On \((h,k)\) manifolds with asymptotic phase. (English) Zbl 0894.34040
The authors introduce the notion of \((h,k)\) manifolds and give conditions under which the property of being a manifold with asymptotic phase holds. These conditions allow the construction of a transformation of the variational equation of a nonlinear nonautonomous system with \((h,k)\) dichotomies, into an almost decoupled quasilinear system. The class of admissible \(h\) and \(k\) functions is also briefly analyzed.
In particular nonautonomous systems of the form \(\dot x = f(t,x)\), \(x \in \mathbb{R}^n\) are considered. The notion of an \((h,k)\) trichotomy for a linear system is introduced. This notion was suggested by M. Pinto [J. Math. Anal. Appl. 195, No. 1, 16-31 (1995; Zbl 0847.34058)] and generalizes the well-known concepts of ordinary und exponential trichotomies (or dichotomies). When applied to the variational equation of the nonlinear system with respect to a given solution, this leads to \((h,k)\) manifolds (and integral manifolds) of the nonlinear system. Then the authors develop the Aulbach-Coppel-Knobloch (ACK) transformation of the variational equation, bringing it into a quasilinear form, which turns out to be appropriate for giving conditions in order to ensure that the manifold has an asymptotic phase [see B. Aulbach, Nonlinear Anal., Theory Methods Appl. 6, 817-827 (1982; Zbl 0509.58034); Continuous and discrete dynamics near manifolds of equilibria, Lect. Notes Math. 1058. Berlin etc.: Springer-Verlag (1984; Zbl 0535.34002); J. Math. Anal. Appl. 112, 317-327 (1985; Zbl 0595.34060)] for the exposition in the usual hyperbolic case. Finally the family of admissible functions \(h\) and \(k\) is studied, and by means of an example the authors show that a large class of functions is allowed, pertaining to systems not to be classified as being of exponential type.
In particular nonautonomous systems of the form \(\dot x = f(t,x)\), \(x \in \mathbb{R}^n\) are considered. The notion of an \((h,k)\) trichotomy for a linear system is introduced. This notion was suggested by M. Pinto [J. Math. Anal. Appl. 195, No. 1, 16-31 (1995; Zbl 0847.34058)] and generalizes the well-known concepts of ordinary und exponential trichotomies (or dichotomies). When applied to the variational equation of the nonlinear system with respect to a given solution, this leads to \((h,k)\) manifolds (and integral manifolds) of the nonlinear system. Then the authors develop the Aulbach-Coppel-Knobloch (ACK) transformation of the variational equation, bringing it into a quasilinear form, which turns out to be appropriate for giving conditions in order to ensure that the manifold has an asymptotic phase [see B. Aulbach, Nonlinear Anal., Theory Methods Appl. 6, 817-827 (1982; Zbl 0509.58034); Continuous and discrete dynamics near manifolds of equilibria, Lect. Notes Math. 1058. Berlin etc.: Springer-Verlag (1984; Zbl 0535.34002); J. Math. Anal. Appl. 112, 317-327 (1985; Zbl 0595.34060)] for the exposition in the usual hyperbolic case. Finally the family of admissible functions \(h\) and \(k\) is studied, and by means of an example the authors show that a large class of functions is allowed, pertaining to systems not to be classified as being of exponential type.
Reviewer: S.Siegmund (Augsburg)
MSC:
| 34C30 | Manifolds of solutions of ODE (MSC2000) |
| 34C45 | Invariant manifolds for ordinary differential equations |
Keywords:
trichotomies; integral manifolds; asymptotic phase; asymptotic equivalence of solutions; invariant manifoldsReferences:
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