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引用次数: 0
摘要
我们将最近开发的离散几何奇异扰动理论扩展到非正态双曲系统。我们的主要工具是塔肯斯嵌入定理,它提供了一种用形式向量场的时间-1映射来近似特定映射动态的方法。首先,我们证明了所谓的还原映射(它支配着常态双曲系统中慢流形附近的慢动力学)可以用连续时间几何奇异扰动理论中出现的还原矢量场的时间-1映射局部逼近。在非正态双曲系统中,我们证明了具有单能线性部分的快慢映射的动力学可以由同一维度的快慢矢量场诱导的时间-1映射局部逼近,该矢量场具有相应类型的无能奇点。后一结果用于描述:(i) 具有规则折叠、跨临界和干草叉类型非正则奇点的二维快慢图的局部动力学;(ii) 具有临界流形的规则接触或折叠子流形的 n 维快慢图中(可能是高维)局部中心流形上的动力学。一般来说,我们的结果表明,可以通过使用形式嵌入定理来描述快慢图中一大类重要奇点附近的动力学,这些定理允许用具有法向双曲性损失的快慢向量场的时间-1映射来逼近它们。
Extending discrete geometric singular perturbation theory to non-hyperbolic points
We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-1 map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high-dimensional) local center manifold in n-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.
期刊介绍:
Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest.
Subject coverage:
The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal.
Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena.
Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.