# zbMATH — the first resource for mathematics

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