# Exponential synchronization of memristor-based recurrent neural networks with multi-proportional delays

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## Abstract

This paper focuses on the exponential synchronization of memristor-based recurrent neural networks with multi-proportional delays. Act as a vital mathematical model, the system with proportional delays has been widely popular in several scientific fields, such as biology, physics systems as well as control theory. In the sense of Filippov solutions, we receive a novel sufficient condition based on the theories of set-valued maps and differential inclusions, by constructing a proper Lyapunov functional and taking advantage of inequality techniques. Here, the condition is easy to be verified by algebraic methods. A couple of numerical examples and their simulations are given to illustrate the correctness and effectiveness of the obtained results.

## Keywords

Memristor-based neural networks Proportional delay Exponential synchronization Feedback control Lyapunov functional Filippov solution## Notes

### Acknowledgements

The work is supported by the National Science Foundation of China (No. 61374009), Project training of backbone teachers in colleges and universities of Tianjin (No. 043-135205GC38).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Chua L (1971) Memristor-the missing circuit element. IEEE Trans Circuit Theory 18(5):507–519CrossRefGoogle Scholar
- 2.Strukov D, Snider G, Stewart D, Williams R (2008) The missing memristor found. Nature 453(7191):80–83CrossRefGoogle Scholar
- 3.Zhang C, Shang J, Xue W, Tan H (2016) Convertible resistive switching characteristics between memory switching and threshold switching in a single ferritin-based memristor. Chem Commun 52(26):4828–4831CrossRefGoogle Scholar
- 4.Cho K, Lee S, Eshraghian K (2015) Memristor-CMOS logic and digital computational components. Microelectron J 46(3):214–220CrossRefGoogle Scholar
- 5.Sun Z, Chen X, Zhang Y, Li H, Chen Y (2012) Nonvolatile memories as the data storage system for implantable ECG recorder. ACM J Emerg Technol Comput Syst 8(2):1–16CrossRefGoogle Scholar
- 6.Jo S, Chang T, Ebong I, Bhadviya B (2010) Nanoscale memristor device as synapse in neuromorphic systems. Nano Lett 10(4):1297–1301CrossRefGoogle Scholar
- 7.Shin S, Kim K, Kang S (2011) Memristor applications for programmable analog ICs. IEEE Trans Nanotechnol 10(2):266–274CrossRefGoogle Scholar
- 8.Adhikari S, Yang C, Kim H, Chua L (2012) Memristor bridge synapse-based neural network and its learning. IEEE Trans Neural Netw Learn Syst 23(9):1426–1435CrossRefGoogle Scholar
- 9.Shen Y, Miao P, Huang Y, Shen Y (2015) Finite-time stability and its application for solving time-varying Sylvester equation by recurrent neural network. Neural Process Lett 42(3):763–784CrossRefGoogle Scholar
- 10.Stanimirovic P, Zivkovic I, Wei Y (2015) Recurrent neural network for computing the Drazin inverse. IEEE Trans Neural Netw Learn Syst 26(11):2830–2843MathSciNetCrossRefGoogle Scholar
- 11.Qin S, Xue X (2015) A two-layer recurrent neural network for nonsmooth convex optimization problems. IEEE Trans Neural Netw Learn Syst 26(6):1149–1160MathSciNetCrossRefGoogle Scholar
- 12.Wen S, Zeng Z, Huang T, Chen Y (2013) Passivity analysis of memristor-based recurrent neural networks with time-varying delays. J Frankl Inst 350(8):2354–2370MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Wang H, Duan S, Li C, Wang L, Huang T (2017) Exponential stability analysis of delayed memristor-based recurrent neural networks with impulse effects. Neural Comput Appl 28(4):669–678CrossRefGoogle Scholar
