13 problems

Julia-set characterization conjecture. If the boundary of ${{\mathrm{M}}_{CH}^T}$ consists only of $P$-starting separatrices, then ${{\mathrm{M}}_{CH}^T}$ coincides with

A nonlinear dynamical system may have a bounded invariant set whose basin of attraction has small measure and complicated structure, so that selecting an initial point in the basin…

Consider the stages of changing global dynamical structure under perturbations of increasing magnitude. Morse decomposition conjecture. We conjecture that this series of changes wi…

Exponential-dependence conjecture. An exponential dependence is unavoidable at this level of generality.

Consider the quasiperiodically forced dynamical system and the current setting of parameters. The outer dark-green regions are the regions identified numerically in the analysis, a…

Let $T_a$ be the fixed transformation, and let $f$ be a real-analytic function on either the real line or the circle: … Assume that $f$ is not affine. A set is jointly $T_a$- and…

Let $\mathcal{S}$ be the nontrivial symmetric invariant set of the map $G_{\varepsilon,3}$, and let $\varepsilon$ denote the coupling parameter. Invariant-sets-inside-$\mathcal{S}$…

Symmetric invariant-set conjecture. At every $\varepsilon_n$, new symmetric invariant sets appear; these sets are contained in $\mathcal{S}$ and are symmetric. The conjecture is mo…

Second asymmetric-set conjecture. There exists such a polyhedron $P_3$ for which $\mathcal{A}_2$ is symmetric with respect to $S_1,S_2$ and $S_4$, asymmetric with respect to…

Let $G_{\varepsilon,3}:\mathbb{T}^3\to\mathbb{T}^3$ be the piecewise affine map of the four-site globally coupled system, with coupling parameter $\varepsilon$, and let $S_G$ denot…

Let $G_{\varepsilon,3}:\mathbb{T}^3\to\mathbb{T}^3$ be the piecewise affine map describing the four-site system, and let $\varepsilon^*$ be the critical coupling value at which the…

Consider the scaled Hénon map … with $a=1.3$ and $b=0.3$, and its maximal invariant set. The stable and unstable manifolds associated with this invariant set may exhibit tangencies…

Let $f:\overline{{\mathbb R}^n}\to\overline{{\mathbb R}^n}$ be quasiregular with $\operatorname{deg}(f)>K_I(f)$. Let $E(f)$ be the exceptional set. A set $X$ is completely invarian…