Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback
著者
書誌事項
Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback
（Fields Institute monographs, 11）
American Mathematical Society, c1999
- タイトル別名
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An unstable set for delayed positive feedback
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
This book contains recent results about the global dynamics defined by a class of delay differential equations which model basic feedback mechanisms and arise in a variety of applications such as neural networks. The authors describe in detail the geometric structure of a fundamental invariant set, which in special cases is the global attractor, and the asymptotic behavior of solution curves on it. The approach makes use of advanced tools which in recent years have been developed for the investigation of infinite-dimensional dynamical systems: local invariant manifolds and inclination lemmas for noninvertible maps, Floquet theory for delay differential equations, a priori estimates controlling the growth and decay of solutions with prescribed oscillation frequency, a discrete Lyapunov functional counting zeros, methods to represent invariant sets as graphs, and Poincare-Bendixson techniques for classes of delay differential systems.Several appendices provide the general results needed in the case study, so the presentation is self-contained. Some of the general results are not available elsewhere, specifically on smooth infinite-dimensional center-stable manifolds for maps. Results in the appendices will be useful for future studies of more complicated attractors of delay and partial differential equations.
目次
Introduction The delay differential equation and the hypotheses The separatrix The leading unstable set of the origin Oscillation frequencies Graph representations Dynamics on $\overline W$ and disk representation of $\overline W \cap S$ Minimal linear instability of the periodic orbit $\mathcal O$ Smoothness of $W \cap S$ in case $\mathcal O$ is hyperbolic Smoothness of $W \cap S$ in case $\mathcal O$ is not hyperbolic The unstable set of $\mathcal O$ contains the nonstationary points of bd$W$ bd$W$ contains the unstable set of the periodic orbit $\mathcal O$ $H \cap \overline W$ is smooth near $p_0$ Smoothness of $\overline W$, bd$W$ and $\overline W \cap S$ Homeomorphisms from bd$W$ onto the sphere and the cylinder Homeomorphisms from $\overline W$ onto the closed ball and the solid cylinder Resume Equivalent norms, invariant manifolds, Poincare maps and asymptotic phases Smooth center-stable manifolds for $C^1$-maps Smooth generalized center-unstable manifolds for $C^1$-maps Invariant cones close to neutrally stable fixed points with 1-dimensional center spaces Unstable sets of periodic orbits A discrete Lyapunov functional and a-priori estimates Floquet multipliers for a class of linear periodic delay differential equations Some results from topology Bibliography Index.
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