Divergence and summability of normal forms of systems of differential equations with nilpotent linear part
Schäfke, Reinhard ; Canalis-Durand, Mireille
HAL, hal-00144869 / Harvested from HAL
Text on HAL
We consider the prenormal form of a generic perturbation of the Hamiltonian system given by $h(x,y)=y^2-x^3$. The associated formal normal form is the system $\dot{x}=2y+2x\Delta^*$, $\dot{y}=3x^2+3y\Delta^*$, where $\Delta^*=A_0(h)+xA_1(h)$. We show that $A_0$, $A_1$ and the normalizing transformations are divergent, but $k$-summable, where the number $k$ depends on the first nonzero terms of $A_0$ and $A_1$.
Publié le : 2004-07-05
Classification:  normal form,  differential equation,  divergence,  summability,  34M,  [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] 
@article{hal-00144869,
     author = {Sch\"afke, Reinhard and Canalis-Durand, Mireille},
     title = {Divergence and summability of normal forms of systems of differential equations with nilpotent linear part},
     journal = {HAL},
     volume = {2004},
     number = {0},
     year = {2004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-00144869}
}

Schäfke, Reinhard; Canalis-Durand, Mireille. Divergence and summability of normal forms of systems of differential equations with nilpotent linear part. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00144869/