carr J, Dyson J. The matrix functional differential equation y′ ( x )= Ay ( λx )+ By ( x ) [J]. Proceedings of the Royal Society of Edinburgh Section A, 1976, 75 (1): 5-22.

［96］Kuang Y, Feldstein A. Monotonic and oscillatory solution of a linear neutral delay equation with infinite lag [J]. SIAM Journal on Mathematical Analysis, 1990, 21: 1633-1641.

［97］Buhmann M, Iserles A. Stability of the discretized pantograh differential equation [J]. Mathematics of Computation, 1993. 60 (202): 757-589.

［98］Iserles A, Liu Y. On neutral functional-differential equations with proportional delays [J]. Journal of Mathematical Analysis and Applications, 1993, 207 (1): 73-95.

［99］Iserles A. On the generalized pantograph functional differential equation [J]. European Journal of Applied Mathematics, 1993, 4 (1): 1-38.

［100］Iserles A. On nonlinear delay-differential equations [J]. Transactions of the American Mathematical Society, 1994, 344: 441-477.

［101］Iserles A, Liu Y. On pantograph integro-differential equations [J]. Journal of Integral Equations and Applications, 1994, 6: 213-237.

［102］Liu Y K. Asymptotic behavior of functional-differential equations with proportional time delays [J]. European Journal of Applied Mathematics, 1996, 7: 11-30.

［103］Liu Y K. Numerical investigation of the pantograph equation [J]. Applied Numerical Mathematics, 1997, 24 (2-3): 309-317.

［104］Si J G, Cheng S S. Analytic solutions of a functional differential equation with proportional delays [J]. Bulletin of the Korean Mathematical Society, 2002, 39 (2): 225-236.

［105］Van Brunt B, Marshall J C, Wake G C. Holomorphic solutions to pantograph type equations with neutral fixed points [J]. Journal of Mathematical Analysis and Applications, 2004, 295 (2): 557-569.

［106］Čermák Jan. On the differential equation with power coefficients and proportional delays [J]. Tatra Mountains Mathematical Publications, 2007, 38: 57-69.

［107］Čermák Jan. On a linear differential equation with a proportional delay [J]. Mathematische Nachrichten, 2007, 280 (5/6): 495-504.

［108］Abazari R, Kilicman. Application of differential transform method on nonlinear integro-differential equations with proportional delays [J]. Neural Computing & Applications, 2014, 24: 391-397.

［109］Balachandran K, Kiruthika S, Trujillo J. Existence of solutions of nonlinear fractional pantograph equations [J]. Acta Mathematica Scientia, 2013, 33B (3): 712-720.

［110］Yu Y. Global exponential convergence for a class of neutral functional differential equations with proportional delays [J]. Mathematical Methods in the Applied Science, 2016, 39 (15): 4520-4525.

［111］Gan S. Exact and discretized dissipativity of the pantograph equation [J]. Journal of Computational and Applied Mathematics, 2007, 25 (1): 81-88.

［112］李冬松，刘明珠.多比例延迟微分方程精确解的性质［J］.哈尔滨工业大学学报，2000，32（3）：1-3.

［113］曹婉容.常延迟中立型及比例延迟微分方程稳定性［D］.哈尔滨：哈尔滨工业大学，2001.

［114］陈新德.多比例延迟微分方程的散逸性［J］.数学理论与应用，2008，28（4）：113-117.

［115］白小红.变分迭代法在某些比例延迟微分方程中的应用［J］.数学理论与应用，2010，30（4）：38-41.

［116］关开中，贺小宝.一类具比例时滞的脉冲微分方程解的振动性［J］.南华大学学报：自然科学版，2011，25（3）：58-62.

［117］李慧.比例延迟微分方程稳定性分析［D］.哈尔滨：黑龙江大学，2011.

［118］李杰.对比例时滞Volterra积分泛函方程的配置法［D］.长春：吉林大学，2011.

［119］张家驹.M-矩阵的一些性质［J］.数学年刊，1980，1（1）：47-50.

［120］郭大钧.非线性泛函分析［M］.山东：山东科学技术出版社，2002. v+GMcpuOLCXUo9s9ChUGM3fdQp1Xqh2C41rYPIt8r4m9qAPy7cbKp6532dN/ch9c