The evolution of the stable and unstable manifold of an equilibrium point
The author deals with a Hamiltonian system with two degrees of freedom depending on a parameter \(\nu\) which has an equilibrium point at the origin. Assuming that the eigenvalues of the linearized system are complex for \(\nu<0\) and purely imaginary for \(\nu>0\), the author shows that there exist two-dimensional stable and two-dimensional unstable manifolds for \(\nu<0\) which are gone if \(\nu>0\). The evolution of these submanifolds as \(\nu \to 0\) depends on a certain quantity \(\eta\). In the case \(\eta<0\), the submanifolds detach from the equilibrium as single invariant manifolds, in the case \(\eta>0\) they shrink to the equilibrium and disappear. The proof uses the Sokols'skij normal form of the Hamiltonian \(H=x_2y_1- x_1y_2+\delta {1\over 2} (x^2_1+x^2_2) +\nu\delta {1\over 2}(y^2_1+ y^2_2)+\eta \delta[ {1\over 2}(y^2_1 +y^2_2)]^2\), and a symplectic change to polar coordinates.
Hamiltonian system with two degrees of freedom
scientific article; zbMATH DE number 1287753