30 problems

Minkowski dimension conjecture. Any such spiral trajectory has Minkowski dimension

Global stability conjecture. Suppose all eigenvalues of $A_L$ and $A_R$ have negative real part. If $H(y_1)$ is an asymptotically stable equilibrium of the limiting fast system for…

Consider the SIRS compartmental model with fast information and misinformation spreading, and suppose that the slow flow converges in a damped oscillatory manner towards an endemic…

Consider the family of homoclinic orbits along the segment $\widehat{\mathrm{BE}}_1^P$, the Welander-type large periodic orbit, and the maximally sliding orbit $\gamma^*_1$. Singul…

Damped-nudging conjecture. A damped nudging update can improve the basin of convergence of the optimal-balance iteration.

Extension conjecture. Optimal balance, together with the analysis presented in the paper, should extend beyond systems with symplectic slow manifolds to a much larger class of syst…

Locally invariant manifold conjecture. Locally invariant manifolds of this type do indeed exist, and the resulting dynamics includes complex interactions between oscillators.

Consider the two different maximal canards of the nonsmooth system with reset parameter $k$, and let $k$ tend to $0$. In the corresponding smooth slow-fast systems, there is only o…

Let $k$ be the reset parameter, and consider the different maximal canard cycles in the reset-induced nonsmooth system that are exponentially close in parameter space but well sepa…

Consider system … and suppose it has a canard cycle with $N>2$ resets, as described in Proposition 2. Reset-induced canard-cycle doubling conjecture. In some sufficiently small nei…

Let $varepsilon$ and $delta$ be positive perturbation parameters, and suppose that the assumptions concerning the parameters and reduced flow hold. Assume that the singularities of…

Let the assumptions concerning the parameters, reduced flow, and slow drift hold, and suppose that the singularities of the system are remote. Let $\theta$ denote the system parame…

Let $H(z,\varepsilon)$ be a vector field satisfying Assumptions --, with slow manifold $S_{\varepsilon}$ and linear fast fibers $\mathcal{N}_{\varepsilon}(p)$ at basepoints…

Let $f\in C^{1}(\mathbb{R})$ satisfy conditions (H1)–(H3), and let $\beta$ be the balance point. Consider initial histories satisfying $x(\theta)\in[\chi^{-},\chi^{+}]$ for…

Phase-transition conjecture. Phase transitions may play a central role in the equilibration process between microscopic and macroscopic variables; for example, this may occur in an…

Equation-free approximation conjecture. The function $\tilde f_L$ approximates $\tilde f$, up to a coordinate change from $x_L$ to $x$, for sufficiently small $\delta$ and sufficie…

A toral folded singularity is a folded singularity of the averaged radial-slow system associated with a fold of limit cycles. A toral FSN II is a toral folded singularity that coin…

Consider a slow–fast system with a family of limit cycles and with at least three fast variables. The nontrivial Floquet exponents $\varphi_i$ of a limit cycle satisfy … A fold of…

A toral FSN II is a folded singularity of the averaged radial-slow system that occurs when an ordinary singularity crosses the fold curve, under the conditions … The averaged radia…

A torus canard is a trajectory in a slow–fast system whose rapid oscillations follow a family of limit cycles and whose slow drift passes through a folded singularity of the averag…

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…

Good-system conjecture. Good systems are topologically generic in the space of slow-fast systems.

Generic coexistence conjecture. For a locally topologically generic slow-fast system on the two-torus, there exists a sequence of intervals on the ray $\{\varepsilon>0\}$ accumulat…

Canard-cycle conjecture. For every desired number $l\in\mathbb{N}$ there exists a local topologically generic set of systems such that the slow curve is nondegenerate and simple un…

The delayed FitzHugh–Nagumo system displays canard explosions, period-doubling bifurcations, chaotic small oscillations, mixed-mode oscillations, fast oscillations, bursting, and i…