308 problems

Density conjecture. The set of parameter values $\beta_2$ at which such special degenerate periodic orbits exist is dense in

Arnold's conjecture. Generic finite-parameter families of vector fields on the two-sphere are structurally stable: close families are weakly topologically equivalent.

Let a stable $p$-cycle belong to an invariant closed curve, and consider the three eigendirections of the Jacobian matrix of the $p$th iterate at each point of the cycle. Assume th…

Feigenbaum-number conjecture. The Feigenbaum number exists for fractional maps and fractional difference maps, and has the same value as for regular maps.

Let $H_2(n,N)$ denote the polynomial system whose BKK bounds are listed in the table, and let the resulting polynomial system be the one obtained from the averaged functions at ord…

Let $\chi$ be the fissility parameter and let $\chi_*$ be the bifurcation value. The Ren–Wei branch is the equilibrium branch emerging at this bifurcation. Subcritical pitchfork co…

Let $n\geq 3$, let $N\in\mathds{N}$, and let $\varepsilon>0$ be as in the preceding multiplicity result: for…

Consider the Loud normal form … and let $\Gamma_U$ denote the subset of parameter space left unspecified in the cited bifurcation analysis. A parameter is regular when the correspo…

Consider the $(r,\mathcal S)$-plane, where $r$ is the Bernoulli constant and $\mathcal S$ is the flow force constant. The cuspidal region is the region bounded by the two curves re…

Consider the radial Neumann problem discussed above in dimension $N=1$, with $R=1$, $p=2$, and nonlinearity $g$. For the power nonlinearity $g(s)=s^{q-1}$, the bifurcation branches…

Let $(u_\rho,\psi_{u_\rho})$ denote the stationary solutions constructed in Theorem 3, where the construction uses a sequence of parameters tending to infinity, and let $\lambda$ d…

Non-degeneracy conjecture. All frequency-locking intervals in the family away from $\beta=0$ are non-degenerate. Equivalently, at the boundaries of each phase-locking interval, the…

High-capacitance oscillation conjecture. The surviving upper Hopf bifurcation moves toward

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…

Connected Bone Conjecture. Every bone $B_\pm^{\rm cub}({\bf o})$ for the cubic family is a simple arc.

Newhouse–Palis conjecture. For generic planar diffeomorphisms, all explosions occur through one of two local bifurcations: saddle-node bifurcations or homoclinic bifurcations.

Let a generic family of maps depend on finitely many parameters, and consider the subset of parameter space corresponding to systems with infinitely many sinks or sources. Finite-a…

Consider the singular elliptic problem described above, with potential coefficient $K$ and singular nonlinearity $g$ satisfying the stated hypotheses. The parameters $\beta$ govern…

Infinitely many bifurcation trees conjecture. There are infinitely many bifurcation trees.

Let $F(h)=I_1(h)/I_0(h)$, where $I_0$ and $I_1$ are the Abelian integrals associated with the family of ovals, and let $\lambda\in\mathbb{C}$ be the parameter. Consider the equatio…

Let $h_a$ denote the bifurcation field for asymmetric solutions and $h_s$ the bifurcation field for symmetric solutions, and consider parameters $(\kappa,d)$ for which asymmetric s…

Pitchfork bifurcation conjecture. The triangular relative equilibria $\operatorname{T}_1$ and $\operatorname{T}_2$ bifurcate from $\operatorname{L}_3$ at $\kappa=\kappa_4(\mu)$.

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,…

Consider $C^1$ families of vector fields $\dot{s}=F_\omega(s)$ whose parameter $\omega$ varies in a smooth, connected, $k$-dimensional manifold, typically $\mathbb{R}^k$, with…