11 problems

Let the assumptions of Theorem hold. For $\epsilon>0$, let $(\bm{X}^{(\epsilon)}(t),\bm{w}^{(\epsilon)}(t))$ solve, and let $(\bm{X}^{(0)}(t))$ solve the limiting system with the s…

Let a tropical dynamical system have a tropical singularity at $(u_0,v_0)$ of sink, source, or saddle type. Let $X$ be a neighborhood of $(u_0,v_0)$. Tropical singularity persisten…

Fernández-Real–Ros-Oton conjecture. Suppose that $u:\mathbb{R}^n\to(0,\infty)$ minimizes $I_1$ locally. If $n\leq n_e^*-1$, then $u$ is one-dimensional: in a suitable Euclidean coo…

Let $H(z,\varepsilon)$ be a vector field satisfying Assumptions --, with slow manifold $S_{\varepsilon}$ and linear fast fibers $\mathcal{N}_{\varepsilon}(p)$ at basepoints…

Persistence conjecture. For one fast and two slow variables, the critical set $\mathcal{C}[g]$ is persistent under perturbations for $g\in V_f$ if all folds are non-degenerate, tha…

Let $u$ be the reduced dynamical system … associated with the planar dynamics in the paper, and let $\gamma_0$ be the periodic orbit whose existence is established by Lemma. Floqu…

Let $\alpha$ and $\xi$ be parameters with $\alpha>\xi$, let $\varepsilon>0$ be sufficiently small, and let $\Gamma_0$ denote the singular cycle described by the concatenation of th…

Let $A=(p^*,0)$, $B=(p_-,0)$, and $C=(p_-,s^*)$, where $s^*\approx 1.50815$. Consider the C-shaped homoclinic bifurcation curves in $(p,s)$-parameter space for the FitzHugh–Nagumo…

Let $X_-(b)$ and $X_+(b)$ be the two values of the trajectory quantity at the focus-node bifurcation point $b$, obtained from the two distinguished trajectories considered in the p…

Consider the system … Let $R$ denote the relief function and let $a$ and $b$ be the parameters appearing in the focus-node bifurcation, with $b$ the focus-node bifurcation point. T…

Let $t^{**}$ be the anti-bump and let $t^*_\mu$ be the corresponding bump, with these quantities defined by … Assume hypotheses HFN1 through HFN5. The focus-node persistence conjec…