27 problems

Let $\mathcal{T}_0$ be an analytic Lagrangian Diophantine quasi-periodic torus invariant under a real-analytic Hamiltonian system. A set of tori is accumulated by $\mathcal{T}_0$ i…

In a real-analytic, nearly integrable Hamiltonian system with three or more degrees of freedom, consider the relative measure of the phase space that is free of invariant tori. Arn…

Consider an analytic Hamiltonian system obtained as a perturbation of an integrable Hamiltonian system, whose unperturbed dynamics admit quasi-periodic solutions on invariant tori.…

Let $H(p,q):=\frac{1}{2}p^2+V(q)$ be the classical Hamiltonian on ${\cal P}:=T^*{\bf T}$, where ${\bf T}:={\bf R}^d/{\cal L}$ and $V\in C^\infty({\bf R}^d,{\bf R})$ is ${\cal L}$-p…

Greene's rational-iterate conjecture. Every KAM curve $gamma$ can be constructed as the limit of an appropriate family of $m(n)$-periodic discrete invariant sets:

Consider sufficiently smooth approximations of Coulomb friction in the forced vibro-impact oscillator, and let the Hamiltonian islands and saddle-center bifurcation be those descri…

Let $V_N$ denote the relative phase-space volume of the union of KAM islands for an $N$-particle system at fixed energy density and fixed forcing parameters. Vanishing-of-islands c…

Consider a general Hamiltonian system with $n$ degrees of freedom and action-angle variables $(x,y)$, whose Hamiltonian is $H(x,y)$. Minimum regularity conjecture. The minimum regu…

In a Hamiltonian system with action-angular variables and $n$ degrees of freedom, let $H$ denote the Hamiltonian function. Minimum regularity conjecture. The minimum regularity req…

In the outer, retrograde configuration, let $\mathcal{D}_{{{{{{{{{\rm t} t} t} t} t} t} t} t} s}$ be the $8$-dimensional phase-space region almost completely filled with a positive…

Apsidal-locking conjecture. The planets $\upsilon$Andromedæ c and $\upsilon$Andromedæ d were conjectured to be in such an apsidal locking state shortly after their discovery.

Consider the Hamiltonian system in the source, with $m_0$ denoting the dimension of the periodic angle variable $u$ and with the mean perturbation taken over the $x$-angles. Suppos…

Let $H_0(y)$ be a nondegenerate integrable Hamiltonian with $n$ degrees of freedom, and consider resonant invariant $m$-tori of a sufficiently small perturbation. Kappeler–Pöschel…

Let $\upsilon$ Andromedae c and d be the two planets in the $\upsilon$ Andromedae planetary system, and consider the difference of their arguments of pericenter. Librational-regime…

Consider the $d$-dimensional nonlinear Schrödinger equation with periodic boundary conditions … where $V*$ denotes convolution with the potential $V$. Bourgain's conjecture. Full-d…

Herman's conjecture. Without any non-degeneracy assumptions on $H$, there always exists such a set $\mathcal{K}_r$ with positive Lebesgue measure.

Minimal-regularity conjecture. The class $C^n$ is the minimal regularity assumption for KAM theory to apply in this setting, while still allowing Herman's $C^{n-\epsilon}$ countere…

Let $\varepsilon$ be a complex parameter measuring the perturbation from the symplectic case, and suppose the conformal factor has the form … where $a\in\mathbb{Z}_+$ and…

Whiskered tori are invariant tori with hyperbolic directions in a Hamiltonian system. Arnold's whiskered-torus conjecture. Whiskered tori should be the key geometric structures lea…

Simulation conjecture. There exists a set

Katok–Herman conjecture. K.A.M. theory is perturbation theory for rotations in tori, and only rotations in tori can serve as its linear model.

Arnold's conjecture. In a generic system with three or more degrees of freedom, the measure of the non-torus set has order $\varepsilon$.

Fix an initial value $z(0)=z_0$. The $xy$-plane is divided into disjoint KAM and non-KAM regions, and a KAM region is one of the regions in which the associated trajectories are tr…

KAM existence conjecture. There exist three explicit functions $a,b_d,b_a:\mathbb R^+\to\mathbb R^+$ such that

A posteriori KAM existence conjecture. There exist three explicit functions $a,b_d,b_a:\mathbb R^+\to\mathbb R^+$ such that