Local invariants of differential equations

We consider an analytic system $X=\Phi(X)$ in the neighborhood of the fixed point $X=0$. Depending on the characteristic numbers of the matrix $(\partial\Phi/\partial X)_0$, we define the integer $d\ge0$ as the ldquodimensionrdquo of the normal form or as the ldquomultiplicityrdquo of the resonance. We show that a system with $d=1$, subject to certain additional assumptions, has a finite number of invariants relative to reversible formal changes of variables $X=\Xi(Y)$. All these invariants are the coefficients of some normal form. We touch upon questions concerning invariants of relatively smooth and continuous substitutions.