No abstract

A cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate toWe define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent toOur main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to

No abstract

No abstract

No abstract

No abstract

The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

The paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions. We follow a geometric approach based on the use of C k -normal forms, centre manifolds and (family) blow up, as we did in (Trans. Amer. Math. Soc., to appear). In (Trans. Amer. Math. Soc., to appear) we considered the existence of manifolds of canard solutions for given appropriate boundary conditions. These manifolds need not be smooth at the turning point. In this paper we essentially study the transition time along such manifolds, as well as the divergence integral, providing a structure theorem for these integrals. As a consequence we get a nice structure theorem for the transition equation, governing the canard solutions. It permits to compare different control manifolds and to obtain a precise description of the entry-exit relation of different canard solutions. Attention is also given to the special case in which the canard manifolds are smooth, i.e. when "formal" canard solutions exist.