Fuzzy fractional monodromy and the section-hyperboloid (Q1035245)

A mechanical system with 2 degrees of freedom viewed as a Lagrangian compact fibration of the symplectic 4-dimensional linear space \(\mathbb{R}^4\) is considered. This system has a singular fiber \(\Lambda^0\) at its turn having a unique singular point \(\xi^0\) - of oscillator type with resonant set of frequencies \(\omega=(\omega_1,\omega_2)\). The motions around this fiber generate a group of classes of homotopy-equivalent cycles on a torus. In a certain basis of this group the matrix of the homotopy is some \(2\times 2\) matrix \(M\) defining a monodromy; its parameters \(m_1\) and \((-m_2)\) are relatively primes in the ratio \(\omega_1:\omega_2=m_1:(-m_2)\). The author discusses fractional monodromy when \(m_1m_2\geq 2\) with its so-called fuzziness -- when the continuous dependence property is lost. The paper contains a theorem on fuzzy monodromy and a sketch of its proof.