A discussion on the coexistence of heteroclinic orbit and saddle foci for third-order systems

The coexistence of heteroclinic orbits and saddle foci is concerned with the basic assumption in Shilʼnikov heteroclinic theorem. Two aspects of this discussion are conducted in the paper. Firstly, many third-order systems, which possess exact heteroclinic orbits expressed by pure hyperbolic functions or the combination of hyperbolic and triangle functions and so on, have been constructed. At the same time, the existence of saddle foci is tested and some problems are proposed. Secondly and more importantly, the existence of heteroclinic orbits to saddle foci is studied. The necessary condition for the coexistence of heteroclinic orbits and saddle foci is obtained. Finally, an example is given to show the effectiveness of the results, and some conclusions and problems are presented.