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2.3. A representation formula 27
we see that, as long as for the initial point w0 of the trajectory w, w(t0) = w0 > 0,
then there is no finite escape time. But all trajectories with initial point w(t0)=w0 <0
have finite escape time. Then from (2.23) we infer that w(t) −→ w2 = 1
for all trajectories with initial data w0 > 0. Hence, w2 is asymptotically sta- ble while w1 is unstable. Since w1,w2 correspond to S1,S2, respectively, via the embedding ψ, the corresponding equilibrium points have the same stability properties with respect to the Riccati flow on RP(1).
Even though the equilibrium points and their stability properties corre- spond via ψ, there is an important difference between the two phase portraits. The embedding ψ can map trajectories in R which escape in finite time to trajec- tories in RP(1) which converge. The explanation for this lack of correspondence is that the corresponding trajectories cross the point [0, 1] in the process of con- verging to S2. Since [0, 1] is outside the image of ψ, the correspondence between the trajectories w(t) and S(t) is lost; w(t) passing to infinity at this time while S(t) continues on its asymptotic approach to S2.
It is worth noting that as far as the phase portraits on RP(1) are concerned, Examples 2.3.3 – 2.3.5 are essentially the same. In each case there are two equilibrium points, one of which has a region of attraction which includes every non–equilibrium point. The equivalence of the three phase portraits is explained as follows: Let M be a constant 2×2 matrix, let P be a non – singular 2×2 matrix, and let M˜ := PMP−1. It is then trivial to check that if S(t) is a trajectory of the flow corresponding to the coefficient matrix M, then PS(t) is a trajectory of the flow corresponding to the coefficient matrix M˜ .
In other words, the nonsingular linear transformation P on R2 induces a diffeomorphism of the manifold RP(1) (where P[q,p] is the image of the line [q, p] under the linear transformation P ) which maps trajectories to trajectories.
Since the matrices M in Examples 2.3.3 – 2.3.5 have the same eigenvalues, the three phase portraits are related by such transformations. This brings up the question as to why the three phase portraits on the real line are so different. The answer is that the diffeomorphism of RP(1) induced by P will generally not leave invariant the subset which is the image of the embedding ψ . Consequently, the diffeomorphism cannot be restricted to this subset to obtain a diffeomorphism of the real line. To say this another way, the diffeomorphism induced by P can map the point at infinity [0,1] to a point of the form [1,w] in ψ(R). If [0,1] is an equilibrium point for M, then [1,w] is an equilibrium point for M˜. The Riccati equation corresponding to M˜ has an equilibrium point at w, whereas the