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Stability criteria for certain second-order delay differential equations with mixed coefficients. (English) Zbl 1064.34060

This paper deals with the asymptotic stability of the zero solution of the second-order linear delay differential equation \[ y''(t)= p_1 y'(t)+ p_2y'(t- \tau)+ q_1 y(t)+ q_2 y(t- \tau), \] where \(p_1\), \(p_2\), \(q_1\), \(q_2\) and \(\tau\) are constants. Some practical stability criteria for the zero solution of the equation are given by using Pontryagin’s theory of quasi-polynomials. The criteria can be either easily checked or algorithmically verified, and then several numerical examples are given to illustrate the results.
Reviewer: Jin Zhou (Tianjin)

MSC:

34K20 Stability theory of functional-differential equations
34K06 Linear functional-differential equations
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[1] Arieh, I.; Josef, T., Stability and asymptotic stability of functional-differential equations, J. London Math. Soc., 51, 559-572 (1995) · Zbl 0832.34080
[2] Bellman, R.; Cooke, K. L., Differential-difference Equations (1963), Academic Press: Academic Press New York · Zbl 0115.30102
[3] Boese, F. G., Stability criteria for second-order dynamical systems involving several time delays, SIAM J. Math. Anal., 26, 1306-1330 (1995) · Zbl 0844.34074
[4] Boese, F. G., Stability criteria for second-order dynamical systems involving several time delays, SIAM J. Math. Anal., 26, 1306-1330 (1995) · Zbl 0844.34074
[5] Xu, Bugong, Stability criteria for linear time-invariant systems with multiple delays, J. Math. Anal. Appl., 252, 484-494 (2000) · Zbl 0982.34064
[6] Cahlon, B.; Schmidt, D., On stability of systems of delay differential equations, J. Comput. Appl. Math., 10, 137-158 (1999) · Zbl 0961.34064
[7] B. Cahlon, D. Schmidt, Stability criteria for certain second order delay differential equations, Dyn. Continuous Discrete Impulsive Systems 10 (2003) 593-621.; B. Cahlon, D. Schmidt, Stability criteria for certain second order delay differential equations, Dyn. Continuous Discrete Impulsive Systems 10 (2003) 593-621. · Zbl 1036.34085
[8] Campbell, S. A.; Belair, J.; Ohira, T.; Milton, J., Limit cycles, Tori, and Complex Dynamics in a second-order differential equation with delayed negative feedback, J. Dynamics Differential Equations, 7, 213-236 (1995) · Zbl 0816.34048
[9] Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht/Boston/London · Zbl 0752.34039
[10] Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer New York · Zbl 0425.34048
[11] Hara, T.; Rinko, M.; Morii, T., Asymptotic stability condition for linear differential-difference equations with delays, Dynamic Systems Appl., 6, 493-506 (1997) · Zbl 0892.34066
[12] Hara, T.; Sugie, J., Stability region for systems of differential-difference equations, Funk Ekvac., 39, 69-86 (1996) · Zbl 0860.34045
[13] Hu, G. D.; Mitsui, T., Stability of numerical methods for systems of neutral delay differential equations, BIT, 35, 505-515 (1995) · Zbl 0841.65062
[14] Wenzhang, Huang; Wen Zhang, Huang, On asymptotic stability for linear delay equations, Differential and Integral, 4, 1303-1310 (1991) · Zbl 0737.34054
[15] Karsai, J., On the asymptotic behavior of the solutions of a second order differential equation with small damping, Acta Math. Hungarica, 34, 1721-1723 (1999)
[16] Kolmanovskii, V. B., Stability of some test equations with delay, SIAM J. Math. Anal., 25, 948-961 (1994) · Zbl 0808.34086
[17] MacDonald, N., Biological Delay Systems: Linear Stability Theory (1989), Cambridge University Press: Cambridge University Press Cambridge, New York · Zbl 0669.92001
[18] Pinto, M., Asymptotic solutions for second order delay differential equations, Nonlinear Anal., 28, 1729-1740 (1997) · Zbl 0871.34045
[19] Pontryagin, L. S., On zeros of some transcendental functions, IZV. Akad Nouk SSSR, Ser. Mat., 6, 115-134 (1942), (The English translation is given in Amer. Math. Soc. Transl., Ser. 2, Vol. 1, 1955, pp. 95-110) · Zbl 0063.06306
[20] Shcheglov, V. A., Stability of solutions of a second-order equation with delay, Differential Equations (journal translation), 34, 1721-1723 (1999) · Zbl 0953.34063
[21] C.R. Steele, Studies of the Ear, Lectures in Applied Mathematics, Vol. 17, American Mathematical Society, RI, 1979, pp. 69-71.; C.R. Steele, Studies of the Ear, Lectures in Applied Mathematics, Vol. 17, American Mathematical Society, RI, 1979, pp. 69-71. · Zbl 0424.92006
[22] Stépán, G., Retarded Dynamical Systems: Stability and Characteristic Functions, Research Notes in Math. Series, Vol. 210 (1989), Wiley: Wiley New York · Zbl 0686.34044
[23] Tobias, S. A., Machine Tool Vibrations (1965), Blackie: Blackie London
[24] Xiaoxin, L.; XiaoJun, W., Stability for differential-difference equations, J. Math. Anal. Appl., 174, 84-102 (1991) · Zbl 0780.34052
[25] Kuang, Yang, Delay Differential Equations with Applications in Population Dynamic (1993), Academic Press: Academic Press San Diego, CA · Zbl 0777.34002
[26] Yuanhong, Yu, Stability criteria for linear second order delay differential systems, Acta Math. Appl. Sinica, 4, 109-112 (1988) · Zbl 0668.34071
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