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Kovalevskaya Exponents, Weak Painlevé Property and Integrability for Quasi-homogeneous Differential Systems

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Abstract

We present some necessary conditions for quasi-homogeneous differential systems to be completely integrable via Kovalevskaya exponents. Then, as an application, we give a new link between the weak-Painlevé property and the algebraical integrability for polynomial differential systems. Additionally, we also formulate stronger theorems in terms of Kovalevskaya exponents for homogeneous Newton systems, a special class of quasi-homogeneous systems, which gives its necessary conditions for B-integrability and complete integrability. A consequence is that the nonrational Kovalevskaya exponents imply the nonexistence of Darboux first integrals for two-dimensional natural homogeneous polynomial Hamiltonian systems, which relates the singularity structure to the Darboux theory of integrability.

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Funding

This work is supported by NSFC grant (No. 11771177), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20), NSF grant (No. 20190201132JC) of Jilin Province.

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Authors and Affiliations

  1. School of Mathematics, Sichun University, Chengdu, 610000, China

    Kaiyin Huang

  2. School of Mathematics, Jilin University, Changchun, 130012, China

    Shaoyun Shi & Wenlei Li

  3. State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, 130012, P. R. China

    Shaoyun Shi

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  1. Kaiyin Huang
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  2. Shaoyun Shi
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  3. Wenlei Li
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Correspondence to Kaiyin Huang, Shaoyun Shi or Wenlei Li.

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Huang, K., Shi, S. & Li, W. Kovalevskaya Exponents, Weak Painlevé Property and Integrability for Quasi-homogeneous Differential Systems. Regul. Chaot. Dyn. 25, 295–312 (2020). https://doi.org/10.1134/S1560354720030053

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  • DOI: https://doi.org/10.1134/S1560354720030053

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