Asymptotic recurrence in semiflows with an application to delay-differential equations (Q1591031)

Qualitative properties of the solutions to the problem \[ \dot x(t)=-b \bigl(1+x(t) \bigr)x(t-1),\;t>0,\quad x(t)= \varphi(t),\;-1\leq t\leq 0,\;\varphi(t)\in\Phi,\tag{1} \] are considered; \(\Phi\) unspecified, later \(\Phi=C\). The preliminary statements are correct: 1) If \(b<\pi/2\), \(\varphi(t)\equiv 0\), then \(x(t)\equiv 0\) is locally asymptotically stable, because all eigenvalues of the linear part of (1) have negative real parts. The stability is not global because for \(\varphi(t) \equiv -1\), the degenerate periodic solution \(x(t)\equiv -1\) is unstable. 2) \(b=\pi/2\) is a bifurcation value. 3) If \(b=\pi/2+a\), \(0<a\ll 1\), and \(\varphi(t)\) is appropriately chosen (it is not stated how), then there exist periodic solutions \(x(t)=v(t,b)\) close to \(x(t)\equiv 0\). The stability of \(x=v\) is not examined; no Krasovskij \(V\)-functionals are mentioned. It is also not mentioned that the `size' of the influence domain (``functional basin'') of \(x=v\) depends on the choisce of \(\Phi\). On p. 622 it is claimed that ``as a increases the solutions \(x(t)\) become more chaotic''; \(\varphi(t)\) is not specified. -- It is known that this claim is incorrect [cf. \textit{I. Gumowski}, C. R. Acad. Sci. Paris, Sér. A 268, 157-159 (1969; Zbl 0182.12801)]. At least up to \(a=1\) the \(v(t,b)\) is periodic and asymptotically stable. \(v(t,b)\) becomes merely more and more asymmetric, a part of \(v\) approaching \(x(t)\equiv -1\). -- The claim that for some \(\varphi(t)\) the solution \(x(t)\) becomes `recurrent' needs further elucidation, beyond Birkhoff's description. -- \(b\) has infinity of bifurcation values: \(a=2n\pi\), \(n=0,1, \dots.\) It is not yet known for \(n>0\) whether the Hopf method, or any other well understood one, is powerful enough to determine the corresponding bifurcated solutions, as well as their stability or instability.