Page 35 - Textos de Matemática Vol. 36

2.3. A representation formula 25 The flow [q, p](t) on RP(1) corresponds to the flow w(t) on R via
w(t) = p(t) q(t)
as long as the trajectory remains in ψ(R). Let us now consider some examples.
1 0
Example 2.3.3. Let M = 0 2 . The flow on RP(1) is then given by
q0
1
0 or
0 1
is one of the eigenvectors point of the flow, i.e.,
of M, then we obtain a stationary 0
If p
0
q(t) Mt q0
.
Im p(t) =Im e
p
0
hence
q 0 φ˙ q 0 p φ(t)=M p ,
00
φ˙ q 0 = constant and
1
S1=Im 0 ,S2=Im 1
are two stationary points. In general, in a stationary point of the flow, q(t) q0
holds for some differentiable scalar function φ ̸= 0. With (2.20) this yields
p(t) = p φ(t) 0
has to be an eigenvector of M.
φ p0
To investigate the stability properties of these equilibria, let the initial
point (i.e., initial line) S0 ∈ RP(1) \ S1. Since S0 is not horizontal, we can express it as [r, 1], r ∈ R. The resulting trajectory
Mt r etr e−tr 0 S(t)=Im e 1 =Im e2t =Im 1 −→Im 1 =S2
as t → ∞. Thus, S1 is unstable while S2 is asymptotically stable with domain of attraction consisting of every point other than S1. This describes the phase portrait of the Riccati differential equation on RP(1).
The corresponding Riccati differential equation by (2.21) is w˙ = w .