Analytical invariant manifold

It follows from formula (9.2.9) that if the right-hand side of the system (9.2.6) is analytic, and if all Lyapunov values vanish, then g[x,ip x)) = 0. Hence, since y = (p x) is the solution of the system (9.2.7), it follows that the curve y = ip x) is filled out by the equilibrium states of the system (9.2.6). Thus, it is an invariant manifold of this system. Since it is tangent to y = 0 at O, it is the center manifold by definition. It follows that for the case imder consideration, the system has an analytic center manifold W y = (p(x) which consists of equilibrium states as illustrated in Fig. 9.2.5. 

Theorem 9.3. If all Lyapunov values are equal to zero, then the associated analytic system has an analytic invariant (center) manifold which is filled with closed trajectories around the origin, as shown in Fig. 9.3.3. On the center manifold the system has a holomorphic integral of the type... 

In the original proof the system under consideration was assumed to be analytic. Later on, other simplified proofs have been proposed which are based on a reduction to a non-local center manifold near the separatrix loop (such a center manifold is, generically, 3-dimensional if the stable characteristic exponent Ai is real, and 4-dimensional if Ai = AJ is complex) and on a smooth linearization of the reduced system near the equilibrium state (see [120, 147]). The existence of the smooth invariant manifold of low dimension is important here because it effectively reduces the dimension of the problem. ... 

The question of the construction of a nontrivial smooth SC-metric on S was solved in 1902 by Zoll [190] who explicitly constructed a real-analytic SC rotation surface homeomorphic to 5. A simple generalization of the Zoll construction leads to constructing a family of smooth 5C-metrics on S dependent on the functional parameter and invariant under rotations around a certain axis [I9lj. In his book [192] Blaschke proposed an exquisite modification of the Zoll construction which allows us to construct 5C-manifolds which are homeomorphic to the sphere but are not rotation surfaces. Blaschke surfaces are glued of pieces of different SC rotation surfaces by means of films isometric to parts of the standard sphere. 

From (10.2.5)-(10.2.7) we can conclude that the analytic curve y = (t> x) consists entirely of fixed points and, therefore, is an invariant (center) manifold. 

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