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On the existence of a measure unbounded exponential spectral characteristic set of a linear system with a small parameter at the derivative. (English) Zbl 0919.34050

Here, the linear system \[ \varepsilon \dot{x} = A(t)x, \quad \varepsilon \in (0,1), \quad x \in\mathbb{R}^2, \quad t \geq 1, \] with piecewise continuous bounded coefficients, characteristic exponents \(\lambda_1(A) \leq \lambda_2(A)\) and the Grobman coefficient of inequality \(\sigma_G(A)\) at \(\varepsilon = 1\) is considered.
The authors prove the theorem that for any real numbers \(\lambda_1 < \lambda_2\) and \(\sigma_0 > 2(\lambda_2 - \lambda_1)\), there exists a two-dimensional system with infinitely differentiable bounded coefficients and their derivatives which has the characteristic exponents \(\lambda_i(A) = \lambda_i\), \(i=1,2\), and the Grobman coefficient of inequality \(\sigma_G(A) = \sigma_0\). Moreover, a set described by special inequalities is contained in the spectral sigma-set (which measures somehow the Lyapunov-spectrum under linear perturbations of sufficiently small norm).

MSC:

34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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