Brunovský, Pavol Controlling nonuniqueness of local invariant manifolds. (English) Zbl 0783.58061 J. Reine Angew. Math. 446, 115-135 (1994). Nonuniqueness of local invariant manifolds corresponding to certain spectral decompositions for \(A\) for the differential equation \[ \dot x=Ax+F(x),\quad x \in \mathbb{R}^ n, \] \(F\) being \(C^ r\) with \(F(0)=0\), \(DF(0)=0\), is discussed. It is proved that a unique \(C^ r\) local manifold can be selected by choosing its submanifold satisfying certain a priori estimates. The result is employed to prove for Morse-Smale systems a conjecture of M. Hirsch according to which certain Lipschitz continuous invariant manifolds of strongly cooperative systems are \(C^ 1\) smooth. Reviewer: P.Brunovský (Bratislava) Cited in 8 Documents MSC: 37A30 Ergodic theorems, spectral theory, Markov operators 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 37D15 Morse-Smale systems Keywords:nonuniqueness; local invariant manifolds; spectral decompositions; Morse- Smale systems PDFBibTeX XMLCite \textit{P. Brunovský}, J. Reine Angew. Math. 446, 115--135 (1994; Zbl 0783.58061) Full Text: DOI Crelle EuDML