Marc Diener

Regularizing microscopes and rivers

This paper proposes a generalization of the existing geometric studies of resonance. The Riccati equation associated with any second-order linear equation is extended to any

The steady-state phase distribution of the motor switch complex model of Halobacterium salinarum.

\mathcal{C}^\infty

Ducks and rivers: three existence results

first-order equation. The Morse-critical point is generalized to any “generic” critical point. The resonant solution becomes a general canard solution. The paper explains how to find the regularizing blowup, and shows how classical special functions become enlarged in rivers, i.e., some resurgent solutions of polynomial differential equations. The paper shows a matching principle that connects the slow solutions with these rivers. The method to show the existence of canards is applied for some Union-Jack equations, i.e., equations with a critical point where three smooth curves intersect.

A Pedestrian Proof of the Hopf-Bifurcation Theorem

Steady-state analysis is performed on the kinetic model for the switch complex of the flagellar motor of Halobacterium salinarum (Nutsch et al.). The existence and uniqueness of a positive steady-state of the system is established and it is demonstrated why the steady-state is centered around the competent phase, a state of the motor in which it is able to respond to light stimuli. It is also demonstrated why the steady-state shifts to the refractory phase when the steady-state value of the response regulator CheYP increases. This work is one aspect of modeling in systems biology wherein the mathematical properties of a model are established.

Nonstandard analysis in practice

The beginnings of nonstandard analysis in Prance are strongly related to the study of singular perturbations of the Van der Pol equation [110] and its ducks [29, 12]. This equation is an interesting and simple example of a slow-fast equation, in other words, an equation of the form: where e > 0 is a fixed i-small number, f a near-standard function, and f0 := 0f. We will study here a typical equation of such a kind and this will provide us with the opportunity to look at some of the tools that have been developed for their study.

Chasse au canard (première partie)

Having shown the short-shadow lemma, we apply this result of non standard analysis to two classical problems : small perturbations of the harmonic oscillator and the Hopf-bifurcation.