Canards Existence in the Hindmarsh–Rose model

In two previous papers we have proposed a new method for proving the existence of “canard solutions” on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2016) 381–431; J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh-Rose model.

The genus of regular languages

The paper defines and studies the genus of finite state deterministic automata (FSA) and regular languages. Indeed, an FSA can be seen as a graph for which the notion of genus arises. At the same time, an FSA has a semantics via its underlying language. It is then natural to make a connection between the languages and the notion of genus. After we introduce and justify the the notion of the genus for regular languages, the following questions are addressed. First, depending on the size of the alphabet, we provide upper and lower bounds on the genus of regular languages: we show that under a relatively generic condition on the alphabet and the geometry of the automata, the genus grows at least linearly in terms of the size of the automata. Second, we show that the topological cost of the powerset determinization procedure is exponential. Third, we prove that the notion of minimization is orthogonal to the notion of genus. Fourth, we build regular languages of arbitrary large genus: the notion of genus defines a proper hierarchy of regular languages.

Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models

In a previous paper we have proposed a new method for proving the existence of “canard solutions” for three- and four-dimensional singularly perturbed systems with only onefastvariable which improves the methods used until now. The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with twoslowand twofastvariables. This method enables stating a unique generic condition for the existence of “canard solutions” for such four-dimensional singularly perturbed systems which is based on the stability offolded singularities(pseudo singular pointsin this case) of thenormalized slow dynamicsdeduced from a well-known property of linear algebra. This unique generic condition is identical to that provided in previous works. Application of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables showing the existence of “canard solutions” in such systems.

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

For a piecewise expanding unimodal interval map$f$with unique absolutely continuous invariant probability measure$\mu $, a perturbation$X$, and an observable$\varphi $, the susceptibility function is$\Psi _\varphi (z)= \sum _{k=0}^\infty z^k \int X(x) \varphi '( f^k)(x) (f^k)'(x) \, d\mu $. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps.Comm. Math. Phys.275(2007), 839–859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps.Nonlinearity21(2008), 677–711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer–Simon [Natural boundaries and spectral theory.Adv. Math.226(2011), 4902–4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet.Ann. Sci. Éc. Norm. Supér.(3)66(1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that$\Psi _\varphi (z)$has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of$\Psi _\varphi (z)$, associated with precritical orbits. If the perturbation$X$is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of$\Psi _\varphi (z)$as$z\to 1$exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map (‘linear response formula’). Applying the Wiener–Wintner theorem, we study the singularity type of non-tangential limits of$\Psi _\varphi (z)$as$z\to e^{i\omega }$for real$\omega $. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.