Integral manifolds and a reduction principle in stability theory. II
Rewriting the system studied in part I [Ukr. Math. J. 41, No. 12, 1379- 1384 (1989); translation from Ukr. Mat. Zh. 41, No. 12, 1607-1613 (1989; Zbl 0695.34052); see also the above review] as (3) \(dx/dt=A_ 0(t)x+B(t)y+g_ 0(t,x,y)\), \(dy/dt=C_ 0(t)y+D(t)x+h_ 0(t,x,y)\) and letting \(y=Qx+z\), where the matrix \(Q\) gives the solution \(y=Qx\) of the corresponding homogeneous equation, (3) is transformed into equations of the form (4) \(dx/dt=A(t)x+B(t)z+g(t,x,z),\;dz/dt=C(t)z+h(t,x,z)\). Three theorems are established: I. Assuming the critical case holds and (4) has a non-zero solution, then the original system has a two-parameter integral manifold \(y=\Phi^*(t,x)\) containing the graph of the zero-solution \(\Phi^*(t,0)=0\). II ( Reduction Principle). Under the above assumptions, the non-zero solution of (4) is asymptotically stable if and only if the non-zero solution of (4) with \(y\) replaced by \(Qx\) is asymptotically stable. III. This combines the results of I. and II. Finally, an example is given where these results are applied. [For parts III, IV see the reviews below].
scientific article; zbMATH DE number 6915