论文标题
具有无限多个渐近稳定的单一周期性解决方案的同层面切线
Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions
论文作者
论文摘要
我们认为，在$ \ Mathbb {r}^2 $上的任意$ c^\ inty $ map $ f $的鞍座固定点上的同型轨道，并研究了$ f $具有无限稳定的单一周期性解决方案的无限家族的现象。从古典理论来看，这需要$ f $具有同质性的相切。我们也表明，$ f $也必须满足“全局共振”条件以及与固定点相关的特征值，$λ$和$σ$，以满足$ |λσ| = 1 $。该现象是编码的 - 在情况下为$λσ= -1 $，但在情况下为$λσ= 1 $的codimension-four，因为在这里，在固定点上与$ f $相关的前阶谐振项的系数必须添加到零。我们还确定了足以使现象发生的条件，说明了抽象的地图家族的结果，并显示了数值计算的吸引力盆地。
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 also necessary for $f$ to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, $λ$ and $σ$, to satisfy $|λσ| = 1$. The phenomenon is codimension-three in the case $λσ= -1$, but codimension-four in the case $λσ= 1$ 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.
