Summary
Inclination lemmas (λ-lemmas) serve as tools for the investigation of dynamical systems in the neighborhood of saddle points. They assert convergence of inclinations (slopes of tangents) when the system shifts a given transversal to the stable or unstable manifold towards equilibrium.
We derive estimates of the speed of this convergence, for preimagesg −k (H), k∈N, of a transversalH to the unstable manifold,g is assumed to be aC 2-map in a Banach space, not necessarily reversible, with a saddle pointx=0 and already normalized so that local stable and unstable manifolds are contained in linear spaces.
The estimates are needed particularly in a study of nonlocal bifurcation-from heteroclinic to periodic solutions of the second kind-for parameterized functional differential equations
which describe phase-locked loops
Zusammenfassung
Neigungslemmata (λ-Lemmata) dienen zur Untersuchung dynamischer Systeme in der Nähe von Sattelpunkten. Sie garantieren Konvergenz von Neigungen (Tangentensteigungen), wenn das System gegebene Transversalen zur stabilen oder instabilen Mannigfaltigkeit zum Gleichgewicht hin transportiert.
Wir leiten Abschätzungen der Geschwindigkeit dieser Konvergenz her, für Urbilderg −k (H), k∈N, einer TransversalenH zur instabilen Mannigfaltigkeit.g ist dabei eineC 2-Abbildung in einem Banachraum, nicht notwendig umkehrbar, mit Sattelpunktx=0 und schon normalisiert, so daß lokale stabile und instabile Mannigfaltigkeit in linearen Räumen liegen. Die Abschätzungen werden insbesondere zu einer Untersuchung nichtlokaler Verzweigung-von heteroklinen zu periodischen Lösungen zweiter Art-für parametrisierte Funktionaldifferentialgleichungen
benötigt, die PLL-Schaltungen beschreiben.
Similar content being viewed by others
References
J. K. Hale and X. B. Lin,Symbolic dynamics and nonlinear semiflows. Preprint LCDS # 84-8. Providence (R.I.) 1984.
J. K. Hale, L. T. Magalhaes and W. M. Oliva,An introduction to infinite dimensional dynamical systems-geometric theory. Springer. New York et al. 1984.
J. Palis,On Morse-Smale dynamical systems. Topology8, 385–404 (1969).
J. Palis and W. de Melo,Geometric theory of dynamical systems. Springer. New York et al. 1982.
L. P. Šilnikov,On a Poincaré-Birkhoff problem. Math. USSR — Sbornik3, 353–371 (1967) (Transl, of Mat. Sbornik74 (116), 1967).
L. P. Šilnikov,On the generation of periodic motion from trajectories doubly asympotic to an equilibrium state of saddle type. Math. USSR-Sbornik6, 427–438 (1968) (Transl. of Mat. Sbornik77 (119), 1968).
H. O. Walther,Bifurcation from a heteroclinic solution in differential delay equations. Trans. AMS290, 213–233 (1985).
H. O. Walther,Inclination lemmas with dominated convergence. Research report 85-03, Sem. Angew. Math., ETH Zürich 1985.
H. O. Walther,Bifurcation from a saddle connection in functional differential equations: An approach with inclination Lemmas. Preprint 1986.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Walther, H.O. Inclination lemmas with dominated convergence. Z. angew. Math. Phys. 38, 327–337 (1987). https://doi.org/10.1007/BF00945417
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00945417