Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 645–657

# Asymptotics for Solutions of Harmonic Oscillator with Integral Perturbation

• P. N. Nesterov
Article

## Abstract

We construct the asymptotics for solutions of a harmonic oscillator with integral perturbation when the independent variable tends to infinity. The distinctive feature of the considered integral perturbation is the oscillatory decreasing character of its kernel. We assume that the integral kernel is degenerate. This makes it possible to reduce the initial integro-differential equation to an ordinary differential system. To get the asymptotic formulas for the fundamental solutions of the obtained ordinary differential system, we use a special method proposed for the asymptotic integration of linear dynamical systems with oscillatory decreasing coefficients. By the use of special transformations, we reduce the ordinary differential system to the so-called L-diagonal form. We then apply the classical Levinson’s theorem to construct the asymptotics for the fundamental matrix of the L-diagonal system. The obtained asymptotic formulas allow us to reveal the resonant frequencies, i.e., the frequencies of the oscillatory component of the kernel that give rise to unbounded oscillations in the initial integro-differential equation. It appears that these frequencies differ slightly from the resonant frequencies that occur in the adiabatic oscillator with the sinusoidal component of the time-decreasing perturbation.

## Keywords

asymptotics Volterra integro-differential equations harmonic oscillator oscillatory decreasing kernels method of averaging Levinson’s theorem

## References

1. 1.
Bellman, R., Stability Theory of Differential Equations, New York: McGraw-Hill, 1953.
2. 2.
Burd, V.Sh. and Karakulin, V.A., On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients, Math. Notes, 1998, vol. 64, no. 5, pp. 571–578.
3. 3.
Coddington, E.A. and Levinson, N., Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955.
4. 4.
Nesterov, P.N., Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients, Differ. Equations, 2007, vol. 43, no. 6, pp. 745–756.
5. 5.
Nesterov, P.N. and Agafonchikov, E.N., Specific features of oscillations in adiabatic oscillators with delay, Autom. Control Comput. Sci., 2015, vol. 49, no. 7, pp. 582–596.
6. 6.
Burton, T.A., Volterra Integral and Differential Equations, Amsterdam: Elsevier, 2005.
7. 7.
Eastham, M.S.P., The Asymptotic Solution of Linear Differential Systems, Oxford: Clarendon Press, 1989.
8. 8.
Grace, S.R. and Lalli, B.S., Asymptotic behaviour of certain second order integro-differential equations, J. Math. Anal. Appl., 1980, vol. 76, pp. 84–90.
9. 9.
Harris, W.A., Jr., and Lutz, D.A., Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl., 1975, vol. 51, no. 1, pp. 76–93.
10. 10.
Harris, W.A., Jr., and Lutz, D.A., A unified theory of asymptotic integration, J. Math. Anal. Appl., 1977, vol. 57, no. 3, pp. 571–586.
11. 11.
Levinson, N., The asymptotic nature of solutions of linear systems of differential equations, Duke Math. J., 1948, vol. 15, no. 1, pp. 111–126.
12. 12.
Naulin, R. and Vanegas, C.J., Asymptotic formulas for the solutions of integro-differential equations, Acta Math. Hungar., 2000, vol. 89, no. 4, pp. 281–299.
13. 13.
Nesterov, P., Asymptotic integration of functional differential systems with oscillatory decreasing coefficients, Monatsh. Math., 2013, vol. 171, pp. 217–240.
14. 14.
Wintner, A., The adiabatic linear oscillator, Am. J. Math., 1946, vol. 68, pp. 385–397.
15. 15.
Wintner, A., Asymptotic integration of the adiabatic oscillator, Am. J. Math., 1946, vol. 69, pp. 251–272.
16. 16.
Yang, E.H., Asymptotic behaviour of certain second order integro-differential equations, J. Math. Anal. Appl., 1985, vol. 106, pp. 132–139.