Branching of solutions in a complex domain from the point of view of symbolic dynamics and nonintegrability of multi-dimensional systems
Branching of solutions to analytic ordinary differential equations \[ \frac{dx}{dt}=X(x,t) \] in a complex domain of variation of the variables is considered from the point of view of the methods of symbolic dynamics. This makes it possible to find the conditions for the absence of a nonconstant meromorphic first integral and a meromorphic vector field that commutes with the field of the phase flow and cannot be obtained by multiplying the phase flow by a constant in some neighborhood of some union of closed contours in the extended phase space \(\{x,t\}\). An approach based on perturbation theory is applied, and nonintegrability is established for each fixed (but nonzero) value of the small parameter. The found nonintegrability conditions hold under small perturbations and can easily be applied to a broad class of systems arising in mathematics and physics, including systems to which other methods for proving nonintegrability do not apply or their application involves substantial analytic difficulties.
scientific article; zbMATH DE number 1557420