Totally Bounded Cubic Systems in ℝ²

Abstract

This work describes the behavior of trajectories of a planar cubic system such that

a) all the trajectories are bounded,

b) there exist just two singular points S, O,

c) the system is reversible about the line SO.

The exposition is divided into:

i) there is no heterocline. In particular O is a Poincaré center (Sec. 2.13), a degenerate center (Sec. 2.14), a right pseudo-center (Sec. 2.15), a left pseudo-center (Sec. 2.16);

ii) there exists just one pair of heteroclines. In particular O is a Poincaré saddle (Sec. 2.17), a degenerate saddle (Sec. 2.18), a tangential limit point with indexO=1 (Sec. 2.19);

iii) there exists a pair of bands of heteroclines (Sec. 2.23).

Maintaining the total boundedness, i.e., the boundedness of all the trajectories, the following topics

α ) O is the unique singular point (Sec. 1.5),

β ) invariancy of an ellipse (Sec. 2.20),

γ ) existence of limit cycles (Sec. 3.1)

are also considered.