Generalization of the perron effect whereby the characteristic exponents of all solutions of two differential systems change their sign from negative to positive |
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Authors: | S K Korovin N A Izobov |
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Institution: | 1.Moscow State University,Moscow,Russia;2.Steklov Mathematical Institute,Moscow,Russia |
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Abstract: | We obtain a generalization of the complete Perron effect whereby the characteristic exponents of all solutions change their
sign from negative for the linear approximation system to positive for a nonlinear system with perturbations of higher-order
smallness Differ. Uravn., 2010, vol. 46, no. 10, pp. 1388–1402]. Namely, for arbitrary parameters λ
1 ≤ λ
2 < 0 and m > 1 and for arbitrary intervals b
i
, d
i
) ⊂ λ
i
,+∞), i = 1, 2, with boundaries d
1 ≤ b
2, we prove the existence of (i) a two-dimensional linear differential system with bounded coefficient matrix A(t) infinitely differentiable on the half-line t ≥ 1 and with characteristic exponents λ
1(A) = λ
1 ≤ λ
2(A) = λ
2 < 0; (ii) a perturbation f(t, y) of smallness order m > 1 infinitely differentiable with respect to time t > 1 and continuously differentiable with respect to y
1 and y
2, y = (y
1, y
2) ∈ R
2 such that all nontrivial solutions y(t, c), c ∈ R
2, of the nonlinear system .y = A(t)y + f(t, y), y ∈ R
2, t ≥ 1, are infinitely extendible to the right and have characteristic exponents λy] ∈ b
1, d
1) for c
2 = 0 and λy] ∈ b
2, d
2) for c
2 ≠ 0. |
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Keywords: | |
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