Integral manifolds and a reduction principle in stability theory. II
type: Item
instance of: scholarly article
title: Integral manifolds and a reduction principle in stability theory. II
zbMATH Open document ID: 0751.34026
DOI: 10.1007/BF01057384
author: Ya. S. Baris
author: Olga Lykova
published in: Ukrainian Mathematical Journal
publication date: 1992-06-25T00:00:00Z
review text: 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].
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 6915
zbMATH Keywords: stability
zbMATH Keywords: integral manifold
zbMATH Keywords: Reduction Principle
reviewed by: Emilio Roxin
MaRDI profile type: Publication
cites work: Integral manifolds and a reduction principle in stability theory
full work available at URL: https://doi.org/10.1007/bf01057384
OpenAlex ID: W4244186171
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description: scientific article; zbMATH DE number 6915