Â鶹ŮÀÉ

Education

• Ph D in Mathematics, Universit´e de Bourgogne, 1996 Ìý Ìý

Research Interests

• Bifurcation Theory
• Mathematics in Biology
• Normal Form Theory

Research Description

Bifurcation TheoryÌý: Starting from the time I was a Ph-D student, my interest is with the dynamics that arise after perurbation of degenerate dynamical systems. In particular, I have studied the case where a given vector field possesses a degenerateÌýhomoclinic orbit, See RefÌýÌýÌý We show the existence ofÌýchaos,Ìýstrange attractorsÌýÌýand the existence of infinitely manyÌýhomoclinic doubling cascadesÌý.ÌýÌý ÌýTechniques developed in [1,3] and those in [7,8] (see section ('normal for theory' below) are used to investigate strange attractor with a large entropyÌý. This later results reveals the only knowledge of low order terms in asymptotics are not alwaysÌý sufficient to describe/anticipate the dynamics.

Mathematics in Biology: Classical normal form theory is used to investigated the dynamics of Predator-Prey systems with their bifurcation pattern. A 2-dimensional predator-prey model with five parameters is investigated, adapted from the Volterra-Lotka system by a non-monotonic response function. A description of the various domains of structural stability and their bifurcations is given. The bifurcation structure is reduced to four organising centres of codimension 3. Research is initiated on time-periodic perturbations by several examples of strange attractors. SeeÌý.Ìý In order to classify all possible Morse-Smale portrait, we proceed to a surgeryÌý

applied to limit-cycle and elaborate a classification of the Morse-Smale portraits 'Modulo' the limit cycles. This is of great help to anticipate the dynamics and their bifurcation. ÌýÌýÌýÌý

Normal Form TheoryÌýÌý: Motivated by the research done in homoclinic bifurcations, we are interested in theÌýDulac mapÌýof germ of a vector field. A result is proposed inÌýÌýint he case of a saddle in the three-dimensional space. Other result concern a direct (asymptotic) computation of aÌýlinearisation, conjugating the germ
with its linear part. First results are obtained in the case of a single vector fieldÌýÌýand later in terms of familyÌý. These results ohold in any dimension. AÌýnon mooth normal form theoryÌýis develop (which generalise that ofÌýPoincare-Dulac) that eliminate resonant term by resonant term. Ìý This non smooth normal form carries Dulac or compensator expansion. A Dulac expansion is formed by monomial terms that may contain a specific logarithmic factor. In compensator expansions this logarithmic factor is deformed. InÌýÌýsuch expansions are studied in the frame of quasi periodic bifurcation. We study analytic properties of compensator and Dulac expansions in a single variable.Ìý We first consider Dulac expansions when the power of theÌý logarithm is eitherÌý 0 or 1. Here we construct an explicit exponential scalingÌý in the space of coefficients, which in an exponentially narrow horn, up to rescaling and division, leads to aÌýpolynomial expansion. A similar result holds for the compensator case.Ìý

Recent Publications

• M. Fontaine, W. D. Kalies, V. Naudot, A Reinjected Cuspidal Horseshoe.ÌýDiscrete and Continuous Dynamical Systmem,Ìý(Supp. 2013), 227-226.
• V. Naudot, J. Yang, Quasi-linearization of parameter depending germs of vector fields.ÌýDynamical Systems,Ìý28,Ìýno 2, (2013), 173-186.
• V. Naudot, E. Noonburg, Predator-Prey System with a General Non- Monotonic Response Function,ÌýPhysica D,Ìý Ìý Ìý Ìý 253,Ìý(2013), 1-11.
• Q. Lu, V. Naudot, Bifurcation complexity from orbit-flip homoclinic orbit of weak type. To appear in International Journal of Bifurcation and Chaos. (2015)
• S. Ippolito, V. Naudot, E. Noonburg. Alternative Stable States, Coral Reefs, and Smooth Dynamics with a Kick. To appear in Bulletin of Mathematical Biology. (2015)

Personal Website

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