Loss of stability in Hamiltonian systems that depend on parameters (Q1314197)

The paper is about a class of bifurcations of typical 1-parameter families of Hamiltonian systems \(H_ \varepsilon\). It is assumed that at \(\varepsilon = 0\), an equilibrium position \(x(0)\) has all its eigenvalues purely imaginary \((\pm i\omega_ k)\), with \(\omega_ 1 = \omega_ 2\), and that as \(\varepsilon\) goes through zero, the family of equilibria \(x(\varepsilon)\) sees two of its pairs of imaginary eigenvalues come together and begin to form a quadruple \(\pm\alpha \pm i\beta\). First, a normal form analysis of \(H_ \varepsilon\) is performed, assuming that there are no low-order resonances. In this process, suitable canonically conjugate variables are introduced and a crucial factor for the subsequent main theorem is the coefficient \(A\) of \((q^ 2_ 1 + q^ 2_ 2)^ 2\), where \((q_ i,p_ i,\;i = 1,2)\) roughly correspond to the coordinates in the plane where the bifurcation takes place. This main theorem asserts that for \(A < 0\), there is a hard loss of (metric orbital) stability, roughly meaning that solutions will leave any neighborhood of order 1 (in some measure theoretic sense), while for \(A > 0\) the instability somehow develops gradually. The proof of this theorem is very hard and relies in part on techniques of KAM-theory. Analogous results are further stated for the loss of stability in a similar situation of bifurcation of periodic solutions. Interesting applications are found in the planar restricted three-body problem.