11 problems

Birkhoff's conjecture. The retrograde periodic orbit spans a disk-like global surface of section.

Persistence conjecture. For every small $e>0$ there exists $\mu(e)$ for which the orbit $L_1$ has a homoclinic orbit. Moreover, there is a neighborhood $U$ of the curve…

Triangular stability classification conjecture. If $0<\mu<\mu_R$, then $\operatorname{L}_4$ and $\operatorname{L}_5$ are elliptic for every $\kappa>0$ for which they exist. If…

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,…

Continuation conjecture. These two relative equilibria are continuations of $\operatorname{L}_4$ and $\operatorname{L}_5$.

Existence classification conjecture. There exist $\kappa_i=\kappa_i(\mu)\in(0,\pi^2/4)$ with $\kappa_1<\kappa_2<\kappa_3$ such that the curved restricted three-body problem has exa…

Alekseev's conjecture. The set $OM$ has Lebesgue measure zero.

Consider the restricted three-body problem with strong potential exponent $\alpha\geq 2$, and let the second Lagrange point determine the threshold energy for which the Hill's regi…

Let $K$ be the footpoint projection of an original simple periodic orbit, let $n$ be a positive integer, and let $L$ be the footpoint projection of a new simple periodic orbit born…