Abstract
In this paper, the exponential stability of anti-periodic solutions for inertial neural networks with time delays is investigated. First, by properly chosen variable substitution the system is transformed to first order differential equation. Second, some sufficient conditions which can ensure the existence and exponential stability of anti-periodic solutions for the system are obtained by using Lyapunov method and uniformly converges. Finally, an example is given to illustrate the effectiveness of the results.
Similar content being viewed by others
References
Chua LO, Yang L (1988) Cellular neural networks Theory. IEEE Trans Circuits Syst 35:1257–1272
Chua LO, Yang L (1988) Cellular neural networks: applications. IEEE Trans Circuits Syst 35:1273–1290
Wen S, Zeng Z, Huang T, Zhang Y (2014) Exponential adaptive lag synchronization of memristive neural networks via fuzzy method and applications in pseudorandom number generators. IEEE Trans Fuzzy Syst 22(6):1704–1713
Wen S, Zeng Z, Huang T, Meng Q, Yao W (2015) Lag synchronization of switched neural networks via neural activation function and applications in image encryption. IEEE Trans Neural Netw Learn Syst 26(7):1493–1502
Wen S, Huang T, Zeng Z, Chen Y, Li P (2015) Circuit design and exponential stabilization of memristive neural networks. Neural Netw 63:48–56
Horikawa YO (2009) Bifurcation and stabilization of oscillations in ring neural networks with inertia. Physica D 238:2409–2418
Badcock KL, Westervelt RM (1987) Dynamics of simple electronic neural networks. Physica D 28:305–316
Tani J (1992) Proposal of chaotic steepest descent method for neural networks and analysis of their dynamics. Electron Commun Japan 75(4):62–70
Tani J, Fujita M (1992) Coupling of memory search and mental rotation by a nonequilibrium dynamics neural network. IEICE Trans Fund Electron Commun Comput Sci E 75A(5):578–585
Tani J (1996) Model-based learning for mobile robot navigation from the dynamical systems perspective. IEEE Trans Syst Man Cybern 26(3):421–436
Li CG, Chen GR, Liao XF, Yu JB (2004) Hopf bifurcation and chaos in a single inertial neuron model with time delay. Eur Phys J B 41:337–343
Liu Q, Liao XF, Wang GY, Wu Y (2006) Research for Hopf bifurcation of an inertial two-neuron system with time delay. In: IEEE Proceeding GRC, pp 420–423
Liu Q, Liao XF, Yang DG, Guo ST (2007) The research for Hopf bifurcation in a single inertial neuron model with external forcing. In: IEEE Proceeding GRC, pp. 528–533
Liu Q, Liao X, Liu Y, Zhou S, Guo S (2009) Dynamics of an inertial two-neuron system with time delay. Nonlinear Dyn 58:573–609
Wheeler DW, Schieve WC (1997) Stability and chaos in an inertial two-neuron system. Physica D 105:267–284
Liu Q, Liao XF, Guo ST, Wu Y (2009) Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation. Nonlinear Anal Real World Appl 10:2384–2395
Cao J, Wana Y (2014) Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays. Neural Netw 53:165–172
Yu S, Zhang Z, Quan Z (2015) New global exponential stability conditions for inertial Cohen–Grossberg neural networks with time delays. Neurocomputing 15:1446–1454
Zhang Z, Quan Z (2015) Global exponential stability via inequality technique for inertial BAM neural networks with time delays. Neurocomputing 15:1316–1326
Qi J, Li C, Huang T (2015) Stability of inertial BAM neural network with time-varying delay via impulsive control. Neurocomputing. doi:10.1016/j.neucom.2015.02.052
Ke YQ, Miao CF (2011) Stability analysis of BAM neural networks with inertial term and time delay. WSEAS Trans Syst 10(12):425–438
Ke YQ, Miao CF (2013) Stability and existence of periodic solutions in inertial BAM neural networks with time delay. Neural Comput Appl 23:1089–1099
Ke YQ, Miao CF (2013) Stability analysis of inertial Cohen–Grossberg-type neural networks with time delays. Neurocomputing 117:196–205
Ke YQ, Miao CF (2014) Exponental stability of periodic solutions for inertial Cohen–Grossberg-type neural networks. Neural Netw World 4:377–394
Miao CF, Ke YQ (2014) Exponential stability of periodic solutions for inertial type BAM Cohen–Grossberg neural networks. Abstract Appl Anal 857341:19
Zhao C, Fan Q, Wang W (2010) Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying coefficients. Neural Process Lett 31(3):259–267
Xu C, Zhang Q (2014) Existence and exponential stability of anti-periodic solutions for a high-order delayed cohen-grossberg neural networks with impulsive effects. Neural Process Lett 40(3):227–243
Shi PL, Dong LZ (2010) Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses. Appl Math Comput 216(2):623–630
Li YK, Yang L (2009) Anti-periodic solutions for Cohen–Grossberg neural networks with bounded and unbounded delays. Commun Nonlinear Sci Numer Simul 14(7):3134–3140
Shao JY (2008) Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Phys Lett A 372(30):5011–5016
Zhang AP (2013) Existence and exponential stability of anti-periodic solutions for HCNNs with time-varying leakage delays. Adv Differ Equ. doi:10.1186/1687-1847-2013-162
Li YK, Xu EL, Zhang TW (2010) Existence and stability of anti-periodic solution for a class of generalized neural networks with impulsives and arbitrary delays on time scales. J Inequal Appl 2010(1):1–19
Pan LJ, Cao JD (2011) Anti-periodic solution for delayed cellular neural networks with impulsive effects. Nonlinear Anal Real World Appl 12(6):3014–3027
Xu C, Zhang Q (2015) Existence and global exponential stability of anti-periodic solutions for BAM neural networks with inertial term and delay. Neurocomputing 153:108–116
Abdurahman A, Jiang H (2015) The existence and stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effects. Neurocomputing 149:22–28
Xu C, Zhang Q (2014) On anti-periodic solutions for Cohen–Grossberg shunting inhibitory neural networks with time-varying delays and impulses. Neural Comput 26(10):2328–2349
Li Y, Yang L, Wu W (2015) Anti-periodic solution for impulsive BAM neural networks with time-varying leakage delays on time scales. Neurocomputing 149:536–545
Wang Q, Fang Y, Li H, Su L, Dai B (2014) Anti-periodic solutions for high-order Hopfield neural networks with impulses. Neurocomputing 138:339–346
Xu C, Zhang Q (2015) Anti-periodic solutions in a ring of four neurons with multiple delays. Int J Comput Math 92(5):1086–1100
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ke, Y., Miao, C. Anti-periodic Solutions of Inertial Neural Networks with Time Delays. Neural Process Lett 45, 523–538 (2017). https://doi.org/10.1007/s11063-016-9540-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11063-016-9540-z