Energy function for an $\Omega$-stable flow with a saddle connection on a sphere

In this paper the class of simplest not rough $\Omega$-stable flows on a sphere is considered. We call simplest not rough $\Omega$-stable flow an $\Omega$-stable flow with least number of fixed points, a single separatrix connecting saddle points and without limit cycles. For such flows we design the Morse energy function.

Well known that the Morse-Smale flows, introduced for the first time on a plane by A. A. Andronov and L. S. Pontryagin have finite number of hiperbolic fixed points and closed trajectories, and its non-wandering set does not contain other elements. Besides, such flows does not have separatrices connecting saddle points.

Morse-Smale flows which does not have limit cycles (they are called the gradient-like flows), as Smale showed, in suitable metrix they are gradient-like flows generated by some Morse function. Then this function decrease along non-singular trajectories of a flow and its fixed points are exactly the fixed points of a flow. Thus, it was the first example of designing so called energy function for a dynamical system, i.e a smooth function decreasing along wandering trajectories and whose singular points set is equal with the non-wandering set of a system.
K. Meyer generalised the Smale's result and constructed energy function for an arbitrary Morse-Smale flow. As such flow has periodic trajectories in general case, an energy function could not be a Morse function but its generalization – a Morse-Bott function with points of first degeneracy degree along limit cycles.

In this work we make a first step to generalise Meyer's results to flows which are not structural stable. Precisely, we consider the class of simplest $\Omega$-stable flows with separatrices connecting saddle points on a two-dimensional sphere and we show that any such flow has its Morse energy function. Obviously, this work is going to be a foundation for next generalisation of Smale's and Meyer's results.

Let us denote by $S^2$ a two-dimensional sphere with a metric $d$ and by $G$ the class of $\Omega$-stable flows $f^t$ on $S^2$ whose non-wandering set consists of six fixed points: two sinks $\omega_1$ and $\omega_2$, two sources $\alpha_1$ and $\alpha_2$ and two seddle points $\sigma_1$ and $\sigma_2$ with a common connecting separatrix.

Теорема. There is energy Morse function for each flow from the class $G$.