Computation of the maximal lower Perron exponent of a linear system

Abstract

Consider the linear system = A(t)x, x E R ~, t > O, (1A) with bounded piecewise continuous coefficients and some linearly independent system Xk(t) = [xl(t),... ,xk(t)] of solutions x~(t) e R ~, k < n. For the Lyapunov characteristic exponentA[.], we have the following well-known relation proved by Lyapunov himself [1, p. 27]: m a x A [ C l X 1 "~- " " " ~- CkXk] ---- ,~ [ X k ] , c E R k c = We encounter the problem as to whether a similar relation is valid for the lower Perron exponent [2] r[.] of the same linear combination of solutions of system (1A) (or arbitrary linearly independent continuous vector functions defined on the positive half-line). Note that [3, 4] (see also [5]) there exists a maximal lower exponent maxceR~ ~ [Xkc] of the subspace Xk(t)c, c E R k, of solutions of system (1A); it is realized on a set of full Lebesgue k-measure in this subspace.