The two-dimensional attractor of a differential equation with state-dependent delay.

*(English)*Zbl 1016.34075The state-dependent delay equation
\[
x'(t)=-\mu x(t)+f(x(t-r)),\quad r=r(x(t)), \tag{1}
\]
is considered. Here, \(\mu >0\), \(f\) and \(r\) are smooth real functions, \(r(0)=1\), and \(f\) satisifies the negative feedback condition \(x f(x)<0\) for all \(x\neq 0\). In addition, it is assumed that \(f\in C^1 (\mathbb{R}, \mathbb{R})\), \(f'<0\) and \(\sup f<\infty\) provided \(r(u)>0\) for all \(u\in \mathbb{R}\). Equation (1) is a well motivated equation, with interest in many applications. As it is pointed out by the authors, the goal of the paper is to describe the asymptotic behaviour of the slowly oscillating solutions to equation (1) – that is, solutions whose zeros are spaced at distanges larger than one –, in the spirit of the monograph by H.-O. Walther [Mem. Am. Math. Soc. 544 (1995; Zbl 0829.34063)], where the case \(r\equiv 1\) in (1) is addressed. However, the generalization of Walther’s results to the general case of (1) is far from being trivial.

In this very well organized paper, the authors prove carefully all necessary details to get similar conclusions to those obtained by Walther. They make excellent additional contributions to the investigation of equation (1), by proving interesting versions of some known results for the case of constant delay. Moreover, their extensive list of references is an excellent guide for recent works in state-dependent delay equations.

A key point in the paper is the choice of an appropriate phase space. In this regard, it is shown the existence of positive constants \(A,B,R,K\) such that the solutions to (1) generate a semiflow \(F\) on the compact set \[ L_k=\left\{\varphi\in X:\varphi([-R,0])\subset [-B,A],\;\left|\frac{\varphi(t)-\varphi(s)}{t-s}\right|\leq K \text{ for } -R\leq s<t\leq 0\right\}, \] where \(X=C([-R,0],\mathbb{R})\), with the usual supremum norm. Then it is proven that \(F\) leaves invariant the subset \(S\) of \(L_k\) with at most one sign change on all subintervals of \([-R,0]\) of length one.

The main results of the paper establish that the induced semiflow on \(S\) has a global attractor \(\mathcal A\) consisting of \(0\) and the segments \(x_t\) of the globally defined slowly oscillating solutions \(x:\mathbb{R}\to [-B,A]\) to equation (1), and, if \({\mathcal A}\neq \{ 0\}\), then \(\mathcal A\) is homeomorphic to the two-dimensional closed unit disc and the unit circle corresponds to a slowly oscillating periodic orbit. In addition, a Poincaré-Bendixson-type theorem is derived for the \(\alpha\)- and \(\omega\)- limit sets of phase curves in \(\mathcal A\).

In this very well organized paper, the authors prove carefully all necessary details to get similar conclusions to those obtained by Walther. They make excellent additional contributions to the investigation of equation (1), by proving interesting versions of some known results for the case of constant delay. Moreover, their extensive list of references is an excellent guide for recent works in state-dependent delay equations.

A key point in the paper is the choice of an appropriate phase space. In this regard, it is shown the existence of positive constants \(A,B,R,K\) such that the solutions to (1) generate a semiflow \(F\) on the compact set \[ L_k=\left\{\varphi\in X:\varphi([-R,0])\subset [-B,A],\;\left|\frac{\varphi(t)-\varphi(s)}{t-s}\right|\leq K \text{ for } -R\leq s<t\leq 0\right\}, \] where \(X=C([-R,0],\mathbb{R})\), with the usual supremum norm. Then it is proven that \(F\) leaves invariant the subset \(S\) of \(L_k\) with at most one sign change on all subintervals of \([-R,0]\) of length one.

The main results of the paper establish that the induced semiflow on \(S\) has a global attractor \(\mathcal A\) consisting of \(0\) and the segments \(x_t\) of the globally defined slowly oscillating solutions \(x:\mathbb{R}\to [-B,A]\) to equation (1), and, if \({\mathcal A}\neq \{ 0\}\), then \(\mathcal A\) is homeomorphic to the two-dimensional closed unit disc and the unit circle corresponds to a slowly oscillating periodic orbit. In addition, a Poincaré-Bendixson-type theorem is derived for the \(\alpha\)- and \(\omega\)- limit sets of phase curves in \(\mathcal A\).

Reviewer: Eduardo Liz (Vigo)

##### MSC:

34K25 | Asymptotic theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |