13 problems

Lum–Chua conjecture. Continuity across the switching line prevents the appearance of more limit cycles.

Han–Zhang conjecture. Such a system has no more than two limit cycles.

Consider the $2\pi$-periodic scalar differential equation … where … A non-constant periodic solution is a limit cycle when it is isolated among the periodic solutions. The five-lim…

Let … where $\mathbf{x}=(x\ y)^T$, $\mathbf{A}^\pm=(a_{ij}^\pm)$ are constant $2\times 2$ matrices, and $\mathbf{b}^\pm=(b_1^\pm,b_2^\pm)^T$ are constant vectors in $\mathbb{R}^2$.…

Finite convergence conjecture. If $T$ is symmetric and positive definite, then the semi-smooth Newton iteration converges after finitely many steps.

Subcritical transitory-canard bifurcation conjecture. In the subcritical case, the system possesses a saddle-node bifurcation of transitory canard cycles for $k=1$.

Transitory canard cycles conjecture. There exist values of $\varepsilon>0$ and $k\approx 1$ such that the corresponding system exhibits one, two, or three canard limit cycles in a…

Two-crossing-periodic-orbits conjecture. System $(\mathrm{DPWL})$ has at most two crossing periodic orbits.

A discontinuous piecewise linear system in the plane has two regions separated by a straight line. A crossing limit cycle is a limit cycle that crosses the separating line. Two-cro…

Planar discontinuous piecewise linear differential systems with two linearity regions separated by a straight line consist of two linear differential systems joined across that lin…

Consider discontinuous piecewise linear differential systems in the plane with two pieces separated by a straight line, and consider the class in which nested limit cycles surround…

A continuous piecewise linear vector field in the plane is a vector field whose two linear pieces fit continuously across their separating straight line. A limit cycle is an isolat…

Braga–Mello conjecture. Given $n\in\mathbb{N}$, there is a piecewise linear system with two zones in the plane with exactly $n$ limit cycles.