8 problems

Let $H$ be the Hamiltonian and let $\gamma$ denote the relevant family of cycles. For each $k\in\mathbb{N}$, let $\nu_{H,\gamma}(k)$ be the Noetherianity index giving the optimal b…

Let $H$ be the Hamiltonian and let $cgamma$ be the associated family of cycles, with $n_r$ the universal Noetherianity index at level $r$ and $\nu_r$ the corresponding optimal Noet…

Optimal orbit-depth bound conjecture. The orbit depth is an optimal bound for the length of Melnikov functions: the length $\ell$ can attain the orbit depth $k$.

Orbit-depth and Melnikov-length conjectures. (i) For any polynomial $F$ and any non-trivial cycle $\gamma$ of $F$, either the depth is unbounded, or it is $1$, or $2$. (ii) For any…

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…

Let $F\in\mathbb{C}[x,y]$ be a polynomial, let $\gamma\in\pi_1(\Gamma_t,p_0)$ be a loop, and let $k=k(F,\gamma)$ be its orbit depth. For a deformation of $dF=0$, denote by…

Finiteness conjecture. For any $\gamma$, the torsion-free orbit depth $k$ and the orbit depth $\kappa$ are finite.

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.