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Dr. Vincent Naudot

Dr. Rindy Anderson
  • Associate Professor
  • Department of Mathematical Sciences
  • Boca Raton - SE43, Room 278


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

Research Interests

Research Description

Bifurcation Theory : 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 degenerate homoclinic orbit, See Ref  [1,2,3,4,5,10].  We show the existence of chaosstrange attractors [1, 3] and the existence of infinitely many homoclinic doubling cascades [2,4].    Techniques developed in [1,3] and those in [7,8] (see section ('normal for theory' below) are used to investigate strange attractor with a large entropy [10]. This later results reveals the only knowledge of low order terms in asymptotics are not always  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. See [12,13].  In order to classify all possible Morse-Smale portrait, we proceed to a surgery 

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.     

Normal Form Theory  : Motivated by the research done in homoclinic bifurcations, we are interested in the Dulac map of germ of a vector field. A result is proposed in [6] int he case of a saddle in the three-dimensional space. Other result concern a direct (asymptotic) computation of a linearisation, conjugating the germ
with its linear part. First results are obtained in the case of a single vector field [7,8] and later in terms of family [17]. These results ohold in any dimension. A non mooth normal form theory is develop (which generalise that of Poincare-Dulac) that eliminate resonant term by resonant term.   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. In [18] such expansions are studied in the frame of quasi periodic bifurcation. We study analytic properties of compensator and Dulac expansions in a single variable.  We first consider Dulac expansions when the power of the  logarithm is either  0 or 1. Here we construct an explicit exponential scaling  in the space of coefficients, which in an exponentially narrow horn, up to rescaling and division, leads to a polynomial expansion. A similar result holds for the compensator case. 

Recent Publications

  • M. Fontaine, W. D. Kalies, V. Naudot, A Reinjected Cuspidal Horseshoe. Discrete and Continuous Dynamical Systmem, (Supp. 2013), 227-226.
  • V. Naudot, J. Yang, Quasi-linearization of parameter depending germs of vector fields. Dynamical Systems28, no 2, (2013), 173-186.
  • V. Naudot, E. Noonburg, Predator-Prey System with a General Non- Monotonic Response Function, Physica D,  253, (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)

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