Minimal number of periodic orbits for non-singular Morse–Smale flows in odd dimension

We consider the couples $(M, \Phi )$ where $M$ is an odd-dimensional compact manifold with boundary, endowed with a non-singular Morse–Smale flow $\Phi $, satisfying some given homological boundary information. We compute, in terms of that information, a number $p_{\rm min}$ such that any non-singular Morse–Smale flow $\Phi $ on any manifold $M$ satisfying the given abstract homological data must have at least $p_{\rm min}$ closed periodic orbits.

Moreover, we provide, for any initial homological data, a non-singular Morse–Smale model $(M_0, \Phi _0)$ for which $p_{\rm min}$ is attained. In the general case of a couple $(M, \Phi )$ satisfying the given homological information, such a number $p_{\rm min}$ is a lower bound.

The algorithm underlying this computation is based on optimization theory in network flows and transport systems.