17 problems

Let $D$ and $\sigma$ be parameters, and consider the linearization of the steady-state solutions with in-phase and anti-phase modes. The oscillations of the second cell centered at…

Let $\Gamma_0$ be the singular orbit defined in Assumption 5, and consider the corresponding discontinuity in the associated one-dimensional piecewise-affine map. Canard-explosion…

Consider the three-timescale dynamical system discussed in the paper, with positive timescale parameters $varepsilon$ and $delta$, and let the bifurcation diagram depend on the mod…

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…

In the stochastic three-species predator-prey model, let $N$ denote the number of small-amplitude oscillations between two spikes, and consider the distribution of $N$ for sample p…

An amplitude-modulated bursting (AMB) solution is a torus-canard-induced mixed-mode oscillation whose rapidly oscillating waveform has an envelope with a specified number of small…

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

The delayed FitzHugh–Nagumo system exhibits small-amplitude chaos and mixed-mode oscillations near a dynamic Bogdanov–Takens point. The existence conjecture. The existence of such…

For solutions entering the fold region from the attracting slow manifold or nearby, define the rotation number as the number of small loops made in the fold region. Slow-drift rota…

The model exhibits small-amplitude chaos in simulations, and the saddle-focus equilibrium has an unstable manifold whose normal hyperbolicity may fail away from the equilibrium. Ch…

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 $\bar\mu$ 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…

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…