Detecting alien limit cycles near a Hamiltonian 2-saddle cycle
From the authors' abstract and text: The paper aims at providing an example of a cubic Hamiltonian 2-saddle cycle which after bifurcation can give rise to an alien limit cycle; this is a limit cycle which is not controlled by a zero of the related abelian integral. To guarantee the existence of an alien limit cycle one can verify generic conditions on the abelian integral and on the transition map associated to the connections of the 2-saddle cycle. In this paper, a general method is developed to compute the first and the second derivative of the transition map along a connection between two saddles. Next, a concrete generic Hamiltonian 2-saddle cycle is analyzed using these formulas to verify the generic relation between the second-order derivative of both transition maps, and a calculation of the Abelian integral. The principal result in this paper establishes the presence of this bifurcation phenomenon of alien limit cycles in the unfolding of the quadratic Hamiltonian system with two centers and two heteroclinic loops \[ \begin{matrix}\dot{x}=-H_y+\varepsilon[\bar{\mu}_3xy +\bar{\mu}_4y^2x+H(x-\frac{\sqrt3\pi}{8}xy)],\\ \dot{y}=H_x+\varepsilon y(\bar{\mu}_1+\bar{\mu}_2x),\end{matrix} \] where the Hamiltonian \(H\) is given by \(H(x,y)=y(x^2+\frac{1}{12}y^2-1)\). Verification of the generic conditions which guarantee the presence of alien limit cycle that bifurcates from the 2-saddle cycle lying in \(\{y\leq 0\}\) is quite involved.
alien limit cycle perturbation of Hamiltonian system
scientific article; zbMATH DE number 5648548