Description
Book SynopsisContains 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. This book describes the geometric structure of a fundamental invariant set, which in special cases is the global attractor.
Table of ContentsIntroduction 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.
