64 problems

Let $f$ satisfy $f(0)=0$ and $f'(0)>0$. For the characteristic equation at $x=0$, … write $\theta=rf'(0)-\gamma$ and let $W_m$, $m\in\mathbb Z$, denote the $m$-th branch of the Lam…

Let $r$ be the parameter in Wright's equation, and consider solutions $y$ satisfying $y(t)>-1$. Wright proved that every such solution converges to $0$ as $t\to\infty$ when…

Let $f\in C^3(\mathbb R)$ satisfy the hypotheses $(\mathbf H)$: $xf(x)<0$ for $x\ne0$ with $f'(0)<0$; $f$ has at most one critical point $x^*$, with $f(x)>C$ for some…

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…

Let $f\bin C^3(\mathbb{R},\mathbb{R})$ satisfy (H1)–(H3), and consider the Mackey–Glass type delay equation … Its equilibrium is $e(t)\equiv 0$. Local-to-global stability conjectur…

Typicality conjecture. The repeating resonance structure for large $\tau$ is a typical phenomenon of periodically forced delay differential equations.

Asymptotic stability conjecture. The entire region $a<0$ and $a<b<-a$ is asymptotically stable for every $q\in(0,1)$.

Let $\nu$ be the invariant measure in Corollary for the stochastic Wright equation, and let $\sigma_0$ denote the noise parameter. Uniqueness conjecture. In the presence of noise,…

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…

Consider the delayed population model with fixed delay $\tau\in\mathbb{N}$ and positive equilibrium $N^*_+$. Theorem-based analysis gives conditions under which $N^*_+$ is locally…

Let $r(\mu)$ denote the low-density consumer growth rate as a function of the reproductive delay $\tau$. Suppose there are two critical values $0<\tau_*\leq\tau^*$ such that…

Generalization conjecture. The results on the existence and properties of a discrete Lyapunov function for cyclic systems can be further generalized to these time-dependent-delay s…

Let the branch of maximum-period orbits in the two-parameter bifurcation diagram terminate at a point on the branch of fold bifurcations, and consider the homoclinic orbits describ…

Instability conjecture. The regions bounded by the curves $Gamma_1$ and $Gamma_2$ are unstable.

For the positive-feedback delay equation studied in the paper, let $\Lambda(\tau)$ denote the critical semicycle length for $\tau\in(1/e,1)$. Critical semicycle length problem. Det…

Let $x$ solve the first-order delay differential equation … where $0\leq p(t)\leq 1$ and $0\leq t-\tau(t)\leq 2.75+\ln 2$ for $t\in\mathbb{R}$. An oscillatory solution is a solutio…

Mutual-stabilization conjecture. The four-variable case may be similar to the two-variable case if the agents mutually stabilize.

Let $l>-\frac{1}{2}$, and let $z_l$ be the real-root curve defined near $k=l+\frac{3}{2}$ by … Here $I_l$ denotes the maximal interval on which $z_l$ is defined. Maximal-interval c…

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

The Mackey–Glass delay differential equation is being considered in a parameter regime where a sequence of subcritical period-doubling bifurcations has been observed, followed by a…

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…

Consider the systems referred to as the delayed consensus models with proportional and reactive anticipation, denoted by (in the source) systems … . The question concerns whether a…

Let an autonomous periodic ordinary differential equation have a hyperbolic periodic orbit with an odd number of real Floquet multipliers larger than one. The odd-number limitation…

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…