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2.3. A representation formula 29 The corresponding Riccati differential equation is
w˙ = b(1 + w2) (2.26)
with the general real solution
w(t) = tan(bt + t0).
Although tan(bt + t0) is a π -periodic meromorphic function, this solution does
b
not yield a periodic trajectory. Indeed, the solution has finite escape time π − t0 , 0 ≤ t0 ≤ π, which is when S(t) crosses the point at infinity which
It is useful to interpret the periodic solution S(t) from the point of view that the flow on RP(1) is a linear flow on R2 as given by (2.25) composed with the projection Π. The trajectory of the linear flow in (2.25) is not periodic if a is not zero. S(t) is obtained from this trajectory by applying the projection Π which identifies all points in R2 which are on the same line through the origin. Thus, the radial distance from the origin becomes irrelevant and a periodic solution which is independent of the value of a results.
In Example 2.3.6, the periodic solution on RP(1) does not correspond to a periodic solution of the Riccati equation on R but rather to a solution which escapes in finite time. This is necessarily the case for a (time-invariant) scalar Riccati equation which can never have a periodic solution. This can be explained equivalently by noting that RP(1) is a circle, so every periodic solution on RP(1) must traverse every point of RP(1) including the point [0, 1] at infinity. In higher dimensions, this topological obstruction to periodic solutions is not present. Un- der typical conditions, the periodic solutions on the higher-dimensional analogies of RP(1) will indeed correspond (via an embedding ψ) to periodic solutions of the Riccati differential equation.
In addition, in higher dimensions, the periodic motion exhibited in Ex- ample 2.3.6 can occur in several independent coordinates simultaneously. This results in invariant sets for the Riccati equation which are products of circles; i.e., invariant tori. Under reasonable assumptions, the phase portrait of the Ric- cati differential equation is characterized by the presence of a finite number of invariant tori of various dimensions. The 0-dimensional tori are the equilib- rium points, the 1-dimensional tori are the isolated periodic orbits, while the higher-dimensional tori contain trajectories which are either periodic or almost periodic. Furthermore, every other solution either escapes in finite time or con- verges to one of the invariant tori.
Examples 2.3.3 – 2.3.5 showed that the equilibrium points of the flow S(t) are the stable or unstable invariant manifolds which always can be characterized by letting t → ∞ or t → −∞ (time inversion). If these limit points are in ψ(R), then they are also stationary points of the corresponding Riccati differential equation and hence are solutions of the algebraic Riccati equation associated
2bb 2 is [0, 1].