22 problems

Let $x$ be a slowly oscillating periodic solution (SOPS) of Wright's equation … with parameter $\alpha$ and period length $L(\alpha)$. Monotonicity conjecture. The period length of…

Periodic-solution conjecture. Under these assumptions, system $$ has a nontrivial slowly oscillating periodic solution.

Let … with parameter $\alpha>\tfrac{\pi}{2}$. A slowly oscillating periodic solution (SOPS) is a periodic solution $y(t)$ for which, up to time translation, there exist…

Supercritical degenerate Hopf-bifurcation conjecture. At the parameter value $h=h_H$ and solution $w=w^{c_1(h_H),d}$, the canonical equation, and hence the original car-following e…

Consider the scalar delay differential equation … under conditions (W), with the trivial solution and its local asymptotic stability understood as in the paper. Smith and Wright's…

Consider the Wright delay differential equation … Assume that all conditions (H) on $f$ are satisfied. Generalized Wright conjecture. If $-f'(0)<\pi/2$, then … for every solution…

Global-stability conjecture. The natural equilibrium $X_+^{**}$ is globally asymptotically stable relative to $\mathcal P_+$: $\mathcal P_+$ is positively invariant, $X_+^{**}$ is…

Consider the normalized delay differential equation referred to as, with solution $u(t;\beta,w_0)$ for parameter $\beta$ and initial velocity $w_0$. Let the comparison properties b…

Two-delay minimal spectral abscissa conjecture. For arbitrary $a_1,a_2\in\mathbb{R}$, the minimal spectral abscissa is

Consider the competition system … with positive initial data, basic reproduction numbers $\mathcal{R}_0^{(1)}>1$ and $\mathcal{R}_0^{(2)}>1$, boundary equilibria $E_1,E_2$, and coe…

Let $s\in[1,2]$ be arbitrary. Consider the delay differential equation (1), with coefficient $c(t)$ and delay function $\Greekmath 011C (t)$ satisfying the assumptions in (10a), an…

Consider the delay differential equation (1), with coefficient $c(t)$ and delay function $\Greekmath 011C (t)$ satisfying the assumptions in (10a). Assume that $c(t)\leq 0$ for…

Let $f\in C^{1}(\mathbb{R})$ satisfy conditions (H1)–(H3), and let $\beta$ be the balance point. Consider initial histories satisfying $x(\theta)\in[\chi^{-},\chi^{+}]$ for…

Let $\alpha>0$ and let $U(\alpha)$ be the forward extension by the semiflow of a local unstable manifold at zero for Wright's equation. Write … Let $\overline{U(\alpha)}$ denote th…

Continuity and real-analyticity conjecture. The function $\mathfrak{I}_{\mathcal{G}}(\sigma)$ is continuous in $\sigma$ and is real-analytic except at integers and half-integers.

Let $\alpha>\tfrac{\pi}{2}$, and consider the family of slowly oscillating periodic solutions (SOPS) of Wright's equation … An isola is a connected component of the SOPS family tha…

Let $u(t)=0$ be the steady state of a state-dependent delay differential equation, and suppose its expansion has terms through a given order involving only constant delays, with al…

Let $\mathfrak{D}_{n}$ denote the set of parameter pairs $(\tau,c)\in\mathbb{R}_+^2$ for which equation (17) admits a non-monotone wavefront, and let $c^*(\tau)$ and…

Let $I$ be the parameter interval for which the delay equation has no stable steady state or stable periodic solution. For $\phi\in C([-1,0],\mathbb{R})$, let $v_\beta(t;\phi)$ sol…

Let $I$ be the parameter interval for which the delay equation has no stable steady state or stable periodic solution. For $\phi\in C([-1,0],\mathbb{R})$, let $v_\beta(t;\phi)$ sol…

Zero-mean conjecture. For every such $\phi$ and every $\beta\in I$, $\mu(\beta;\phi)=0$. The authors state that this is unproved for irregular solutions; it holds for symmetric per…

Assume that $a\geq 0$ and that $f$ is an $S$-map. Let $K$ be the unique fixed point of $f$. Equation (1) has an equilibrium at $K$; a global attractor attracts the phase space, and…