KAM theory for some dissipative systems
KAM theory for some dissipative systems
复制标题
一些耗散系统的 KAM 理论
DOI：
--
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
R. Llave
中科院分区：
文献类型：
--
作者：
R. Calleja;A. Celletti;R. Llave
耗散系统在几个物理模型中扮演着非常重要的角色，尤其是在天体力学中，耗散驱动自然卫星和人造卫星的运动，导致它们的轨道迁移、共振态等。因此，需要发展理论来确保结构的存在，如不变环面或周期轨道，并设计有效的计算方法。
在这项工作中，我们集中讨论了耗散系统的特殊情况下不变环面的存在性，这类系统被称为共形辛系统，它具有将辛形式变换为自身的倍数的性质。为了给出共形辛系统的明确例子，我们将给出两种不同的模型：离散系统，标准映射，和连续系统，自旋-轨道问题。在这两种情况下，我们都将考虑守恒和耗散形式，这将有助于突出辛动力学和共形辛动力学之间的差异。
对于这样的耗散系统，我们将以后验形式给出一个KAM定理。证明的方法是基于最初在[39]中针对辛情形发展的几何恒等式。该方法除了简化了KAM定理的证明外，还提供了一种已经实现的非常有效的算法。结合一个有效的数值算法和一个后验定理，我们有一个非常有效的方法来提供接近最优的严格估计。
事实上，该方法给出了一个判据(索博列夫爆破判据)，允许用数字计算击穿。我们将回顾这种方法以及J.Greene方法的扩展，并给出在保守和耗散标准映象中的结果。计算接近崩溃，可以发现新的数学现象，如“束崩溃机制”。
Dissipative systems play a very important role in several physical models, most notably in Celestial Mechanics, where the dissipation drives the motion of natural and artificial satellites, leading them to migration of orbits, resonant states, etc. Hence the need to develop theories that ensure the existence of structures such as invariant tori or periodic orbits and device efficient computational methods.
In this work we concentrate on the existence of invariant tori for the specific case of dissipative systems known as "conformally symplectic" systems, which have the property that they transform the symplectic form into a multiple of itself. To give explicit examples of conformally symplectic systems, we will present two different models: a discrete system, the standard map, and a continuous system, the spin-orbit problem. In both cases we will consider the conservative and dissipative versions, that will help to highlight the differences between the symplectic and conformally symplectic dynamics.
For such dissipative systems we will present a KAM theorem in an a-posteriori format. The method of proof is based on extending geometric identities originally developed in [39] for the symplectic case. Besides leading to streamlined proofs of KAM theorem, this method provides a very efficient algorithm which has been implemented. Coupling an efficient numerical algorithm with an a-posteriori theorem, we have a very efficient way to provide rigorous estimates close to optimal.
Indeed, the method gives a criterion (the Sobolev blow up criterion) that allows to compute numerically the breakdown. We will review this method as well as an extension of J. Greene's method and present the results in the conservative and dissipative standard maps. Computing close to the breakdown, allows to discover new mathematical phenomena such as the "bundle collapse mechanism".
共形辛系统须状 KAM 环面的存在性
影响因子：
1.7
作者：
Calleja, Renato C;Celletti, Alessandra;de la Llave, Rafael
通讯作者：
de la Llave, Rafael