We consider a homoclinic orbit to a saddle fixed point of an arbitrary $ C^\infty $ map $ f $ on $ \mathbb{R}^2 $ and study the phenomenon that $ f $ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory this requires $ f $ to have a homoclinic tangency. We show it is also necessary for $ f $ to satisfy a 'global resonance' condition and for the eigenvalues associated with the fixed point, $ \lambda $ and $ \sigma $, to satisfy $ |\lambda \sigma| = 1 $. The phenomenon is codimension-three in the case $ \lambda \sigma = -1 $, but codimension-four in the case $ \lambda \sigma = 1 $ because here the coefficients of the leading-order resonance terms associated with $ f $ at the fixed point must add to zero. We also identify conditions sufficient for the phenomenon to occur, illustrate the results for an abstract family of maps, and show numerically computed basins of attraction.

中文翻译：

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具有无限多个渐近稳定单轮周期解的同调切线

我们考虑到在$ \ mathbb {R} ^ 2 $上任意$ C ^ \ infty $地图$ f $的鞍固定点的同宿轨道，并研究$ f $具有无限渐近稳定的单族的现象。全面的定期解决方案。根据经典理论，这要求$ f $具有同斜切线。我们表明，$ f $还必须满足“全局共振”条件，与固定点相关的特征值$ \ lambda $和$ \ sigma $，还必须满足$ | \ lambda \ sigma | = 1 $。在$ \ lambda \ sigma = -1 $的情况下，该现象是共维数3，但是在$ \ lambda \ sigma = 1 $的情况下，该现象是共维数4，因为在这里与$ f $相关的前导共振项的系数在固定点必须加零。我们还确定了足以使现象发生的条件，