# Asymptotic Stability of Solutions to One Class of Nonlinear Second-Order Differential Equations with Parameters

G.V. Demidenko, K.M. Dulina, I.I. MatveevaWe consider a class of nonlinear second-order ordinary differential equations with parameters. Differential equations of such type arise when studying oscillations of an 'inversed pendulum' in which the pivot point vibrates periodically. We establish conditions under which the zero solution is asymptotically stable. We obtain estimates for the attraction domain of the zero solution and establish estimates for the decay rate of solutions at infinity. Obtaining the results, we use a criterion for asymptotic stability of the zero solution to systems of linear ordinary differential equations with periodic coefficients. The criterion is formulated in terms of solvability of a special boundary value problem for the Lyapunov differential equation on the interval. The estimates of the attraction domain of the zero solution and estimates for the decay rate of the solutions at infinity are established by the use of the norm of the solution to the boundary value problem.Full text

- Keywords
- second-order differential equations, periodic coefficients, asymptotic stability, Lyapunov differential equation.
- References
- 1. Demidenko G.V., Matveeva I.I. On Stability of Solutions to Linear Systems with Periodic Coefficients. Siberian Math. J., 2001, vol. 42, no. 2, pp. 282-296.

2. Demidenko G.V., Matveeva I.I. On Asymptotic Stability of Solutions to Nonlinear Systems of Differential Equations with Periodic Coefficients. Selcuk J. Appl. Math., 2002, vol. 3, no. 2, pp. 37-48.

3. Demidenko G.V., Matveeva I.I. On Stability of Solutions to Quasilinear Periodic Systems of Differential Equations. Siberian Math. J., 2004, vol. 45, no. 6, pp. 1041-1052.

4. Demidenko G.V., Matveeva I.I. On Numerical Study of Asymptotic Stability of Solutions to Linear Periodic Differential Equations with a Parameter J. Comput. Math. Optim., 2009, vol. 5, no. 3, pp. 163-173.

5. Malkin I.G. Teoriya ustoychivosti dvizheniya [Theory of Stability of Motion]. Moscow, Nauka, 1966.

6. Daletskiy Yu.L., Kreyn M.G. Ustoychivost' resheniy differentsial'nykh uravneniy v banakhovom prostranstve [Stability of Solutions of Differential Equations in Banach Space]. Moscow, Nauka, 1970.

7. Yakubovich V.A., Starzhinskiy V.M. Lineynye differentsial'nye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya [Linear Differential Equations with Periodic Coefficients and Their Applications]. Moscow, Nauka, 1972.

8. Andreev Yu.N. Upravlenie konechnomernymi lineynymi ob''ektami [Control of Finite Dimensional Linear Objects]. Moscow, Nauka, 1976.

9. Rozenvasser E.N. Pokazateli Lyapunova v teorii lineynykh sistem upravleniya [Lyapunov Exponents in the Theory of Linear Control Systems]. Moscow, Nauka, 1977.

10. Bodunov N.A., Kotchenko F.F. On the Dependence of Stability of Linear Periodic Systems on the Period [O zavisimosti ustoychivosti lineynykh periodicheskikh sistem ot perioda]. Differentsial'nye uravneniya [Differential Equations], 1988, vol. 24, no. 2, pp. 338-341.

11. Bogolyubov N.N. The Theory of Perturbations in Nonlinear Mechanics [Teoriya vozmushcheniy v nelineynoy mekhanike]. Sbornik trudov Instituta stroitel'noy mekhaniki AN USSR [Proc. Inst. Struct. Mech.], 1950, no. 14, pp. 9-34.

12. Kapitsa P.L. Dynamic Stability of a Pendulum with a Vibrating Point of Suspension [Dinamicheskaya ustoychivost' mayatnika pri koleblyushcheysya tochke podvesa]. ZhETF [Journal of Experimental and Theoretical Physics], 1951, vol. 21, no. 5, pp. 588-597.

13. Bogolyubov N.N., Mitropol'skiy Yu.A. Asimptoticheskie metody v teorii nelineynykh kolebaniy [Asymptotic Methods in the Theory of Nonlinear Oscillations]. Moscow, Fizmatgiz, 1963.

14. Mitropol'skiy Yu.A. Metod usredneniya v nelineynoy mekhanike [The Method of Averaging in Nonlinear Mechanics]. Kiev, Naukova Dumka, 1971.