Firstly we take the function h x

6. Periodic orbits in case m2>0m2>0; Λ>0Λ>0; λ>0λ>0; k=−1k=−1
In this AM095 section we describe the case in which explicit formulas for periodic orbits are found. Let l?:=min?Ψ−1/4l?:=min?Ψ−1/4. Here we present
Theorem 4.
Let  m2>0m2>0;  Λ>0Λ>0;  k=−1k=−1. Then the Ω-limit set of any bounded positive half trajectory is located in the plane   x1=0;y1=0 x1=0;y1=0 . In addition, we suppose that  λ>0λ>0. We establish that1) if  l<l?l<l?then there are no compact invariant sets in level sets with centromere l;2) if  l=l?l=l?then there are two equilibrium points denoted as  E±E±which are equilibrium points of the 1st type;3) if  l>l?l>l?then there are two periodic orbits in level sets with this l.
Proof.
As a result, 1) in case λ>0λ>0; l>l?l>l? ovals C+C+; C−C− are limit cycles in invariant sets M+∩H−1(l)M+∩H−1(l) and M−∩H−1(l)M−∩H−1(l) respectively; 2) the system (2), with λ>0λ>0, has no other types of compact invariant sets apart from equilibrium points and periodic orbits.