Existence and Stability Analysis of Fractional Order BAM Neural Networks with a Time Delay

Yuping Cao¹, Chuanzhi Bai²

¹Department of Basic Courses, Lianyungang Technical College, Lianyungang, China.
²Department of Mathematics, Huaiyin Normal University, Huaian, China.

DOI: 10.4236/am.2015.612181                

Abstract

Based on the theory of fractional calculus, the contraction mapping principle, Krasnoselskii fixed point theorem and the inequality technique, a class of Caputo fractional-order BAM neural networks with delays in the leakage terms is investigated in this paper. Some new sufficient conditions are established to guarantee the existence and uniqueness of the nontrivial solution. Moreover, uniform stability of such networks is proposed in fixed time intervals. Finally, an illustrative example is also given to demonstrate the effectiveness of the obtained results.

Keywords

BAM Neural Networks, Caputo Fractional-Order, Existence, Fixed Point Theorems, Uniform Stability

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York.
[2]	Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
[3]	Soczkiewicz, E. (2002) Application of Fractional Calculus in the Theory of Viscoelasticity. Molecular and Quantum Acoustics, 23, 397-404.
[4]	Kulish, V. and Lage, J. (2002) Application of Fractional Calculus to Fluid Mechanics. Journal of Fluids Engineering, 124, 803-806. http://dx.doi.org/10.1115/1.1478062
[5]	Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 24, Elsevier Science B.V., Amsterdam.
[6]	Sabatier, J., Agrawal, Om.P. and Machado, J. (2007) Theoretical Developments and Applications. Advance in Fractional Calculus. Springer, Berlin.
[7]	Arena, P., Caponetto, R., Fortuna, L. and Porto, D. (1998) Bifurcation and Chaos in Noninteger Order Cellular Neural Networks. International Journal of Bifurcation and Chaos, 8, 1527-1539. http://dx.doi.org/10.1142/S0218127498001170
[8]	Arena, P., Fortuna, L. and Porto, D. (2000) Chaotic Behavior in Noninteger Order Cellular Neural Networks. Physical Review E, 61, 776-781. http://dx.doi.org/10.1103/PhysRevE.61.776
[9]	Yu, J., Hu, C. and Jiang, H. (2012) α-Stability and α-Synchronization for Fractional-Order Neural Networks. Neural Networks, 35, 82-87. http://dx.doi.org/10.1016/j.neunet.2012.07.009
[10]	Wu, R., Hei, X. and Chen, L. (2013) Finite-Time Stability of Fractional-Order Neural Networks with Delay. Communications in Theoretical Physics, 60, 189-193. http://dx.doi.org/10.1088/0253-6102/60/2/08
[11]	Boroomand, A. and Menhaj, M.B. (2009) Fractional-Order Hopfield Neural Networks. In: Köppen, M., Kasabov, N. and Coghill, G., Eds., Advances in Neuro-Information Processing, Springer, Berlin, 883-890.
[12]	Kaslik, E. and Sivasundaram, S. (2012) Nonlinear Dynamics and Chaos in Fractional-Order Neural Networks. Neural Networks, 32, 245-256. http://dx.doi.org/10.1016/j.neunet.2012.02.030
[13]	Chen, L., Chai, Y., Wu, R., Ma, T. and Zhai, H. (2013) Dynamic Analysis of a Class of Fractional-Order Neural Networks with Delay. Neurocomputing, 111, 190-194. http://dx.doi.org/10.1016/j.neucom.2012.11.034
[14]	Cao, Y. and Bai, C. (2014) Finite-Time Stability of Fractional-Order BAM Neural Networks with Distributed Delay. Abstract and Applied Analysis, 2014, Article ID: 634803.
[15]	Song, C. and Cao, J. (2014) Dynamics in Fractional-Order Neural Networks. Neurocomputing, 142, 494-498. http://dx.doi.org/10.1016/j.neucom.2014.03.047
[16]	Kosto, B. (1988) Bi-Directional Associative Memories. IEEE Transactions on Systems, Man, and Cybernetics, 18, 49-60. http://dx.doi.org/10.1109/21.87054
[17]	Cao, J. and Wang, L. (2002) Exponential Stability and Periodic Oscillatory Solution in BAM Networks with Delays. IEEE Transactions on Neural Networks, 13, 457-463. http://dx.doi.org/10.1109/72.991431
[18]	Arik, S. and Tavsanoglu, V. (2005) Global Asymptotic Stability Analysis of Bidirectional Associative Memory Neural Networks with Constant Time Delays. Neurocomputing, 68, 161-176. http://dx.doi.org/10.1016/j.neucom.2004.12.002
[19]	Park, J. (2006) A Novel Criterion for Global Asymptotic Stability of BAM Neural Networks with Time Delays. Chaos, Solitons and Fractals, 29, 446-453. http://dx.doi.org/10.1016/j.chaos.2005.08.018
[20]	Bai, C. (2008) Stability Analysis of Cohen-Grossberg BAM Neural Networks with Delays and Impulses. Chaos, Solitons and Fractals, 35, 263-267. http://dx.doi.org/10.1016/j.chaos.2006.05.043
[21]	Raja, R. and Anthoni, S.M. (2011) Global Exponential Stability of BAM Neural Networks with Time-Varying Delays: The Discrete-Time Case. Communications in Nonlinear Science and Numerical Simulation, 16, 613-622. http://dx.doi.org/10.1016/j.cnsns.2010.04.022
[22]	Meyer-Base, A., Roberts, R. and Thummler, V. (2010) Local Uniform Stability of Competitive Neural Networks with Different Timescales under Vanishing Perturbations. Neurocomputing, 73, 770-775. http://dx.doi.org/10.1016/j.neucom.2009.10.003
[23]	Arbi, A., Aouiti, C. and Touati, A. (2012) Uniform Asymptotic Stability and Global Asymptotic Stability for Time-Delay Hopfield Neural Networks. IFIP Advances in Information and Communication Technology, 381, 483-492. http://dx.doi.org/10.1007/978-3-642-33409-2_50
[24]	Rakkiyappan, R., Cao, J. and Velmurugan, G. (2014) Existence and Uniform Stability Analysis of Fractional-Order Complex-Valued Neural Networks with Time Delays. IEEE Transactions on Neural Networks and Learning Systems, 26, 84-97.
[25]	Zhang, S. (2006) Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations. Electronic Journal of Differential Equations, 2006, 1-12.
[26]	Krasnoselskii, M. (1964) Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, Elmsford.