Two Contrasting Examples of Multidimensional Differential Systems with Lyapunov Extreme Instability

A. A. Bondarev

Matematičeskie zametki, Tome 115 (2024) no. 1, pp. 24-42.

Résumé

Using specific examples, we constructively show that, in dimensions greater than $1$, the Lyapunov extreme instability of a differential system, i.e., the property that the phase curves of all nonzero solutions starting sufficiently close to zero leave any prescribed compact set, does not imply that these solutions go arbitrarily far away from zero in the sense of Perron or in the upper limit sense as $t\to\infty$. Namely, we construct two Lyapunov extremely unstable systems such that all solutions of the first system tend to zero, while the solutions of the second system are divided into two types: all nonzero solutions starting in the closed unit ball tend to infinity in the norm, and all the other solutions tend to zero. Further, both systems constructed in the paper have zero first approximation along the zero solution.

Détail

Keywords: differential system, Lyapunov stability, Perron stability, upper limit stability, nonlinear system, asymptotic properties of solutions.

Bibliographie

[1] B. F. Bylov, R. E. Vinograd, D. M. Grobman, V. V. Nemytskii, Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti, Nauka, M., 1966 | MR

[2] A. F. Filippov, Vvedenie v teoriyu differentsialnykh uravnenii, Editorial URSS, M., 2004

[3] I. N. Sergeev, “Opredelenie ustoichivosti po Perronu i ee svyaz s ustoichivostyu po Lyapunovu”, Differents. uravneniya, 54:6 (2018), 855–856

[4] I. N. Sergeev, “Opredelenie verkhnepredelnoi ustoichivosti i ee svyaz s ustoichivostyu po Lyapunovu i ustoichivostyu po Perronu”, Differents. uravneniya, 56:11 (2020), 1556–1557

[5] A. A. Bondarev, “Odin primer neustoichivoi sistemy”, Differents. uravneniya, 55:6 (2019), 899

[6] A. A. Bondarev, “Primer polnoi, no ne globalnoi neustoichivosti po Perronu”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2021, no. 2, 43–47 | MR | Zbl

[7] A. A. Bondarev, “Suschestvovanie vpolne neustoichivoi po Lyapunovu differentsialnoi sistemy, obladayuschei perronovskoi i verkhnepredelnoi massivnoi chastnoi ustoichivostyu”, Differents. uravneniya, 58:2 (2022), 147–152 | MR

[8] A. A. Bondarev, “O suschestvovanii differentsialnoi sistemy s lyapunovskoi globalnoi neustoichivostyu, vse resheniya kotoroi stremyatsya k nulyu pri neogranichennom roste vremeni”, Differents. uravneniya, 58:8 (2022), 1011–1019 | MR

[9] A. A. Bondarev, “An example of contrasting combination to stability and instability properties in even-dimensional spaces”, Mem. Differ. Equ. Math. Phys., 87 (2022), 25–36 | MR

[10] I. N. Sergeev, “Opredelenie i nekotorye svoistva ustoichivosti po Perronu”, Differents. uravneniya, 55:5 (2019), 636–646 | DOI | MR

[11] A. A. Bondarev, I. N. Sergeev, “Primery differentsialnykh sistem s kontrastnymi sochetaniyami lyapunovskikh, perronovskikh i verkhnepredelnykh svoistv”, Dokl. RAN. Matem., inform., prots. upr., 506 (2022), 25–29 | DOI

[12] I. N. Sergeev, “Lyapunovskie, perronovskie i verkhnepredelnye svoistva ustoichivosti avtonomnykh differentsialnykh sistem”, Izv. IMI UdGU, 56 (2020), 63–78 | DOI

[13] I. N. Sergeev, “Opredelenie svoistva krainei neustoichivosti nulevogo resheniya differentsialnoi sistemy”, Differents. uravneniya, 59:6 (2023), 858–859 | MR

[14] I. N. Sergeev, Lektsii po differentsialnym uravneniyam, Izd-vo Mosk. un-ta, M., 2019