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26 Chapter 2. Global aspects
It has a unique equilibrium point at the origin, and this equilibrium point is unstable. Every non–trivial solution goes to infinity as t tends to infinity. This describes the phase portrait of the Riccati differential equation on R.
The difference between the two phase portraits is easily explained. RP(1) contains an extra point, namely the point [0, 1] which is not in the image ψ(R). This point is the extra equilibrium point of the flow on RP(1). The asymptotic stability of this equilibrium point is reflected in the phase portrait of the Riccati equation on R by the fact that all the non-constant solutions approach infinity asymptotically. In light of this, it is natural to regard the extra point [0, 1] as a point at infinity which has been added to the real line.
2 0
Example 2.3.4. Redefine M = 0 1 . The equilibrium points S1 and S2 re-
main unchanged, but their stability properties are reversed. The corresponding Riccati differential equation is
w˙ = − w
which has the origin as a global asymptotically stable equilibrium point. Since the origin corresponds to S1 via the embedding ψ, we see that here S1 is asymp- totically stable while S2 becomes unstable.
1 1
Example 2.3.5. Redefine M = 0 2 . The equilibrium points of the flow on
RP(1) are the lines spanned by the eigenvectors of M, namely 1 1
S1=Im 0 ,S2=Im 1
with the corresponding eigenvalues λ1 = 1, λ2 = 2. The resulting trajectory (in
RP(1))
t 1 2t 1 S(t)=Im(ec1 0 +e c2 1 ),c1,c2∈R,
incasethatc2 ̸=0goestoS2 ast→∞,since
2t 1 c1 −t1 1 1 S(t)=Im(e c2[ 1 +c e 0 ])=Im[ 1 +o(1)]−→Im 1 =S2.
2
Hence, all points outside S1 are attracted by S2 which is asymptotically stable.
The corresponding Riccati differential equation is w˙ = w − w 2
with equilibrium points w1 = 0 and w2 = 1. From the representation
c2e2t 1
if c2 ̸=0,
w(t)= c1et +c2e2t = 1+ c1 e−t, c2
(2.23)