- 14.Zhang G, Shen Y, Yin Q, Sun J (2015) Passivity analysis for memristor-based recurrent neural networks with discrete and distributed delays. Neural Netw 61(1):49–58CrossRefzbMATHGoogle Scholar
- 15.Chandrasekar A, Rakkiyappan R, Li X (2016) Effects of bounded and unbounded leakage time-varying delays in memristor-based recurrent neural networks with different memductance functions. Neurocomputing 202(16):67–83CrossRefGoogle Scholar
- 16.Pecora L, Carroll T (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821–824MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Wang Q, Yu S, Li C, Lü J, Fang X (2016) Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans Circuits I Regul Pap 63(3):401–412MathSciNetCrossRefGoogle Scholar
- 18.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):1493–1502MathSciNetCrossRefGoogle Scholar
- 19.Du W, Zhang J, Li Y, Qin S (2016) Synchronization between different networks with time-varying delay and its application in bilayer coupled public traffic network. Math Probl Eng 2016(2):1–11MathSciNetGoogle Scholar
- 20.Garzagonzalez E, Posadascastillo C, Rodriguezlinan A, Hernandez C (2016) Chaotic synchronization of irregular complex network with hysteretic circuit-like oscillators in hamiltonian form and its application in private communications. Rev Mex Fis 62(1):51–59MathSciNetGoogle Scholar
- 21.Wu X, Zhao X, Lü J, Tang L, Lu J (2016) Identifying topologies of complex dynamical networks with stochastic perturbations. IEEE Trans Control Netw Syst 3(4):379–389MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Wang J, Feng J, Xu C, Chen Michael ZQ, Zhao Y, Feng J (2016) The synchronization of instantaneously coupled harmonic oscillators using sampled data with measurement noise. Automatica 66:155–162MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Arenas A, Diaz-Guilera A, Kurths J, Moreno Y, Zhou C (2008) Synchronization in complex networks. Phys Rep 469(3):93–153MathSciNetCrossRefGoogle Scholar
- 24.Liu H, Cao M, Wu C, Lu J, Tse C (2015) Synchronization in directed complex networks using graph comparison tools. IEEE Trans Circuits I Regul Pap 62(4):1185–1194MathSciNetCrossRefGoogle Scholar
- 25.Li Y, Wu X, Lu J, Lü J (2016) Synchronizability of duplex networks. IEEE Trans Circuits II Express Briefs 63(2):206–210Google Scholar
- 26.Yang X, Cao J, Long Y, Rui W (2010) Adaptive lag synchronization for competitive neural networks with mixed delays and uncertain hybrid perturbations. IEEE Trans Neural Netw 21(10):1656–1667CrossRefGoogle Scholar
- 27.Chen W, Lu X, Zheng W (2015) Impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks. IEEE Trans Neural Netw Learn Syst 26(4):734–748MathSciNetCrossRefGoogle Scholar
- 28.Han M, Zhang Y (2016) Complex function projective synchronization in drive-response complex-variable dynamical networks with coupling time delays. J Frankl Inst 353(8):1742–1758MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Zhou W, Zhou X, Yang J, Liu Y, Zhang X, Ding X (2016) Exponential synchronization for stochastic neural networks driven by fractional Brownian motion. J Frankl Inst 353(8):1689–1712MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Tong D, Zhang L, Zhou W, Zhou J, Xu Y (2016) Asymptotical synchronization for delayed stochastic neural networks with uncertainty via adaptive control. Int J Control Autom 14(3):706–712CrossRefGoogle Scholar
- 31.Gan Q, Lv T, Fu Z (2016) Synchronization criteria for generalized reaction-diffusion neural networks via periodically intermittent control. Chaos 26(4):1–11MathSciNetCrossRefzbMATHGoogle Scholar
- 32.Hu C, Yu J, Jiang H (2014) Finite-time synchronization of delayed neural networks with Cohen–Grossberg type based on delayed feedback control. Neurocomputing 143(16):90–96CrossRefGoogle Scholar
