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On the Baire class of Lyapunov exponents of nonhomogeneous linear systems. (Russian) Zbl 0743.34052
The system of differential equations (1) \(\dot x=A(t)x+f(t)\), \(x\in R^ n\), \(t\in R_ +\) is considered, where \(A:R_ +\to\hbox{End} R^ n\) is bounded with the norm \(\| A\|=\sup_{t\in R_ +}\sup_{| x|=1}| A(t)x|\) \(f:R_ +\to R^ n\) are bounded with the norm \(\| f\|=\sup_{t\in R_ +}| f(t)|\). The Lyapunov exponent is defined by \(\lambda_ j(A,f)=\inf_{L\in U_{n-j+1}(R^ n)}\sup_{x\in L}\overline{\lim}_{t\to\infty}{1\over t}\ln| X_{A,f}(t,0)x|\), where \(U_{n-j+1}(R^ n)\) denotes the set of all \((n-j+1)\)-dimensional affine subspaces of \(R^ n\), \(X_{A,f}(\centerdot,\centerdot)\) is the Cauchy operator of (1). It is shown that for all \(n\geq 2\) \(j\in\{1,\ldots,n+1\}\) the set of Lyapunov exponents of nonhomogeneous linear systems \(\lambda_ j\) is not a function of the Baire first class.

MSC:
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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