45 problems

Periodic-solution conjecture. If all solutions are uniform-bounded and uniform ultimate-bounded, then there exists a nonzero periodic solution with period $T$, which need not be un…

Rabinowitz's minimal periodic solution conjecture. Under these hypotheses, for every prescribed positive period, the system possesses a non-constant solution having that period as…

Let a soliton equation have a quasi-periodic background, and consider a short-range initial perturbation whose soliton phase shifts add up to zero. Soliton-only asymptotics conject…

Pugh's conjecture. This equation should have at most $k$ solutions.

Large-measure periodic-solution conjecture. For $M=0$, periodic solutions should exist for a large-measure set of values of the amplitudes.

Consider an ODE of the form of equation with a bistable function $f(y)$. Five-solution conjecture. Such an ODE has at most five $1$-periodic solutions. This conjecture concerns the…

Let $r(x,t)$ be a fixed resource environment satisfying (H0), and consider the corresponding time-periodic forager-exploiter system. A non-constant positive periodic solution is a…

Global-attractor conjecture. When a periodic solution exists, it should serve as a global attractor: trajectories are expected to converge to this unique periodic orbit.

Let $s_2$ denote the second segment of the numerically continued branch of periodic solutions in the one-dimensional Thomas model. A saddle-node bifurcation is a bifurcation at whi…

Let the numerically continued branch of periodic solutions in the one-dimensional Thomas model be the branch whose rigorously verified portion is described in the paper, and let th…

Conjectured six-body family. There is a family of solutions of this system parametrized by $\theta(T)=\theta_1$, for values of $\theta_1$ ranging from $\frac{\pi}{6}$ to $\pi$.

Altshuller multiplier conjecture. If such a multiplier exists, then there exists a nonzero periodic solution with period $T$, which need not be unique or attracting.

Let a stochastic time-periodic Newtonian system be a Newtonian system with periodic coefficients and stochastic forcing, and interpret recurrence through periodicity in distributio…

Perturbation-technique conjecture. The perturbation technique used in the cited work should work well for this model when studying the Turing instability of the symmetric periodic…

Reducible-tree conjecture. The infinite reducible tree and its rescalings according to the relevant scaling provide an effective description for the structure of time-periodic solu…

Existence conjecture. Space- and time-periodic solutions to the Euler equations should exist.

The quintic Swift–Hohenberg (SH) equation is the stationary planar equation obtained by including a quintic nonlinearity in the Swift–Hohenberg model. A localized pattern is a spat…

Consider the delayed elapsed-time model with refractory period $\sigma$ and delay kernel $\alpha$. Let $N(t)$ denote the discharging flux and $X(t)$ the total activity. Refractory-…

Let $N(t)$ be the discharging flux and $X(t)$ the total activity in the delayed equation, with delay parameter $d$. Periodicity conjecture. In the excitatory example, both $N$ and…

Let $N(t)$ be the discharging flux and $X(t)$ the total activity in the delayed equation, with delay parameter $d$. Periodic-profile conjecture. Unlike the instantaneous model, the…

Consider solutions $v\colon\mathbb{R}\to\mathbb{R}$ of the fractional equation … that satisfy … Here $P$ is the nonlocal operator … with the kernel $K$ defined by the fractional co…

Consider the heat equation on a finite interval with time-periodic Robin-type boundary conditions. The paper expects that solutions will generally become strongly asymptotically pe…

Consider a dynamical system with balanced loss and gain, whose equations of motion have no velocity-mediated interaction. Decompose its equations into linear and nonlinear parts, a…

Asymptotic label conjecture. There exists an index $j\notin I$ such that

Let $\mathcal L(t)$ be the time-periodic operator obtained by linearizing the cubic-quintic complex Ginzburg–Landau equation about a periodically stationary solution, let…