55 problems

Lum–Chua conjecture. Continuity across the switching line prevents the appearance of more limit cycles.

Han–Zhang conjecture. Such a system has no more than two limit cycles.

For each degree $d$, let $\mathcal S_d$ be the collection of polynomial vector fields in the plane of degree $d$. Write $F_\mu$ for the vector field whose coefficients are encoded…

Let $g(x)=x$, and let $f(x)$ be a polynomial of degree $n\geq 1$. Consider the generalized Liénard equation … Here, $[r]$ denotes the integer part of $r$. Lins–de Melo–Pugh conject…

Multiplicity-$(n+1)$ limit-cycle conjecture. There is an open subset $W$ of this surface in the induced topology such that, for every $(\lambda_1,\ldots,\lambda_n)\in W$, a typical…

Consider the Liénard system … where $F(x)$ is a polynomial of odd degree $m$, and let $N$ be determined by $m=2N+1$ or $m=2N+2$. Lins–Pugh–de Melo conjecture. The system has at mos…

Let the non-smooth system be the system denoted by … g(x,y,varepsilon)=alphai h'(x)+varepsilon y quadtext{if }xin(ai,a{i+1}],quad i=0,1,ldots,n, … with this function $g$ has no lim…

Consider the system … with the parameter normalized to $\epsilon=1$ by a linear change of coordinates. Two-limit-cycle conjecture. The resulting system for $\epsilon=1$ has at most…

Consider the $2\pi$-periodic scalar differential equation … where … A non-constant periodic solution is a limit cycle when it is isolated among the periodic solutions. The five-lim…

Let … where $\mathbf{x}=(x\ y)^T$, $\mathbf{A}^\pm=(a_{ij}^\pm)$ are constant $2\times 2$ matrices, and $\mathbf{b}^\pm=(b_1^\pm,b_2^\pm)^T$ are constant vectors in $\mathbb{R}^2$.…

Let $\mathscr P$ be the period annulus of the center at $(0,\frac{1}{2})$ of the quadratic differential system considered in the source, with parameters $(a_0,b_0)$, and let the sh…

Let $a_1$ and $a_2$ be parameters satisfying … In this parameter region, the system has no small limit cycles and at most four large limit cycles. Two-large-limit-cycle conjecture.…

Consider system … is exactly $3$. This conjecture is motivated by the system's symmetry and by numerical simulations showing one large limit cycle when there are two small limit cy…

Let $X=(P,Q)$ be a planar polynomial vector field of degree $n$, let $\pi(X)$ denote its number of limit cycles, and define the Hilbert number by … where $\mathcal{X}_n$ is the fam…

Let $L,C,R$ be the parameters of system … admits a unique stable limit cycle containing $\mathcal{S}$, $\mathcal{A}$, and $\mathcal{B}$ in its interior on the subdomain (iv)$_+$. T…

Let the classical Liénard system be … where $F(x)=\sum_{i=0}^{n}a_i x^i$ is a polynomial and $[\cdot]$ denotes the integer function. Liénard's conjecture. The classical Liénard sys…

Let $m\geq 6$ be even, let $f(x)$ be a polynomial of degree $n$, and consider the discontinuous system … where the zero set of $\operatorname{sgn}(g_m(x,y))$ is the union of $m/2$…

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…

Consider the quadratic three-dimensional systems referred to as systems … is $12$. The paper reports that, among the systems examined, only one reaches this bound; the conjecture i…

Four-limit-cycle conjecture. There exist parameter values for which the unique positive equilibrium is asymptotically stable and is surrounded by four limit cycles $\Gamma_0$,…

Let $X(\rho)$ be the non-negative random variable obtained as the almost-sure limit of the number of limit cycles in the disk $\mathbb{D}_{\rho}$ for the perturbed random center-fo…

A planar system is considered in the setting of the paper, with the limit cycles described there and with parameters satisfying suitable conditions. Existence conjecture. There exi…

Let $H(n)$ denote the upper bound for the number of limit cycles bifurcating from the period annulus around the origin, over all polynomial perturbations $p^{\pm}(x,y)$ and…

A discontinuous piecewise linear system in the plane has two regions separated by a straight line. A crossing limit cycle is a limit cycle that crosses the separating line. Two-cro…

Consider two-dimensional piecewise-linear continuous ordinary differential equation systems in scenarios for which an attracting annulus has been proved to exist. Lum and Chua's co…