Limit cycles for multidimensional vector fields. The elliptic case
Consider the \((m+2)\)-dimensional system of real ordinary differential equations \[ \dot x=H_y+ \mathbf{z}\cdot\mathbf{R}+ \varepsilon P, \qquad \dot y=-H_x+\mathbf{z}\cdot \mathbf{S}+\varepsilon Q, \qquad \dot{ \mathbf z}=\mathbf{Az}+ \varepsilon\mathbf{b}, \] where \(\varepsilon\) is a small parameter, \(H=H(x,y)=y^2-x^3+3x\) is the Hamiltonian, \(P=P(x,y;\mathbf{z},\varepsilon)\) and \(Q=Q(x,y;\mathbf{z},\varepsilon)\) are scalar functions, \({\mathbf A}(x,y;\mathbf{z})\) is a matrix function, and \({ \mathbf R}={\mathbf R}(x,y;\mathbf{z}),\) \({\mathbf S}={\mathbf S}(x,y;\mathbf{z})\) and \({\mathbf b}={\mathbf b}(x,y;\mathbf{z},\varepsilon)\) are vector functions. All these functions are polynomial in \((x,y)\) and analytic in \((\mathbf{z},\varepsilon)\) near \(({\mathbf 0},0).\) Notice that when \(\varepsilon=0\) the above system has, in the plane \({\mathbf z}={\mathbf 0}\), a continuum of periodic orbits \(\gamma_h\) given by \(\{(x,y): H(x,y)=h,\, x<1 \},\) for \(-2<h<2.\) The authors study which of these periodic orbits remains for the perturbed system when \(\varepsilon\) is small enough. As a first step, the authors prove that, if Re\(| \mu_i(x,y)| >2\sqrt3,\,i=1\ldots,m,\) where \(\mu_i(x,y)\) are the eigenvalues of \({\mathbf A}(x,y;{\mathbf 0}),\) then the perturbed system has, for \(\varepsilon\) small enough, a \(C^1\) invariant surface \(L_\varepsilon\) given by an equation of the form \({\mathbf z}=\varepsilon {\mathbf g}(x,y)+ O(\varepsilon^2),\) where \({\mathbf g}\) satisfies the differential equation \(d{\mathbf g}/dt:={\mathbf g}_xH_y-{\mathbf g}_yH_x={\mathbf A g}+{\mathbf b},\) and all expressions are evaluated at \( {\mathbf z}={\mathbf 0}\). The key point for the proof of the above result is the notion of normal hyperbolicity developed by Hirsch, Pugh and Shub. Then, it is proved that the simple zeroes of the Poincaré-Pontryagin-Melnikov-Arnold integral defined for \(h\in(-2,2)\) as \[ I(h)=\int_{\gamma_h}(Q+{\mathbf g}\cdot {\mathbf S})\,dx-(P+{\mathbf g}\cdot{\mathbf R})\,dy, \] give rise to limit cycles of the perturbed system contained in \(L_\varepsilon.\) The rest of the paper is devoted to try to give un upper bound on the number of zeroes of \(I(h)\) in terms of \(n=\max(\deg{P},\deg Q,\deg {\mathbf R},\deg{\mathbf S},\deg {\mathbf b}),\) for a fixed \({\mathbf A.}\) This is the most difficult part of paper. As a first step, the authors are able to give a representation of \(I\) in the form \(I(h)=(P_1(h)I_1(h)+\cdots+P_k(h)I_k(h) )/P_0(h),\) where \(P_0,P_1,\ldots, P_k\) are real polynomials of degree at most a \(K({\mathbf A})(n+1),\) where \(K({\mathbf A})\) is a constant which just depends on \({\mathbf A,}\) \(I_i(h)\) are real functions defined in \((-2,2)\) and \(k\leq m+2.\) Their proof follows ideas of the study of Petrov on the hyperelliptic integrals. From this representation of \(I\), they are able to prove that, when \(m=1\) and \({\mathbf A}\) is a constant matrix, satisfying the above hypotheses on its eigenvalues, then the number of zeros of \(I(h)\) in the interval \((-2,2)\) is bounded by \(C(n+1),\) for some constant \(C.\) The authors also comment that their proof of this last result can not be easily extended either to the case \(m>1\) or to the case \({\mathbf A}\) not constant.
Poincaré-Pontryagin-Melnikov-Arnold integral
scientific article; zbMATH DE number 1924863