Étude de stabilité d'un système non linéaire de dimension 4 en combustion et généralisation à une classe de problèmes homogènes de degré 2. (Study of the stability of a nonlinear system of dimension 4 in combustion and generalization to a class of homogeneous problems of second order) (Q1082519)

We consider a nonlinear differential system modelling a two-step reaction in a plane premixed flame. Of special interest is the behaviour of the system in a neighbourhood of the space variable bound \(+\infty\). An autonomous and second order homogeneous system approximates it here, for which the boundary condition \(0\in {\mathbb{R}}^ 4\) appears to be a degenerate fixed point. More generally, we study the stability of the equilibrium point \(0\in {\mathbb{R}}^ n\) for the system \[ (1)\quad \ddot x=F(x),\quad x\in {\mathbb{R}}^ n, \] F being a second order homogeneous function. The homogeneity property allows us to reduce the dimension of the system to 2n-1 (instead of 2n) by means of a change of the unknowns x, ẋ, and variable. The stationary points of the reduced system are usually hyperbolic, and their asymptotic analysis can be lifted back to get a stability theorem for (1). These results are illustrated by the analysis of the combustion problem. For a special value of a physical real parameter, a bifurcation phenomenon occurs.