Primary-canard and exponentially narrow transition conjecture

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Let μˉ(ε)\bar\mu(\varepsilon) be a parameter function for the canonical three-dimensional system, and let μˉ0=−a(a+b)2b2\bar\mu_0=-\frac{a(a+b)}{2b^2}. Primary-canard transition conjecture. There exists a function μˉ(ε)\bar\mu(\varepsilon) such that, for μˉ=μˉ(ε)\bar\mu=\bar\mu(\varepsilon), the system has a primary S1DS_{1D} canard; the transition from mixed-mode oscillations to spiking occurs within an exponentially small interval O(e−c/ε)O(e^{-c/\varepsilon}), with c>0c>0, about μˉ(ε)\bar\mu(\varepsilon); and μˉ(0)=μˉ0=−a(a+b)2b2\bar\mu(0)=\bar\mu_0=-\frac{a(a+b)}{2b^2}. This combines the conjectured persistence and uniqueness of the singular canard with the predicted exponentially narrow transition.

References

Primary source

Jozsi Jalics, Martin Krupa and Horacio G. Rotstein, “A novel canard-based mechanism for mixed-mode oscillations in a neuronal model”, arXiv:0804.0829 (2008).

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