×

Limit cycle bifurcations for a kind of Hamilton systems of degree three. (Chinese. English summary) Zbl 1399.37033

Summary: By using the Picard-Fuchs equation method, we obtain an upper bound of the number of zeros of abelian integrals \(I\left(h \right)={\oint_{{\Gamma_h}}}g\left({x, y} \right)dx-f\left({x, y} \right)dy\), where \({\Gamma_h}\) is the closed orbit defined by \(H\left({x, y} \right)={x^2}+{y^2}+2xy+\frac{1}{a} (x^4+y^4)=h\), \(a > 0\), \(h \in (0, +\infty)\), \(f (x, y)\) and \(g (x, y)\) are real polynomials in \(x\) and \(y\) of degree \(n\). Therefore, we get the upper bound of the number of limit cycles of this system.

MSC:

37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
PDFBibTeX XMLCite