Bifurcation into invariant tori at points of resonance

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,

参考文章(6)
Oscar E. Lanford, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens Springer, Berlin, Heidelberg. pp. 159- 192 ,(1973) , 10.1007/BFB0060566
David Ruelle, Bifurcations in the presence of a symmetry group Archive for Rational Mechanics and Analysis. ,vol. 51, pp. 136- 152 ,(1973) , 10.1007/BF00247751
Yieh-Hei Wan, Computation of the Stability Condition for the Hopf Bifurcation of Diffeomorphisms on $\mathbb{R}^2 $ SIAM Journal on Applied Mathematics. ,vol. 34, pp. 167- 175 ,(1978) , 10.1137/0134013
V. I. Arnol'd, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields Functional Analysis and Its Applications. ,vol. 11, pp. 85- 92 ,(1977) , 10.1007/BF01081886
G. Iooss, D. D. Joseph, Bifurcation and stability of nT-periodic solutions branching from T-periodic solutions at points of resonance Archive for Rational Mechanics and Analysis. ,vol. 66, pp. 135- 172 ,(1977) , 10.1007/BF00248631
DAVID RUELLE, FLORIS TAKENS, On the Nature of Turbulence tsac. pp. 57- 84 ,(1971) , 10.1142/9789812833709_0002