- 33.Li N, Cao J (2015) New synchronization criteria for memristor-based networks: adaptive control and feedback control schemes. Neural Netw 61:1–9CrossRefzbMATHGoogle Scholar
- 34.Bao H, Park JH, Cao J (2015) Matrix measure strategies for exponential synchronization and anti-synchronization of memristor-based neural networks with time-varying delays. Appl Math Comput 270:543–556MathSciNetGoogle Scholar
- 35.Mathiyalagan K, Anbuvithya R, Sakthivel R, Park JH, Prakash P (2016) Non-fragile \(H_{\infty }\) synchronization of memristor-based neural networks using passivity theory. Neural Netw 74:85–100CrossRefGoogle Scholar
- 36.Bao H, Park JH, Cao J (2015) Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn 82:1343–1354MathSciNetCrossRefzbMATHGoogle Scholar
- 37.Gao J, Zhu P, Alsaedi A, Alsaadi F, Hayat T (2017) A new switching control for finite-time synchronization of memristor-based recurrent neural networks. Neural Netw 86:1–9CrossRefGoogle Scholar
- 38.Cao J, Li R (2017) Fixed-time synchronization of delayed memristor-based recurrent neural networks. Sci China Inform Sci 60(3):1–15MathSciNetGoogle Scholar
- 39.Bao H, Park JH, Cao J (2016) Exponential synchronization of coupled stochastic memristor-based neural networks with time-varying probabilistic delay coupling and impulsive delay. IEEE Trans Neural Netw Learn Syst 27(1):190–201MathSciNetCrossRefGoogle Scholar
- 40.Zhang G, Hu J, Shen Y (2015) New results on synchronization control of delayed memristive neural networks. Nonlinear Dyn 81(3):1167–1178MathSciNetCrossRefzbMATHGoogle Scholar
- 41.Fox L, Mayers D, Ockendon J, Tayler A (1971) On a functional differential equation. IMA J Appl Math 8(3):271–307CrossRefzbMATHGoogle Scholar
- 42.Dovrolis C, Stiliadis D, Ramanathan P (1999) Proportional differentiated services: delay differentiation and packet scheduling. Comput Commun Rev 29(4):109–120CrossRefGoogle Scholar
- 43.Maneyama Y, Kubo R (2014) QoS-aware cyclic sleep control with proportional-derivative controllers for energy-efficient PON systems. IEEE/OSA J Opt Commun Netw 6(11):1048–1058CrossRefGoogle Scholar
- 44.Zhou L (2016) Delay-dependent exponential stability of recurrent neural networks with Markovian jumping parameters and proportional delays. Neural Comput Appl 28(1):765–773Google Scholar
- 45.Zheng C, Li N, Cao J (2015) Matrix measure based stability criteria for high-order neural networks with proportional delay. Neurocomputing 149(3):1149–1154CrossRefGoogle Scholar
- 46.Zhou L (2013) Delay-dependent exponential stability of cellular neural networks with multi-proportional delays. Neural Process Lett 38(3):347–359CrossRefGoogle Scholar
- 47.Zhou L, Chen X, Yang Y (2014) Asymptotic stability of cellular neural networks with multiple proportional delays. Appl Math Comput 229(5):457–466MathSciNetzbMATHGoogle Scholar
- 48.Zhou L, Zhang Y (2016) Global exponential periodicity and stability of recurrent neural networks with multi-proportional delays. ISA Trans 60:89–95CrossRefGoogle Scholar
- 49.Zhou L (2015) Delay-dependent exponential synchronization of recurrent neural networks with multiple proportional delays. Neural Process Lett 42(3):619–632CrossRefGoogle Scholar
- 50.Zhou L (2015) Novel global exponential stability criteria for hybrid BAM neural networks with proportional delays. Neurocomputing 161(15):99–106CrossRefGoogle Scholar
- 51.Zhou L, Zhao Z (2016) Exponential stability of a class of competitive neural networks with multi-proportional delays. Neural Process Lett 44(3):651–663CrossRefGoogle Scholar
- 52.Filippov A (1988) Differential equations with discontinuous right-hand sides. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar