Abstract
We prove the existence of a perturbed two-dimensional system of ordinary differential equations such that its linear approximation has arbitrarily prescribed negative characteristic exponents, the perturbation is of arbitrarily prescribed higher order of smallness in a neighborhood of the origin, all of its nontrivial solutions are infinitely extendible to the right, and the whole set of their Lyapunov exponents is contained in the positive half-line, is bounded, and has positive Lebesgue measure. In the general case, we also obtain explicit representations of the exponents of these solutions via their initial values.
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References
Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 1930, vol. 32, no. 5, pp. 702–728.
Leonov, G.A., Khaoticheskaya dinamika i klassicheskaya teoriya ustoichivosti dvizheniya (Chaotic Dynamics and the Classical Motion Stability Theory), Moscow; Izhevsk: Regular and Chaotic Dynamics, 2006.
Korovin, S.K. and Izobov, N.A., On the Perron sign change effect for Lyapunov characteristic exponents of solutions of differential systems, Differ. Equations, 2010, vol. 46, no. 10, pp. 1395–1408.
Korovin, S.K. and Izobov, N.A., Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values, Differ. Equations, 2010, vol. 46, no. 11, pp. 1537–1551.
Korovin, S.K. and Izobov, N.A., Multidimensional analog of the two-dimensional Perron effect of sign change of characteristic exponents for infinitely differentiable differential systems, Differ. Equations, 2012, vol. 48, no. 11, pp. 1444–1460.
Izobov, N.A. and Il’in, A.V., Finite-dimensional Perron effect of change of all values of characteristic exponents of differential systems, Differ. Equations, 2013, vol. 49, no. 12, pp. 1476–1489.
Il’in, A.V. and Izobov, N.A., Infinite version of the Perron value change effect for characteristic exponents of differential systems, Dokl. Math., 2014, vol. 90, no. 1, pp. 435–439.
Izobov, N.A. and Il’in, A.V., Perron effect of infinite change of values of characteristic exponents in any neighborhood of the origin, Differ. Equations, 2015, vol. 51, no. 11, pp. 1413–1424.
Il’in, A.V. and Izobov, N.A., Version of the Perron effect of change of characteristic exponents of solutions with initial data on finitely many points and lines, Differ. Uravn., 2016, vol. 52, no. 8, pp. 1139–1140.
Izobov, N.A. and Il’in, A.V., Sign reversal of the characteristic exponents of solutions of a differential system with initial data on finitely many points and lines, Differ. Equations, 2016, vol. 52, no. 11, pp. 1389–1402.
Leonov, G.A. and Kuznetsov, N.V., Time-varying linearization and the Perron effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2007, vol. 17, no. 4, pp. 1079–1107.
Kozlov, V.V., On the mechanism of stability loss, Differ. Equations, 2009, vol. 45, no. 4, pp. 510–519.
Kozlov, V.V., Stabilization of unstable equilibria by time-periodic gyroscopic forces, Dokl. Akad. Nauk, 2009, vol. 429, no. 6, pp. 762–763.
Izobov, N.A., Lyapunov Exponents and Stability, Cambridge: Cambridge Sci., 2012.
Izobov, N.A., The number of characteristic and lower indices of an exponentially stable system with high-order perturbations, Differ. Equations, 1988, vol. 24, no. 5, pp. 510–519.
Volkov, I.A. and Izobov, N.A., On connectedness components of the set of characteristic exponents for a differential system with perturbations of a higher order., Dokl. Akad. Nauk BSSR, 1989, vol. 33, no. 3, pp. 197–200.
Barabanov, E.A. and Volkov, I.A., The structure of the set of Lyapunov characteristic exponents for exponentially stable quasilinear systems, Differ. Equations, 1994, vol. 30, no. 1, pp. 1–15.
Izobov, N.A., Stability with respect to a linear approximation, Differ. Equations, 1986, vol. 22, no. 10, pp. 1132–1147.
Gelbaum, B.R. and Olmsted, J.M.H., Counterexamples in Analysis, San Francisco: Holden-Day, 1964. Translated under the title Kontrprimery v analize, Moscow: Mir, 1967.
Il’in, V.A. and Poznyak, E.G., Osnovy matematicheskogo analiza (Foundations of Mathematical Analysis), Pt. 1, Moscow: Fizmatlit, 2005.
Barabanov, E.A., Structure of the set of lower Perron indices of a linear differential system, Differ. Equations, 1986, vol. 22, no. 11, pp. 1261–1270.
Izobov, N.A., The set of lower exponents of a linear differential system, Differ. Uravn., 1965, vol. 1, no. 4, pp. 469–477.
Izobov, N.A., The set of lower exponents of positive measure, Differ. Uravn., 1968, vol. 4, no. 6, pp. 1147–1149.
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Original Russian Text © N.A. Izobov, A.V. Il’in, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 11, pp. 1427–1439.
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Izobov, N.A., Il’in, A.V. Continual version of the Perron effect of change of values of the characteristic exponents. Diff Equat 53, 1393–1405 (2017). https://doi.org/10.1134/S0012266117110015
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DOI: https://doi.org/10.1134/S0012266117110015