31 problems

In a smooth integrable dynamical system, let the system be defined near a nondegenerate singular point, meaning a singular point satisfying the paper's nondegeneracy condition for…

Let ${\cal F}^{(r)}(q,p)$ and ${\cal F}^{(2r-2)}(q,p)$, for $r=3,4,\ldots$, be perturbed harmonic oscillator Hamiltonians with potentials $V^{(r)}$ and $V^{(2r-2)}$ of degrees $r$…

Let $\xi$ be a braid word, and let $\zeta^*$ denote the introduced normal form of a word $\zeta$. A word $\zeta_1$ is a geodesic form of $\zeta$ when it has minimal length among wo…

Let $H$ be a locally analytic Hamiltonian function that is analytically integrable in the generalized Liouville sense, and let $(\hat{x}_i,\hat{y}_i,\hat{z}_j)$ be the normalizing…

Let $L$ be a real-analytic three-dimensional $\operatorname{gl}$-regular Nijenhuis operator given in the form $L_{\operatorname{comp1}}$. Let $f(x,y,z)$, $g(x,y,z)$ and $h(x,y,z)$…

Let $\mu$ be a complex number that is not a root of unity and $|\mu|=1$. A number $\mu$ is Brjuno when it satisfies the Brjuno condition referred to in the source. Let $Q_{\gamma}$…

Let $K_{\mathrm{CNF}}(I,J_2,\dots,J_n)$ be a truncated classical normal form near an index-1 saddle, where $I$ is the reaction integral and $J_k$ are the bath actions. On the norma…

Normal-form conjecture. In the second form, $u$ and $v$ satisfy restriction (a); in the third form, they satisfy restriction (b); and in the fourth form, they satisfy both restrict…

Consider the original Treschev problem, with coefficients $q_{2n}$ and a Diophantine rotation number. Here, $k>0$ is arbitrary and $C(k)$ denotes a constant depending on $k$. Tresc…

Formal linearization conjecture. The existence of $n-1$ functionally independent analytic strong first integrals implies that there exists a random coordinate transformation

Dolgachev–Duncan's conjecture. In characteristic $2$, every cubic surface can be written in Emch normal form. This conjecture concerns the existence of a uniform normal form in the…

Let … denote the spherical water waves equation. Lifespan conjecture. If the Diophantine equation … , the lifespan goes beyond $\varepsilon^{-(1+\alpha)}$ as $\varepsilon\to0$. Fin…

Let $f$ be a $C^k$ vector field with $k\ge 4$, and let $g$ be the saddle-node normal form in Theorem. At parameter value $\mu=0$, suppose the corresponding normal-form coefficient…

Let $F$ be an integrable two-degree-of-freedom Hamiltonian system with a parabolic orbit, and consider the smooth normal form and structural stability results established for a nei…

Convergence conjecture. If the BNF of $H$ is convergent and satisfies the non-degeneracy assumption that its frequency map is a local diffeomorphism, then there is a convergent nor…

Let a nilpotent linear part be $(n_1,\ldots,n_b)$-reducible, with total dimension $n$, and let $G_{\nabla_{\bar{\mathsf{M}}}}(1,t)$ denote the generating function of the kernel of…

Resummation conjecture. Resummation techniques for non-convergent series may be profitable in this setting.

Alternate normal form conjecture. Any such unfolding is orbitally conjugate to an unfolding of the form

Existence conjecture. For every given suitable system, there should exist a coordinate transformation establishing a nearby system that possesses an exact nonlinear normal mode.

Convergence conjecture. If the eigenvalues additionally satisfy good Diophantine properties, then the formal Taylor series defining the function $f$ is locally convergent. The form…

Summability conjecture. There is a normalizing transformation that is summable with respect to $y^2-x^3$.

Let $u(t)=0$ be the steady state of a state-dependent delay differential equation, and suppose its expansion has terms through a given order involving only constant delays, with al…

Consider a state-dependent delay differential equation and its expansion in which the terms through a specified order involve only constant delays, with the remaining terms collect…

Consider bifurcations of families of 1D-tori in generic fully coupled systems, and the normal-form results obtained for parametrized families of quasi-periodically forced Hamiltoni…

Let $F(x)=Ax+f(x)$ be a locally hyperbolic diffeomorphism near $0\in\mathbb{R}^n$, where $f(x)=O(|x|^2)$ is $C^\infty$ (respectively, analytic). Let $B$ be a real logarithm of $A$,…