On the partial instability of the zero solution of nonlinear systems to the first approximation

P. A. Shamanaev

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 3, pp. 280-293.

Résumé

Sufficient conditions for instability with respect to a part of the variables of the zero solution of a nonlinear system in the linear approximation are obtained. The results are presented when the right-hand of the system under study is presented both in the most general form and in the form of a vector polynomial. The results are given for the cases when the right-hand of the system under study is presented both in the most general form and in the form of a vector polynomial. As a first approximation, a linear system of ordinary differential equations with a constant matrix is taken, whose eigenvalues may have zero real parts. Moreover, algebraic and geometric multiplicities of these eigenvalues may not coincide. The approach is based on establishing some correspondence between the solutions of the system under study and its linear approximation. If such correspondence exists, solutions of such systems starting in a sufficiently small neighborhood of zero have some identical component-wise asymptotic properties. In particular, this article focuses on solution instability with respect to some variables, which is one of such properties. Conditions are given for the case when the instability properties of the zero solution of one system are preserved upon transition to another system. The paper gives an example of instability with respect to a part of variables of the zero solution of a nonlinear system, whose linear approximation matrix contains one positive, one negative and one zero eigenvalue, and algebraic and geometric multiplicities of the zero eigenvalue do not coincide.

Détail

Keywords: ordinary differential equations, partial instability, uniform local componen-twise asymptotic equivalence, first approximation

Bibliographie

[1] A. M. Lyapunov, Issledovanie odnogo iz osobennykh sluchaev zadachi ob ustoichivosti dvizheniya [Study of one of the special cases of the problem of stability of motion], Leningr. State University Press, Leningrad, 1963, 116 pp. (In Russ.) | MR

[2] I. G. Malkin, “$\ddot{\mathrm{U}}$ber die Slabilit$\ddot{\mathrm{a}}$t der Bewegung im Sinne von Liapounoff.”, Mat. Sbornik, 45:1 (1938), 47–101 (In Russ.)

[3] V. V. Rumyantsev, “Ob ustoychivosti dvizheniya po otnosheniyu k chasti peremennykh [On motion stability with respect to a part of variables]”, Vestnik of Moscow University. Series. Mathematics. Mechanics. Astronomy. Physics. Chemistry, 1957, no. 4, 9-16 (In Russ.) | MR

[4] V. V. Rumyantsev, A. S. Oziraner, Ustoichivost i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh [Stability and stabilization of motion with respect to a part of variables], Nauka Publ., Moscow, 1987, 253 pp. (In Russ.) | MR

[5] V. I. Vorotnikov, Ustojchivost dinamicheskih sistem po otnosheniyu k chasti peremennyh [Stability of dynamical systems with respect to a part of variables], Nauka, M., 1991, 288 pp. (In Russ.) | MR

[6] A. S. Oziraner, “Ob asimptoticheskoy ustoychivosti i neustoychivosti otnositelno chasti peremennykh [On asymptotic stability and instability with respect to a part of the variables]”, Applied Mathematics and Mechanics [J. Appl. Math. Mech.], 37:4 (1973), 659-665 (In Russ.) | MR | Zbl

[7] V. P. Prokopiev, “Ob ustoychivosti dvizheniya otnositel'no chasti peremennykh v kriticheskom sluchae odnogo nulevogo kornya [On the stability of motion with respect to a part of variables in the critical case of one zero root]”, Applied Mathematics and Mechanics [J. Appl. Math. Mech.]., 39:3 (1975), 422-426 (In Russ.) | MR

[8] I. G. Malkin, Teoriya ustoychivosti dvizheniya [Theory of stability of motion], Nauka Publ., Moscow, 1966, 533 pp. (In Russ.) | MR

[9] A. S. Oziraner, “Ob ustoychivosti dvizheniya v kriticheskom sluchayakh [On stability of motion in critical cases]”, Applied Mathematics and Mechanics, 39:3 (1975), 415-421 (In Russ.) | MR | Zbl

[10] V. N. Shchennikov, “O chastichnoy ustoychivosti v kriticheskom sluchae 2k chisto mnimykh korney [On partial stability in the critical case of 2k purely imaginary roots]”, Differential and integral equations: Methods of topological dynamics. Gor'kiy: Gor'kiy state university named after N. I. Lobachevsky, 1985., 46-50 (In Russ.)

[11] V. N. Shchennikov, “Issledovanie ustoychivosti po chasti peremennykh differentsial'nykh sistem s odnorodnymi pravymi chastyami [Investigation of the stability with respect to a part of the variables of differential systems with homogeneous right-hand sides]”, Differential Equations, 20:9 (1984.), 1645-1649 | MR | Zbl

[12] P. A. Shamanaev, O. S. Yazovtseva, “The sufficient conditions of local asymptotic equivalence of nonlinear systems of ordinary differential equations and its application for investigation of stability respect to part of variables”, Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 19:1 (2017), 102-115 (In Russ.) | DOI | Zbl

[13] P. A. Shamanaev, O. S. Yazovtseva, “The sufficient conditions for polystability of solutions of nonlinear systems of ordinary differential equations”, Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 20:3 (2018), 304-317 (In Russ.) | DOI | Zbl

[14] P. A. Shamanaev, O. S. Yazovtseva, “Studying the equilibrium state stability of the biocenosis dynamics system under the conditions of interspecies interaction”, Vestnik Mordovskogo universiteta [Mordovia University Bulletin journal], 28:3 (2018), 321-332 (In Russ.) | DOI

[15] P. A. Shamanaev, “On the stability of the zero solution with respect to a part of variables in linear approximation”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 19:3 (2023), 374-390 (In Russ.) | DOI | MR

[16] E. V. Voskresenskiy, Asimptoticheskie metody: teoriya i prilozheniya [Asymptotic methods: theory and applications], Middle Volga Mathematical Society Publ., Saransk, 2000, 300 pp. (In Russ.) | MR

[17] E. V. Voskresenskiy, Metody sravneniya v nelineynom analize [Comparison methods in nonlinear analysys], Saratovsky University Press, Saratov, 1990, 224 pp. (In Russ.) | MR

[18] B. F. Bylov, R. E. Vinograd, D. M. Grobman, V. V. Nemytskii, Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti [Theory of Lyapunov Exponents and Its Applications to Stability Problems], Nauka Publ., Moscow, 1966, 576 pp. (In Russ.) | MR