Abstract
Various Perron stability properties of a two-dimensional differential system are studied. It is proved that, generally speaking, the complete Perron instability does not imply the global Perron instability, which may seem at first glance. It turns out that it is possible to construct a counterexample even with an infinitely differentiable right-hand side and a zero matrix of the first approximation at zero. The system under consideration is nonlinear.
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REFERENCES
I. N. Sergeev, ‘‘Definition of Perron stability and its relation with Lyapunov stability,’’ Differ. Uravn. 54, 855–856 (2018).
A. A. Bondarev, ‘‘One example of unstable system,’’ Differ. Uravn. 55, 899 (2019).
I. N. Sergeev, ‘‘Definition and some properties of Perron stability,’’ Differ. Equations 55, 620–630 (2019). doi 10.1134/S0012266119050045
I. N. Sergeev, Lectures on Differential Equations (Mosk. Gos. Univ., Moscow, 2019).
I. N. Sergeev, ‘‘Dependence and independence of the Perron and Lyapunov stability properties on the system phase domain,’’ Differ. Equations 55, 1294–1303 (2019). doi 10.1134/S0012266119100045
ACKNOWLEDGMENTS
The author expresses his gratitude to I.N. Sergeev for valuable comments that contributed to a significant improvement of the paper.
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Translated by I. Tselishcheva
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Bondarev, A.A. An Example of Complete but Not Global Perron Instability. Moscow Univ. Math. Bull. 76, 78–82 (2021). https://doi.org/10.3103/S0027132221020030
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DOI: https://doi.org/10.3103/S0027132221020030