Nonintegrability of dynamical systems with heteroclinic orbits
We consider general n-dimensional systems of differential equations
having an (n−2)-dimensional, locally invariant manifold on which
there exist equilibria connected by heteroclinic orbits for n ≥ 3. The
system may be non-Hamiltonian and have no saddle-centers, and the
equilibria are allowed to be the same and connected by a homoclinic
orbit. Under additional assumptions, we prove that the monodromy group
for the normal variational equation, which is represented by
components of the variational equation normal to the locally invariant
manifold and defined on a Riemann surface, is diagonalizable or
infinitely cyclic if the system is real-meromorphically integrable in
the meaning of Bogoyavlenski. We apply our theory to a
three-dimensional volume- preserving system describing the streamline
of a steady incompressible flow with two parameters, and show that it
is real-meromorphically nonintegrable for almost all values of the two
parameters.

Last modified: Tue Feb 28 09:03:13 JST 2017