Stability in a Hamiltonian system
A Hamiltonian system (1) \(\dot x_ j=-\partial H/\partial y_ j\), \(\dot y_ j=\partial H/\partial x_ j\) \((j=1,2)\) where \[ Hu(1-u\rho_ 1+2Re(x_ 1+iy_ 1)^{q_ 1}(x_ 2+iy_ 2)^{q_ 2}), \] u\(=\alpha \rho_ 1+\rho_ 2\), \(\rho_ j=x^ 2_ j+y^ 2_ j\), \(q_ j\geq 3\) are integers, \(\alpha =-q_ 2/q_ 1\) is considered. Theorem. System (1) possesses the following properties. A) In any neighbourhood of the origin every surface defined by \(H=const\) contains invariant tori of (1), B) The origin is not stable in the sense of Lyapunov.
scientific article; zbMATH DE number 4013090