Computer assisted proofs of two-dimensional attracting invariant tori for ODEs
Computer assisted proofs of two-dimensional attracting invariant tori for ODEs
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DOI：
10.3934/dcds.2020162
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
M. Capinski;Emmanuel Fleurantin;J. D. M. James
中科院分区：
文献类型：
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作者：
M. Capinski;Emmanuel Fleurantin;J. D. M. James
本文研究三维耗散常微分方程组吸引不变环面的存在性和正则性问题。我们的主要结果是一个建设性的计算机辅助证明方法，适用于明确的问题，在非微扰制度。我们得到可验证的吸引子的正则性的下限在环面上的膨胀率与环面附近的收缩率的比率。我们分别考虑两个重要的情况下，旋转和共振环。在旋转的情况下，我们得到\开始{document}$ C^k $\end{document}的嵌入的正则性的下界。在共振情况下，我们验证了圆环面的存在，它们只是\开始{document}$C^0 $\end{document}，既不是星形也不是Lipschitz。
This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain \begin{document}$ C^k $\end{document} lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only \begin{document}$ C^0 $\end{document} and neither star-shaped nor Lipschitz.