11 problems

The regularized fold-fold system has locally unique periodic orbits near the Hopf bifurcation and near the Canard curve. Periodic-orbit continuation conjecture. The “small” curves…

Let a slow-fast system on the two-torus be a generic system of the type considered in the context, and suppose the special family … has, for a sequence of intervals…

Let $J$ and $\tau$ be the parameters of the delayed FitzHugh–Nagumo system, and let $y=\pm 2/3$ denote the branch of folds. The system has a subcritical canard explosion along this…

For fixed $\varepsilon>0$, define the rotation number as the number of small loops made by a solution while passing through the fold region; secondary canards form boundary curves…

Let $\bar\mu(\varepsilon)$ be a parameter function for the canonical three-dimensional system, and let $\bar\mu_0=-\frac{a(a+b)}{2b^2}$. Primary-canard transition conjecture. There…

Let $\bar\mu_0=-\frac{a(a+b)}{2b^2}$ be the parameter value corresponding to the polynomial singular canard $\bar\gamma_{\rm poly}$. Transition-locus conjecture. $\bar\mu_0$ is the…

Let $\bar\gamma_{\bar\mu}$ be the special solution of the rescaled system, and consider its intersections with the nullcline $\bar y=\bar x^2$. Transversality conjecture. The least…

Let $\bar\mu$ denote the parameter in the rescaled system and let $\gamma_{\rm poly}$ be the algebraic singular-canard solution. Transition-limit conjecture. The $\bar\mu$ value co…

In the simplified three-time-scale system, a special solution exists for a particular parameter value. Parameter-uniqueness conjecture. Based on numerical evidence, this parameter…

Consider the three-time-scale model with a singular canard represented by the explicitly found polynomial vector solution. Uniqueness conjecture. The mentioned polynomial solution…

The model has a singular primary canard solution, namely the special solution separating mixed-mode oscillations from spiking. Canard-limit conjecture. The singular canard solution…