17 problems

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…

In a dynamical system with invariant tori, consider the regime in which some invariant tori have been destroyed. Chaotic-dynamics conjecture. It is generally conjectured that chaot…

The paper considers Hamiltonian systems with isoenergetic degeneracy and examples of such systems, for which the hypotheses of Theorem D may not hold directly. Modified applicabili…

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…

Kim–Ostlund conjecture. The spiral field should give the generalization of the noble numbers for two-dimensional maps: robust two-tori should have rotation vectors associated with…

Noble-number conjecture. The most robust invariant circles have noble rotation numbers.

Let $(\tau_1,\tau_2,K_\omega)$ be parameters for the perturbed system, and consider regions of partial mode-locking associated with invariant circles on the torus. Bogdanov–Takens…

Let a dynamical system be reversible with respect to an involution of type $(n,m)$, with $m\geq 1$ and $n\geq 2$, and let it have a symmetric Kronecker $n$-torus with a Diophantine…

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 a family of 1D-tori undergo a bifurcation, and consider the gaps in the families of 1D-tori arising from that bifurcation. Exponential-gap conjecture. These gaps should become…

A quaternion polynomial differential equation is a differential equation with quaternion-valued polynomial dynamics; its degree is the degree of the polynomial. An invariant torus…

Resonant-tori multiplicity conjecture. Under a convexity assumption on $H_0$, for any $r\leq n-1$ and for most families of unperturbed tori whose frequency vectors are resonant wit…

Let $d=2$, and consider invariant tori with rotation vectors in $\mathbb{R}^2$. An integral basis of a cubic field is an integral basis of a real algebraic field of degree $3$, vie…

Consider invariant circles of a twist map, each with a rotation vector, and call a rotation vector noble when it belongs to the noble class associated with the golden mean. MacKay'…

Let the spiral mean be a simple cubic irrational, and consider invariant tori of the volume-preserving map with rotation vectors associated with integral bases of the cubic field…

Let $DC(\tau)$ denote the Diophantine condition on the tangential and normal frequencies that guarantees persistence of lower-dimensional invariant tori for integrable Hamiltonian…