A Proof of the Jacobian Conjecture on Global Asymptotic Stability

Let f∈C~1(R~2, R~2), f(0)=0. The Jacobian Conjecture states that if for any x∈R~2,the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution ofthe differential equation x=f(x) is globally asymptotically stable. In this paper we prove that theconjecture is true.

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Bibliographic Details
Published inActa mathematica Sinica. English series Vol. 17; no. 1; pp. 119 - 132
Main Authors Chen, Peng Nian, He, Jian Xun, Qin, Hua Shu
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 2001
Subjects
Asymptotic methods
differential
Eigenvalues
equation;Global
injectivity
Plane
stability;Global
Studies
Systems science
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Summary:Let f∈C~1(R~2, R~2), f(0)=0. The Jacobian Conjecture states that if for any x∈R~2,the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution ofthe differential equation x=f(x) is globally asymptotically stable. In this paper we prove that theconjecture is true.
Bibliography:Peng Nian CHEN;Jian Xun HE;Hua Shu QIN Division of Mathematics, China Institute of Metrology, Hangzhou 310034, P. R. China Department of Systems Science, Xiamen University, Xiamen 361005, P. R. China Institute of Systems Science, Academia Sinica, Beijing 100080, P. R. China
Plane differential equation;Global stability;Global injectivity
Let f∈C~1(R~2, R~2), f(0)=0. The Jacobian Conjecture states that if for any x∈R~2, the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is globally asymptotically stable. In this paper we prove that the conjecture is true.
11-2039/O1
ISSN:1439-8516
1439-7617
DOI:10.1007/s101140000098