Global attraction for Wright-type delayed feedback
Let $f \in C^3(\mathbb{R}, \mathbb{R})$ satisfy $f(0) = 0$ and $u f(u) < 0$ for $u \ne 0$. Assume that $f$ is bounded below and has at most one critical point, which, if it exists, is a local extremum, and that the Schwarzian derivative $$(Sf)(u) = \frac{f^{\prime\prime\prime}(u)}{f'(u)} - \frac{3}{2}\left(\frac{f''(u)}{f'(u)}\right)^2$$ is negative at every $u$ with $f'(u) \ne 0$. Consider $$x'(t) = f(x(t-1)) \quad (t \ge 0), \qquad x(s) = \phi(s) \quad (-1 \le s \le 0),$$ with a continuous initial history $\phi : [-1, 0] \to \mathbb{R}$. The zero solution of the linearized equation $x'(t) = f'(0)\, x(t-1)$ is exponentially stable exactly when $0 < -f'(0) < \pi/2$. Liz, Pinto, Robledo, Trofimchuk and Tkachenko proved that every solution tends to $0$ when $-f'(0) \le 3/2$. Díaz, Hasík, Kopfová and Trofimchuk extended this to $-f'(0) \le 37/24$ for decreasing $f$, with interval-arithmetic computations, and to $-f'(0) \le \pi/2$ for decreasing $f$ with $f''(0) = 0$. For $f(u) = \alpha(e^{-u} - 1)$ the equation becomes Wright's equation $y'(t) = -\alpha\, y(t-1)\,(1 + y(t))$ in the variable $y = e^{-x} - 1$; Wright proved convergence to $0$ for $\alpha \le 3/2$; Bánhelyi, Csendes, Krisztin and Neumaier extended this to $\alpha \le \pi/2 - 1.9633 \cdot 10^{-4}$ (2014), and van den Berg and Jaquette completed the proof for every $\alpha \in (0, \pi/2]$ (2018).
Problem. If $0 < -f'(0) < \pi/2$, must every such solution satisfy $\lim_{t \to \infty} x(t) = 0$?
Source: Liz, Pinto, Robledo, Trofimchuk and Tkachenko (2003), Conjecture 1.2, generalizing Wright's conjecture (1955).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, delay differential equations, Schwarzian derivatives, Wright's equation
References
@article{liz2003,
author = {Liz, Eduardo and Pinto, Manuel and Robledo, Gonzalo and Trofimchuk, Sergei and Tkachenko, Victor},
title = {Wright type delay differential equations with negative {S}chwarzian},
journal = {Discrete Contin. Dyn. Syst.},
volume = {9},
number = {2},
year = {2003},
pages = {309--321},
doi = {10.3934/dcds.2003.9.309},
eprint = {math/0111285},
archivePrefix = {arXiv},
}
@article{wright1955,
author = {Wright, E. M.},
title = {A non-linear difference-differential equation},
journal = {J. Reine Angew. Math.},
volume = {194},
year = {1955},
pages = {66--87},
doi = {10.1515/crll.1955.194.66},
}
@article{diaz2026,
author = {D{\'i}az, Mauro and Has{\'i}k, Karel and Kopfov{\'a}, Jana and Trofimchuk, Sergei},
title = {Global stability of {W}right-type equations with negative {S}chwarzian},
journal = {SIAM J. Appl. Dyn. Syst.},
volume = {25},
number = {2},
year = {2026},
pages = {964--997},
doi = {10.1137/25M1737298},
eprint = {2502.15965},
archivePrefix = {arXiv},
}
@article{banhelyi2014,
author = {B{\'a}nhelyi, Bal{\'a}zs and Csendes, Tibor and Krisztin, Tibor and Neumaier, Arnold},
title = {Global attractivity of the zero solution for {W}right's equation},
journal = {SIAM J. Appl. Dyn. Syst.},
volume = {13},
number = {1},
year = {2014},
pages = {537--563},
doi = {10.1137/120904226},
}
@article{vandenberg2018,
author = {van den Berg, Jan Bouwe and Jaquette, Jonathan},
title = {A proof of {W}right's conjecture},
journal = {J. Differential Equations},
volume = {264},
number = {12},
year = {2018},
pages = {7412--7462},
doi = {10.1016/j.jde.2018.02.018},
eprint = {1704.00029},
archivePrefix = {arXiv},
}
@misc{aim-generalized-wright-global-attraction,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global attraction for Wright-type delayed feedback}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/322-generalized-wright-global-attraction.md},
note = {Entry 322 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00524, "Global attraction for Wright-type delayed feedback", clopen.solutions.