On the singularity analysis of intersecting separatrices in near- integrable dynamical systems
It has been recently proved by \textit{S.L. Ziglin} [Tr. Mosk. Mat. 0.-va 41, 287-302 (1980; Zbl 0466.70009)] that transversal intersections of separatrices (invariant manifolds) in near-integrable Hamiltonian systems of two degrees of freedom imply the existence of multi-valued solutions with infinitely many Riemann sheets. Ziglin's theorem is illustrated here on a simple example and then extended and applied to non-Hamiltonian, analytic flows \(\dot x=f(x)+\epsilon g(x,t)\), with \(x\equiv (x,y)\) and \(g(x,t)=g(x,t+2\pi)\), which for \(\epsilon =0\) possess the Painlevé property. On the other hand, the theoretically expected logarithmic singularities for \(\epsilon\neq 0\) are obtained explicitly in solutions near the intersecting separatrices. Thus, we conjecture that dynamical systems with the Painlevé property, can have no separatrix intersections and hence no strange attractors, etc. These singularities are then numerically located and found to form certain very interesting ''chimney'' patterns in the complex t-plane, on which they accumulate densely. The upper part of the chimneys (away from the Re t axis) is asymptotically quite insensitive to changes in parameter values or initial conditions. The singularity pattern itself, however, becomes ''denser'' and each chimney is seen to gradually ''collapse'' towards the Re t axis, as the amplitude of the driving term increases and the motion becomes more chaotic.
scientific article; zbMATH DE number 3992976
scientific article