12 problems

Consider the Loud normal form … and let $\Gamma_U$ denote the subset of parameter space left unspecified in the cited bifurcation analysis. A parameter is regular when the correspo…

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 $\Delta$ be the germ of an interior local-bifurcation curve at $\nu=(-2,2)$, and let $\delta_1$ be the germ of an interior local-bifurcation curve at the Loud parameter…

Let $\Delta$ and $\delta_3$ be germs of curves contained in the local bifurcation set of the period function at the interior of the period annulus. A parameter region is one with e…

Let a quadratic center have a period annulus with an outer boundary, and define its criticality at the outer boundary as the maximal number of critical periodic orbits produced the…

Among quadratic centers, let $L_1,L_2,L_3$ be the three Loud parameter values at which the inner-boundary criticality attains its maximum value two. A twin pair consists of a Loud…

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…

Chicone's Loud normal form conjecture. If a quadratic differential system has a center with a period function that is not monotonic, then an affine transformation and a constant re…

The paper studies limit periodic sets in quadratic systems near a nilpotent saddle of multiplicity $3$. In Theorem, the setting is a convex graphic through such a saddle with…

Three-limit-cycle conjecture. At most $3$ limit cycles can exist around one focus point.

Three-zero conjecture. The integral $I(t)$ has at most three zeroes for $t\in(t_c,t_s)=\left(-\frac{1}{12(2-b)},0\right)$, corresponding to the period annulus around the center at…

The systems in cases ${\rm (r0)}$–${\rm (r22)}$ have period annuli whose cyclicity under small quadratic perturbations is being considered. Here, the cyclicity is the maximal numbe…