10 problems

Consider the piecewise-defined system of equations and classify its attracting equilibria as regular, boundary, or virtual according to their locations relative to the switching do…

Let a tropical dynamical system have a hyperbolic crossing cycle $\gamma_0$. Crossing-cycle persistence conjecture. For all $0<\epsilon\ll 1$, the associated smooth system has a li…

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…

Consider the regularized planar piecewise smooth system in case $VI_3^\epsilon$, with the two families of periodic orbits obtained respectively in Proposition and Proposition. Peri…

Braga–Mello conjecture. Given $n\in\mathbb{N}$, there is a piecewise linear system with two zones in the plane with exactly $n$ limit cycles.

Let $n\ge4$ be even. Let $\rho$ and $\phi$ be the game parameters, let $\pi_0$ and $b_{n-2}$ be the quantities defined for the stationary dynamics, and let $G_{n,n-2}$, $E_{n-2}$,…

Let $(\rho,\phi)$ be the parameters of the collective Parrondo game, and let the parameter plane be divided into the regions shown in Figure. For a finite word $w$ in the game symb…

Let $\mathcal{F}^n=\{df=0\}$ be a degree $n$ Hamiltonian foliation, and let $\delta$ be a Hamiltonian identity cycle, meaning an identity cycle of this foliation. A degree $n$ pert…

Uniform boundedness conjecture. For any pair of rational $1$-forms $(\theta,\omega)$ on the real plane $\mathbb R^2$ of degrees $n,m$ respectively, such that $d\theta=0$ and…