7.6. THEORY OF DARK MATTER AND DARK ENERGY 481
Inserting (7.6.41)-(7.6.42) into the first two equations of (7.6.34), we deduce that
(7.6.43)
x′ 1 =−x 1 −1
8
x^21 −1
8
x^22 −3
4
x 1 x 2 +O(|x|^3 ),x′ 2 =−x 2 +1
4
x^21 −x^22 −1
4
x 1 x 2 +O(|x|^3 ).The system (7.6.43) is the system (7.6.34) restricted on the stable manifoldEs. Hence, its
asymptotic behavior represents that of the interaction forceFin (7.6.29).
Step 5. Phase diagram of system (7.6.43).In order to obtain the phase diagram of (7.6.43)
nearx=0, we consider the ratio:x′ 2 /x′ 1 =dx 2 /dx 1. Omitting the termsO(|x|^3 ), we infer from
(7.6.43) that
(7.6.44)
dx 2
dx 1=
x 2 +x^22 +^14 x 2 x 1 −^14 x^21
x 1 +^18 x^22 +^34 x 2 x 1 +^18 x^21.
Letkbe the slope of an orbit reaching tox=0:
k=x 2
x 1
as (x 2 ,x 1 )→ 0.Then (7.6.44) can be expressed as
k=k+k^2 x 1 +^14 kx 1 −^14 x 1
1 +^18 k^2 x 1 +^34 kx 1 +^18 x 1,
which leads to the equation
k^3 − 2 k^2 −k+ 2 = 0.
This equation has three solutions:
k=± 1 , k= 2 ,giving rise to three line orbits:
x 2 =x 1 , x 2 = 2 x 1 , x 2 =−x 1 ,which divide the neighborhood ofx=0 into six invariant regions. It is clear that all physically
meaningful orbits can only be in the following three regions:
(7.6.45)
Ω 1 =
{
−x 2 <x 1 <1
2
x 2 ,x 2 > 0}
,
Ω 2 =
{
1
2
x 2 ≤x 1 ≤x 2 ,x 2 > 0}
,
Ω 3 ={x 2 <x 1 ,x 2 > 0 }.On the positivex 2 -axis (i.e.x 1 = 0 ,x 2 >0), which lies inΩ 1 , the equations (7.6.43) take
the form
x′ 1 =−1
8
x^22 +O(|x|^3 ),x′ 2 =−x 2 −x^22 +O(|x|^3 ).