19 problems

Let $O$ be the collection of vector fields under consideration, let $\mathcal{L}$ be the collection of Lorenz maps that form factor maps for vector fields $V\in O$, and let…

Consider a diatomic Fermi–Pasta–Ulam system in a singular limit whose center manifold remains two-dimensional, while its stable and unstable manifolds are infinite-dimensional. Hom…

The paper considers a piecewise-smooth planar non-autonomous system with four possible geometric scenarios for the local unstable and stable manifolds near a homoclinic trajectory.…

Consider a heteroclinic-cycle type in the reversible fourth-order Hamiltonian system associated with the generalized nonlinear Schrödinger equation, and its constituent periodic or…

Let the coupled nonlinear Schrödinger equations be the system given by … possess transverse super-homoclinic orbits. The conjecture asserts that the global stable and unstable inva…

Let $\gamma_3$ be the homoclinic family, let $\mathfrak{D}$ be the critical curve, and let $m_1,m_2,m_3$ be the mass parameters. Gamma-3 separation conjecture. The $\gamma_3$ famil…

Let $\gamma_2$ be the triple Copenhagen homoclinic family, let $\mathfrak{D}$ be the critical curve, and let $L_0$ and $L_2$ be the relevant libration points. Gamma-2 critical-curv…

Let $\gamma_1$ be the triple Copenhagen homoclinic family and let $\mathfrak{D}$ be the critical curve. Gamma-1 non-continuation conjecture. The $\gamma_1$ family never continues t…

Let $\gamma_{1,2,3}$ denote the short triple Copenhagen homoclinic orbits and let $\mathfrak{D}$ be the critical curve in parameter space. Belyakov-Devaney conjecture. Any paramete…

Let $\gamma_3$ be the triple Copenhagen homoclinic family, let $\mathfrak{D}$ be the critical parameter curve, and consider the homoclinic orbit around $L_3$ on $\mathfrak{D}$. Gam…

Let $\hat{s}_{1,1}$ be the terminal parameter of the $\gamma_1$ homoclinic family along $\ell_1(s)$. Gamma-1 termination conjecture. … Numerical continuation gives the lower bound…

Let $\hat{s}_{3,1}$ and $\hat{s}_{2,1}$ denote the terminal parameters of the $\gamma_3$ and $\gamma_2$ homoclinic families along $\ell_1(s)$. Gamma-3 symmetry conjecture. … This i…

Let $\hat{s}_{2,k}$ denote the terminal parameter for the $\gamma_2$ homoclinic family along the parameter line $\ell_k(s)$, for $k=1,2,3$. Gamma-2 continuation conjecture. … The n…

Let $\hat{s}_{1,k}$ denote the terminal parameter for the continuation of the $\gamma_1$ family along the parameter line $\ell_k(s)$, for $k=1,2,3$. Numerical lower-bound conjectur…

Consider the Hill's lunar problem with homogeneous potential exponent $\alpha\geq 2$, and let the two Lagrange points be its distinguished equilibria. Nonexistence conjecture. Ther…

Let $l_{A=0}$ be the curve forming the right boundary of the region $LA$, and let $O$ be the equilibrium of the Lorenz system. A homoclinic loop to $O$ is a trajectory leaving $O$…

Let $Z$ be a piecewise vector field in $\mega^r$ admitting a sliding Shilnikov orbit $\Gamma$. For every $s\geq r$, let $\mathcal{U}\subset\mega^r$ be a neighbourhood of $Z$. Smoot…

Let $A=(p^*,0)$, $B=(p_-,0)$, and $C=(p_-,s^*)$, where $s^*\approx 1.50815$. Consider the C-shaped homoclinic bifurcation curves in $(p,s)$-parameter space for the FitzHugh–Nagumo…

Let $H_\epsilon$ be a time-reversible Hamiltonian with stable and unstable manifolds parameterized by $\mathbf{\Gamma}^s$ and $\mathbf{\Gamma}^u$, and let $\gamma_h$ be a primary s…