The cusp of order n (Q2640749)

The bifurcation diagram of the family \((1)\quad \dot x=y,\quad \dot y=x^ 2+\lambda_ 0+\lambda_ 1y+\lambda_ 2xy+\lambda_ 3x^ 3y+...\pm x^{\ell}y,\) where \(\ell =[3(n-1)/2]\) (n\(\geq 2)\) and \(\lambda_ i\) are small, is studied. In the case \(\lambda_ 0<0\), where a focus and a saddle coexist, the investigation of the number of limit cycles between the focus and the stable (or instable) manifold of the saddle point is reduced to the study of zeros of \[ (2)\quad \int_{\Gamma_ h}(\nu_ 1y+\nu_ 2xy+\nu_ 3x^ 3y+...\pm x^{\ell}y)dx+o(\delta), \] where \(\lambda_ 0=-\delta^ 4\), \(\lambda_ i=\delta^{2(\ell -i+1)}\nu_ i\) (1\(\leq i\leq \ell -1)\), \(\Gamma_ h\) is an oval of the Hamiltonian \(H(x,y)=y^ 2/2+x-x^ 3/3\) (-2/3\(\leq h\leq 2/3)\) and o(\(\delta\)) is a function of \(\delta\), h and the parameters. The author gives the theorem: Theorem. Let \(\delta =0\). Then the bifurcation diagram of the zeros of (2) is the same as that of the zeros of a polynomial of degree n-1 over a finite interval.