Poincare lemma

This lemma is equivalent to saying that X/G is a rational homology manifold. It holds because we have the Poincare duality isomorphism for the cohomology groups with rational coefficients on X/G. 

Proof of Theorem 5.3. Condition (3.4) makes E locally asymptotically stable. By the Poincare-Bendixson theorem, it is necessary only to show that with condition (3.4) there are no limit cycles. Suppose there were a limit cycle. However, there is at most a finite number of limit cycles and each must contain < in its interior. Hence there is a periodic trajectory V that contains no other periodic trajectory in its interior. Intuitively speaking, r is the trajectory closest to the rest point. The constant term in the formula given in Lemma 5.1 is negative. The corollary shows that P is asymptotically stable. This is a contradiction, since the rest point is asymptotically stable - that is, between the two there must be an unstable periodic orbit. ... 

In this section, we compile some results from nonlinear analysis that are used in the text. The implicit function theorem and Sard s theorem are stated. A brief overview of degree theory is given and applied to prove some results stated in Chapters 5 and 6. The section ends with an outline of the construction of a Poincare map for a periodic solution of an autonomous system of ordinary differential equations and the calculation of its Jacobian (Lemma 6.2 of Chapter 3 is proved). 

In fact, the set of coordinate transformations which keeps the system at /i = 0 in the form (12.2.12) is rather poor. Indeed, a new coordinate (p must satisfy - (Pnew — < ) = 0, hence the difference Pnew must be constant along a trajectory of the system. In particular, it is constant on L. Now, since any orbit on the center manifold tends to L either as t -hoo or as t f — oo, it follows that Pnew — p = constant everywhere on W. Furthermore, since the equation for x in (12.2.12) must remain autonomous, one can show that only autonomous (independent of p) transformations of the variable x are allowed. Indeed, consider first a transformation which is identical at p = 0. By definition, it does not change the Poincare map of the local cross-section 5 v = 0. Therefore, by the uniqueness of the embedding into the flow (Lemma 12.4), if such transformation keeps the system autonomous, it cannot change the right-hand side g. It follows that if Xnew = x at p = 0 then the time evolution of Xnew and the time evolution of x are governed by the same equation which immediately implies that Xnew = x for all p in this case. Since an arbitrary transformation is a superposition of an autonomous... 

Note that Theorem 13.6 follows immediately from this lemma. Here V = Sq, U = Sq, dU = Soo, V U = Sq, and since T is the Poincare map, its stable fixed point in corresponds to a stable limit cycle (the fixed point on 5oo = H ioc o corresponds to the homoclinic loop T by construction). 

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