Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback.

*(English)*Zbl 1004.34002
Fields Institute Monographs. 11. Providence, RI: American Mathematical Society. viii, 245 p. (1999).

The authors study the scalar delay differential equation
\[
\dot x(t)= -\mu x(t)+ f(x(t- 1)),\tag{\(*\)}
\]
where \(\mu\geq 0\) and \(f(x)\) is an increasing bounded \(C^1\) function with \(f(0)= 0\). A set of additional assumptions is imposed such that

(1) Equation \((*)\) has two more equilibria (positive and negative) both stable and attracting;

(2) The corresponding linearization at zero \[ \dot y(t)= -\mu y(t)+ f'(0) y(t-1) \] has at least a three-dimensional unstable subspace.

An invariant subset of the original nonlinear equation is described in great detail (which takes the whole space of 165 pages of the paper). The authors provide the following brief description of a main part of their results:

“Under natural and mild additional conditions the leading 3-dimensional local unstable manifold at the stationary point \(0\) extends in forward time to a smooth solid spindle with singularities at its tip, that are further stationary points both stable and attractive. An invariant smooth disk of solution curves winding from \(0\) towards a bordering unstable periodic orbit splits the spindle into invariant halves each of which is attracted to one of its tips.”

The book also contains a set of eight appendices which represent an independent interest by themselves.

(1) Equation \((*)\) has two more equilibria (positive and negative) both stable and attracting;

(2) The corresponding linearization at zero \[ \dot y(t)= -\mu y(t)+ f'(0) y(t-1) \] has at least a three-dimensional unstable subspace.

An invariant subset of the original nonlinear equation is described in great detail (which takes the whole space of 165 pages of the paper). The authors provide the following brief description of a main part of their results:

“Under natural and mild additional conditions the leading 3-dimensional local unstable manifold at the stationary point \(0\) extends in forward time to a smooth solid spindle with singularities at its tip, that are further stationary points both stable and attractive. An invariant smooth disk of solution curves winding from \(0\) towards a bordering unstable periodic orbit splits the spindle into invariant halves each of which is attracted to one of its tips.”

The book also contains a set of eight appendices which represent an independent interest by themselves.

Reviewer: A.F.Ivanov (Lehman)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34Kxx | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

37Cxx | Smooth dynamical systems: general theory |