A HARTMAN-GROBMAN THEOREM FOR ALGEBRAIC DICHOTOMIES

Chaofan Pan; Manuel Pinto; Yonghui Xia

doi:10.11948/20220260

2022 Volume 12 Issue 6

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Article navigation > Journal of Applied Analysis & Computation > 2022 > 12(6): 2640-2662

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Chaofan Pan, Manuel Pinto, Yonghui Xia. A HARTMAN-GROBMAN THEOREM FOR ALGEBRAIC DICHOTOMIES[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2640-2662. doi: 10.11948/20220260

Citation:

Chaofan Pan, Manuel Pinto, Yonghui Xia. A HARTMAN-GROBMAN THEOREM FOR ALGEBRAIC DICHOTOMIES[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2640-2662. doi: 10.11948/20220260

A HARTMAN-GROBMAN THEOREM FOR ALGEBRAIC DICHOTOMIES

1.
College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China
2.
Departamento de Matemáicas, Universidad de Chile, Santiago, Chile

Corresponding author: Email: xiaoutlook@163.com, yhxia@zjnu.cn.(Y. Xia)
Fund Project: This paper was supported by the National Natural Science Foundation of China under Grant (Nos. 11931016, 11671176), the Natural Science Foundation of Zhejiang Province under Grant (No. LY20A010016) and Grant Fondecyt (No. 1170466), Fondecyt 038-2021-Perú

Abstract

Abstract

Algebraic dichotomy is a generalization of an exponential dichotomy (see Lin [28]). This paper gives a version of Hartman-Grobman linearization theorem assuming that linear system admits an algebraic dichotomy, which generalizes the Palmer's linearization theorem. Besides, we prove that the homeomorphism in the linearization theorem is Hölder continuous (and has a Hölder continuous inverse). Comparing with exponential dichotomy, algebraic dichotomy is more complicate. The exponential dichotomy leads us to the estimates $ \int_{-\infty}^{t}e^{-\alpha(t-s)}ds $ and $ \int_{t}^{+\infty}e^{-\alpha(s-t)}ds $ which are convergent. However, the algebraic dichotomy will leads us to $ \int_{-\infty}^{t}\left(\frac{\mu(t)}{\mu(s)}\right)^{-\alpha}ds $ or $ \int_{t}^{+\infty}\left(\frac{\mu(s)}{\mu(t)}\right)^{-\alpha}ds $, whose the convergence is unknown in the sense of Riemann.
- Algebraic dichotomy /
- linearization /
- Hölder continuous
MSC: 34C41, 34D10, 34D09, 34C40, 34D05

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References

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