ON THE GEOMETRY OF AN ATMOSPHERIC SLOW MANIFOLD
ON THE GEOMETRY OF AN ATMOSPHERIC SLOW MANIFOLD
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DOI：
10.1016/0167-2789(94)00239-m
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发表时间：
1995-07-01
影响因子：
4
通讯作者：
CAMASSA, R
中科院分区：
文献类型：
--
作者：
CAMASSA, R
我们研究了Lorenz提出的一个模型的双曲结构和不变流形，在动力系统理论的框架内引入了大气慢流形的概念，我们使用（全球）几何的观点来解决系统的长时间渐近行为的问题。结果表明，该模型可以减少到一个经典的例子，一个钟摆耦合到一个谐振子。感兴趣的慢流形假设的动力学制度对应于该模型的鞍中心不动点附近的相空间区域，这是以前没有探索。这些相空间区域进行了分析，使用组合的Melnikov型的方法和奇异摄动理论的想法。利用该模型的可逆对称性，推导出梅尔尼科夫理论的推广。这种推广使得我们可以通过简单地计算由Melnikov函数构造的函数的零点来寻找同宿轨道并确定其逼近，从而证明了全局同宿分支的可数无穷大性和混沌动力学的存在性.
We examine the hyperbolic structure and the invariant manifolds of a model proposed by Lorenz to introduce the concept of an atmospheric slow manifold within the framework of dynamical system theory, We address the question of the long time asymptotic behaviour of the system using the (global) geometric point of view. It is shown that the model can be reduced to the classical example of a pendulum coupled to a harmonic oscillator. The dynamical regimes of interest for the slow manifold hypothesis correspond to regions of phase space near the saddle-center fixed point of this model which were not previously explored. These phase space regions are analysed using a combination of Melnikov-type methods and ideas from singular perturbation theory. By using the reversible symmetries of the model, an extension of the Melnikov theory is derived. This extension allows us to find homoclinic orbits and determine their approximation by simply computing the zeros of a certain function, which is constructed in terms of the usual Melnikov function, Countable infinities of global homoclinic bifurcations and existence of chaotic dynamics can be shown to exist by using the new tool.