82 problems

For the SIRS epidemic model with nonlinear incidence rate $\frac{kI^p}{1+\alpha I^q}$, where $p>0$ and $q\ge 0$ are arbitrary real numbers, determine the largest possible codimensi…

Nonlinear instability conjecture. In the range of parameters such that the total myosin mass $M$ of traveling wave solutions decreases as a function of $V$ for small $V$, these sol…

Let the system of equations be forced by time-dependent functions $Q=Q(t)$ and $s_2=s_2(t)$, and let $T_0(t)$ be the specified ramp function. MPT-like transition conjecture. The so…

Let the predator-prey model described in the paper be given, with its parameters and equilibria as defined there. Cyclical-fold conjecture. There exist parameter regimes under whic…

Let the predator-prey model described in the paper be given, with its interior equilibrium point and model parameters as defined there. Multiple-limit-cycle conjecture. There exist…

Let $(n_0,j_0)$ be a pair of natural numbers with $n_0\geq2$, $j_0\geq1$, and set … Suppose that the only natural number solution of the Diophantine equation … is $(j_0,n_0)$. Peri…

The spherical capillary water waves system is considered on the sphere, with small-amplitude time-periodic solutions as the objects of interest. Periodic-solution conjecture. There…

Branch-shape and uniqueness conjecture. The solution branches in cases VII2-1 and VII2-2 of Theorem VII1 have the same behaviour as the branches depicted in the case-IIb and case-V…

Let $\alpha$ be the parameter of the Langford vector field, and let $\alpha_1\approx0.697144898322973$, $\alpha_2\approx0.823$, $\alpha_3\approx0.925$,…

Bifurcation-locus conjecture. The bifurcation locus of the periodgon of this family is precisely $\Sigma$.

Let $f(z,w)=(p(z),q(w))$ be the product of two polynomial maps of degree $d$ in $\mathbb{C}$. The non-density conjecture. If $p$ or $q$ belongs to the bifurcation locus in…

Let $f$ be a mapping (neural network) with sufficiently high dimension $d$, and let $U$ be a bifurcation chain set as in the hyperbolicity violation conjecture. The existence of a…

Collinear stability classification conjecture. $\operatorname{L}_1,\operatorname{L}_2,\operatorname{E}_3$, and $\operatorname{A}_1$ are center-saddles whenever they exist;…

Triangular existence classification conjecture. There exist $\kappa_4(\mu)<\kappa_5(\mu)$ in $(0,\pi^2/4)$ such that there are two triangular relative equilibria,…

Let $M$ be a smooth, orientable, metrizable manifold of dimension at least $2$. A bifurcation ordering conjecture. There exists a class of bifurcations $\{B_\alpha\}_\alpha$ and a…

Geometric localization conjecture. Every coexistence equilibrium at which a Hopf or Bogdanov–Takens bifurcation occurs has prey coordinate in $(x_{\min},x_{\max})$, where…

Let the curves in Figure be energy curves for the problem, with solutions parametrized by their energy and with parameter $\lambda\in\mathbb{R}$. The conjectured joined diagram is…

Consider the piecewise glacial cycles system … exhibits a boundary equilibrium bifurcation. The claim concerns parameter-driven changes in the type of equilibrium, but the supplied…

Let a generic two-parameter glocal family of vector fields have a disconnected large bifurcation support. The basic list consists of the six unavoidable generic one-parameter degen…

Let $\v \in \operatorname{Vect}^{*}_k(\Omega)$, where $\Omega$ is an open subset of $S^2$ or $S^2$ itself, and suppose that $v$ contains only a generic degeneracy of codimension…

Let a glocal family of vector fields be structurally stable, meaning that it is equivalent to any nearby family in the relevant $C^k$ topology. SET conjecture. The family has the s…

Let a glocal family have the extension triviality property (ET), meaning that its glocal extensions are moderately equivalent near the relevant large bifurcation support to trivial…

Let a generic $3$-parameter family have a disconnected large bifurcation support. Three-parameter Cartesian-product conjecture. The family is equivalent either to a Cartesian produ…

Universal bifurcation diagram conjecture. There exists a homotopy

Let $\overline{\phi}\in \overline{K}(\mathbb{R})$ be an admissible kernel perturbation supported in $[-1/2,1/2]$, with $||\overline{\phi}||_1^m$ small. Consider parameters…