### STABILITY AND ASYMPTOTIC PROPERTIES OF A 2n-DIMENSIONAL SYSTEM OF DIFFERENTIAL EQUATIONS

Josef Kalas
1997 Demonstratio Mathematica
The stability and asymptotic properties of a real 2n-dimensional system x' = A(t)x + h(t, x) are studied. Here A(t) is a square block-diagonal matrix with blocks of order two and h(t,x) is a vector function. The method is based on the combination of the technique of complexification and that of vector Lyapunov functions. Stability and asymptotic properties 699 Hence Zj = (a 2 j-l,2j-lX2j-l + 0-2j-l,2j x 2j + h 2 j-l)+ + »( a 2j,2j-l®2i-l + + h2j) = = (o2i-i,2j-i + ia 2 j,2j-i)x2j-i + (a 2
more » ... )x2j-i + (a 2 j-i,2j + ia 2 j t 2j)x 2 j + fj = Z ' 'Z' Z' Z' = (02j-l,2j-l + ia 2 j,2j-\) 3 3 + (a2j-l,2j + ia,2j,2j) 3 3 + fj = = ^[(°2j-l,2j-l + d2j,2j) + K a 2j,2j-1 -0-2j-l,2j)]Zj + + ^[{^j-iaj-i ~ ®2j,2j) + ¿( a 2j,2j-l + a,2j-l,2j)]zj + fj for j = 1,..., n, where i i /, "4" -DEPARTMENT OF MATHEMATICS MASARYK UNIVERSITY Janäckovo Nam. 2a 662 95 BRNO CZECH REPUBLIC