Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov–Takens Bifurcation

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2024-01-25 DOI:10.1137/22m151354x

Maikel M. Bosschaert, Yuri A. Kuznetsov

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Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 410-439, March 2024.
Abstract.This paper provides for the first time correct third-order homoclinic predictors in [math]-dimensional ODEs near a generic Bogdanov–Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt–Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt–Poincaré) to approximate the homoclinic solution near Bogdanov–Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.

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波格丹诺夫-塔肯斯分岔附近同线性预测因子的正则表达式与中心曼菲尔德还原之间的相互作用

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 410-439, March 2024. 摘要.本文首次提供了[math]维 ODEs 在一般 Bogdanov-Takens 分岔点附近的正确三阶同次轨道预测器，可用于开始对出现的同次轨道进行数值延续。要实现这一点，林德斯特-庞加莱方法中的非线性时间变换的高阶时间近似是必不可少的。此外，还严格推导出了正常形式解的近似值与参数相关中心流形上解的近似值之间的正确变换。在波格丹诺夫-塔肯斯点附近采用不同的正则形式（光滑和轨道）、不同的相位条件和不同的扰动方法（正则和林德斯特-平卡莱）来逼近同次元解时，进行了详细的比较。文中举例说明了预测器的正确性。新的同次元预测器是在开源的 MATLAB/GNU Octave continuation 软件包 MatCont 中实现的。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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