11 problems

Alwash–Lloyd composition-center conjecture. All centers of the Abel equation are composition centers. This conjecture concerns the characterization of periodic behavior in Abel equ…

Let $A,B,I,J$ be the convex solutions of the Abel equations … … with the domains and normalizations described in the source: $A$ has a minimum at $x=1/2$, $B$ has a minimum at…

Let $f$ be a polynomial and consider the Abel equation … A center at $x=0$ means that solutions sufficiently close to $0$ satisfy $x(-1)=x(1)$. No-parameter center conjecture. If,…

Composition conjecture for moments. The moment conditions imply the composition condition: there should exist polynomials $\widetilde F$, $\widetilde G$ and $W$, with $F'=f$,…

Let $f=F'$ and $g=G'$ be polynomials, and let $A=[a,b]$. The polynomial Abel equation … has a center on $A$ if every solution with sufficiently small initial value satisfies…

Composition condition conjecture. The sufficient condition given in Theorem 1 is also necessary for a polynomial Abel differential equation to have a center on $[a,b]$.

Consider the higher-order Abel equation … with the composition condition that there is a differentiable function $q$ satisfying $q(\omega)=q(0)$ for some $\omega>0$ and … where the…

Composition-condition conjecture. For a polynomial Abel equation, having a parametric center is equivalent to satisfying the Composition condition (CC).

Moment and Melnikov composition conjecture. Vanishing of all the moments $m_k(P,Q)$ and of all the second Melnikov coefficients $D_j(P,Q)$ implies the Composition Condition.

Composition conjecture. The Center and Composition Sets for any polynomial Abel equation coincide.

Let $P(z)=\int p(z)\,dz$ and $Q(z)=\int q(z)\,dz$, where $p$ and $q$ are the polynomial coefficients in the Abel equation … A center is called reducible if there exist polynomials…