2015-№4(49) Article 16
M.T. Terekhin, E.M. Fulina
The stability of the unperturbed motion in critical cases. Р. 189-205.
UDC 517.925
The paper investigates the stability of the unperturbed motion of second order differential equations with the zero solution of the linear approximation system. The investigations rest on Lyapunov’s theorems of stability, instability, and asymptotic stability of unperturbed motion and on Chataev’s theorem.
The theorems of stability and instability are proved on the basis of the following concepts: the adjoint polynomial, the pseudo root of the adjoint polynomial.
The paper investigates the relationship of the roots of the adjoint polynomials Vn=(x,y) and their derivatives to define the stability and instability of the unperturbed motion. The theory is used to analyze a differential equation. The solution of Sturm is used to define the relationship of the roots of the adjoint polynomials.
Sturm’s method, adjoint polynomial, derivative, pseudo-root of the adjoint polynomial, number of alterations, even and odd numbers
Bibliography:
1. Demidovich, B.P. Lektsii po matematicheskoj teorii ustojchivosti [Text] [Lectures on mathematical stability theory]: Monogr. — M.: Science, 1964. — 720 p.
2. Krasovskij, N.N. Nekotorye zadachi teorii ustojchivosti dvizheniya [Text] [Some problems of the theory of stability of motion]: Monogr. — M.: Fizmatgiz, 1969. — 211 p.
3. Kurosh, А.G. Kurs vysshej algebry [Text] [The course of higher algebra]: Monogr. — M.: Fizmatgiz, 1959. — 431 p.
4. Lyapunov, А.M. Obshhaya zadacha ob ustojchivosti dvizheniya [Text] [The general problem of stability of movement]: Monogr. — L.: Gostekhizdat, 1950. — 471 p.
5. Malkin, I.G. Teoriya ustojchivosti dvizheniya [Text] [Theory of stability of motion]: Monogr. — M.: Science, 1966. — 530 p.
6. Terekhin, M.T. Issledovanie odnogo kriticheskogo sluchaya v zadache ob ustojchivosti dvizheniya [Text] [Study of one critical case in the problem of stability of motion] / M.T. Terekhin // Izvestiya RАEN. Differentsial’nye uravneniya. — News of Russian Academy of natural sciences. Differential equations. — 2001. — N 4. — P. 108–119.