In this study, we are concerned the existence of pseudo almost automorphic (PAA) solutions and globally exponential stability of a Duffing equation system with variable delays. Some differential inequalities and the well-known Banach fixed point theorem are used for the existence and uniqueness of PAA solutions. Also, with the help of Lyapunov functions, sufficient conditions are obtained for globally exponential stability of PAA solutions. Since the PAA is more general than the almost and pseudo almost periodicity, this work is new and complementary compared to previous studies. In addition, an example is given to show the correctness of our results.
Citation: Ramazan Yazgan. An analysis for a special class of solution of a Duffing system with variable delays[J]. AIMS Mathematics, 2021, 6(10): 11187-11199. doi: 10.3934/math.2021649
In this study, we are concerned the existence of pseudo almost automorphic (PAA) solutions and globally exponential stability of a Duffing equation system with variable delays. Some differential inequalities and the well-known Banach fixed point theorem are used for the existence and uniqueness of PAA solutions. Also, with the help of Lyapunov functions, sufficient conditions are obtained for globally exponential stability of PAA solutions. Since the PAA is more general than the almost and pseudo almost periodicity, this work is new and complementary compared to previous studies. In addition, an example is given to show the correctness of our results.
[1] | C. Aouiti, M. S. M'hamdi, A. Touati, Pseudo almost automorphic solutions of recurrent neural networks with time-varying coefficients and mixed delays, Neural Process Lett., 45 (2017), 121-140. doi: 10.1007/s11063-016-9515-0 |
[2] | C. Aouiti, F. Dridi, F. Kong, Pseudo almost automorphic solutions of hematopoiesis model with mixed delays, Comput. Appl. Math., 39 (2020), 87. doi: 10.1007/s40314-020-1118-8 |
[3] | S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, P. Nat. Acad. Sci. USA, 52 (1964), 907-910. doi: 10.1073/pnas.52.4.907 |
[4] | T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations, Orland (FL): Academic Press, 1985. |
[5] | J. K. Hale, Theory of functional differential equations, New York: Springer-Verlag, 1977. |
[6] | T. Diagana, Almost automorphic type and almost periodic type functions in abstract spaces, Springer, Cham, 2013. |
[7] | Y. Kuang, Delay differential equations with applications in population dynamics, New York: Academic Press, 1993. |
[8] | B. M. Levitan, V. V. Zhikov, Almost periodic functions and differential equations, Cambridge, New York: Cambridge University Press, 1978. |
[9] | B. W. Liu, C. Tunç, Pseudo almost periodic solutions for a class of nonlinear Duffing system with a deviating argument, J. Appl. Math. Comput., 49 (2015), 233-242. doi: 10.1007/s12190-014-0835-9 |
[10] | L. Q. Peng, W. T. Wang, Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument, Electron. J. Qual. Theo., 6 (2010), 1-12. |
[11] | G. Y. Wang, S. L. He, A quantitative study on detection and estimation of weak signals by using chaotic duffing oscillators, IEEE T. Circuits Syst. I, Fund. Theory Appl., 50 (2003), 945-953. doi: 10.1109/TCSI.2003.812606 |
[12] | T. J. Xiao, J. Liang, J. Zhang, Pseudo-almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum, 76 (2008), 518-524. doi: 10.1007/s00233-007-9011-y |
[13] | C. J. Xu, M. X. Liao, Existence and uniqueness of pseudo almost periodic solutions for Liénard-type systems with delays, Electron. J. Differ. Eq., 2016 (2016), 1-8. |
[14] | H. Vahedi, G. B. Gharehpetian, M. Karrari, Application of Duffing oscillators for passive islanding detection of inverter-based distributed generation units, IEEE T. Power Deliver., 27 (2012), 1973-1983. doi: 10.1109/TPWRD.2012.2212251 |
[15] | R. Yazgan, C. Tunç, On the almost periodic solutions of fuzzy cellular neural networks of high order with multiple time lags, Int. J. Math. Comput. Sci., 15 (2020), 183-198. |
[16] | R. Yazgan, C. Tunç, On the weighted pseudo almost periodic solutions of Nicholson's blowies equation, Appl. Appl. Math., 14 (2019), 875-889. |
[17] | T. Yoshizawa, Asymptotic behaviors of solutions of differential equations, Qual. Theor., 47 (1987), 1141-1164. |
[18] | E. Yusufoğlu (Agadjanov), Numerical solution of Duffing equation by the Laplace decomposition algorithm, Appl. Math. Comput., 177 (2006), 572-580. |
[19] | G. A. Zakeri, E. Yomba, Exact solutions of a generalized autonomous Duffing-type equation, Appl. Math. Model., 39 (2015), 4607-4616. doi: 10.1016/j.apm.2014.05.010 |
[20] | R. Zivieri, S. Vergura, M. Carpentieri, Analytical and numerical solution to the nonlinear cubic Duffing equation: an application to electrical signal analysis of distribution lines, Appl. Math. Model., 40 (2016), 9152-9164. doi: 10.1016/j.apm.2016.05.043 |
[21] | W. Y. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sin., 13 (1997), 373-380. doi: 10.1007/BF02560018 |
[22] | A. E. Zúñiga, Exact solution of the cubic-quintic Duffing oscillator, Appl. Math. Model., 37 (2013), 2574-2579. doi: 10.1016/j.apm.2012.04.005 |
[23] | A. E. Zúñiga, Solution of the damped cubic-quintic Duffing oscillator by using Jacobi elliptic functions, Appl. Math. Comput., 246 (2014), 474-481. |
[24] | R. Yazgan, On the weighted pseudo almost periodic solutions for Liénard-type systems with variable delays, Mugla J. Sci. Technol., 6 (2020), 89-93. |
[25] | F. Chérif, Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays, J. Appl. Math. Comput., 39 (2012), 235-251. doi: 10.1007/s12190-011-0520-1 |
[26] | M. Amdouni, F. Chérif, C. Tunç, On the weighted piecewise pseudo almost automorphic solutions Mackey-Glass model with mixed delays and harvesting term, Iran. J. Sci. Technol. Trans. Sci., 45 (2021), 619-634. doi: 10.1007/s40995-020-01043-7 |