作者: Yieh-Hei Wan
DOI: 10.1007/BF00250986
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摘要: Let Xu be a family of vector fields on some manifold with real parameters p, and 7 periodic orbit X.o period T for some/2o. A natural way to study the behavior phase portraits ofX u near as # passes through/2o is consider associated PoincarO map ~Pu: V ~ U close Po. Here, stands local cross section through point p in y, denotes small open neighborhood ofp U. For each q V, ~0.(q) first return solution x.( t ) (t > 0) xu(0) = q. instance, fixed ofq~. corresponds T. ~o. order n gives subharmonic X. nT. An invariant circle an orientation preserving torus X.. When spectrum differential D~0.o at lies away from unit complex plane, remain same for/2 #o. On other hand, if Dq~.o has eigenvalue 2(#o) 12(#o) [ 1, then bifurcations are expected. Suppose that D~p. o simple 2(po) 12(/2o)[ 1 (2(#0) + 1). Since all bifurcation phenomena will occur center ~0., without loss generality dimension can assumed two. By implicit function theorem one gets unique of~o, near/2o. Denote by 2(/2) D~0, p,. It well known works SACKER [6] RUELLE & TAKENS [5] that, general,