论文标题
2D和4D保守和耗散标准地图中旋转摩托车的分解
Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps
论文作者
论文摘要
我们通过实施三种不同的方法研究了2D和4D标准地图中旋转不变的Tori的分解。首先，我们通过计算了圆环嵌入和漂移项的Lindstedt系列扩展，分析了带有给定频率的圆环的分析域。 PADé近似值通过在近似值的分母处绘制多项式的极点来提供分析域的形状。其次，我们实施了一种牛顿方法来构建圆环的嵌入；然后，通过查看嵌入的Sobolev规范的爆炸来估算故障阈值。最后，我们实施了Greene方法的扩展，以通过查看周期轨道的稳定性，具有近似于圆环的频率的周期轨道的稳定性，以获取具有不合理频率不变频率的分解阈值的信息。 我们将这些方法应用于2D和4D标准地图。 2D地图可以是保守的（合成性）或耗散性的（更确切地说，是共形的，即具有几何特性的耗散图，以将符号形式转化为本身的倍数）。 4D地图是获得耦合$（i）$（i）$两个符号标准地图，或$（ii）$两个$（II）的两个符号符号标准地图，或$（iii）$ sympletic and sympletic and s simplectic symplectic标准地图。 关于结果，Padé和Newton方法的表现良好，并提供了可靠，一致的结果（尽管我们仅针对符合性和共性合成图实现了牛顿方法）。我们对Greene方法的扩展的实施尚无定论，因为它在计算上昂贵且精致，尤其是在4D非透明图中，这也是由于存在Arnold舌头所致。
We study the breakdown of rotational invariant tori in 2D and 4D standard maps by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Padé approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the approximants. Secondly, we implement a Newton method to construct the embedding of the torus; the breakdown threshold is then estimated by looking at the blow-up of the Sobolev norms of the embedding. Finally, we implement an extension of Greene method to get information on the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus. We apply these methods to 2D and 4D standard maps. The 2D maps can either be conservative (symplectic) or dissipative ( more precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The 4D maps are obtained coupling $(i)$ two symplectic standard maps, or $(ii)$ two conformally symplectic standard maps, or $(iii)$ a symplectic and a conformally symplectic standard map. Concerning the results, Padé and Newton methods perform well and provide reliable and consistent results (although we implemented Newton method only for symplectic and conformally symplectic maps). Our implementation of the extension of Greene method is inconclusive, since it is computationally expensive and delicate, especially in 4D non-symplectic maps, also due to the existence of Arnold tongues.
