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学者姓名：任志华
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Abstract ：
In this paper, we extend the Siegel's theorem for analytically reducing periodic difference system to a linear one under certain small-divisor condition.
Keyword ：
analytic linearization analytic linearization Normal form Normal form periodic difference systems periodic difference systems Siegel's theorem Siegel's theorem
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| GB/T 7714 | Wang, Tian , Ren, Zhihua . A Siegel theorem for periodic difference systems [J]. | JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS , 2021 , 27 (5) : 698-711 . |
| MLA | Wang, Tian 等. "A Siegel theorem for periodic difference systems" . | JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 27 . 5 (2021) : 698-711 . |
| APA | Wang, Tian , Ren, Zhihua . A Siegel theorem for periodic difference systems . | JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS , 2021 , 27 (5) , 698-711 . |
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Abstract ：
We study in this paper a class of smoothly finite determined germs of diffeomorphisms which have infinitely many resonant relations near a nonstrongly 1-resonant fixed point on R-3 and derive the simplest normal forms of such germs with nondegenerated nonlinear parts.
Keyword ：
diffeomorphism diffeomorphism Finite determinacy Finite determinacy nonstrongly 1-resonant nonstrongly 1-resonant normal form normal form
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| GB/T 7714 | Guo, Lingling , Ren, Zhihua . Polynomial Normal Forms for Some Germs of Nonstrongly 1-Resonant Diffeomorphisms [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2015 , 25 (13) . |
| MLA | Guo, Lingling 等. "Polynomial Normal Forms for Some Germs of Nonstrongly 1-Resonant Diffeomorphisms" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 25 . 13 (2015) . |
| APA | Guo, Lingling , Ren, Zhihua . Polynomial Normal Forms for Some Germs of Nonstrongly 1-Resonant Diffeomorphisms . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2015 , 25 (13) . |
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Abstract ：
In this paper we study polynomial normal forms for maps on R-3. We obtain a complete classification of all diffeomorphisms F(x) on R-3 having the Poincare type fixed point O. The simplest resonant polynomial normal forms, namely, normal forms with the possible lowest degree together with the minimal number of parameters, are explicitly given. (C) 2015 Elsevier Masson SAS. All rights reserved.
Keyword ：
Conjugacy Conjugacy Poincare type map Poincare type map Resonance Resonance Smooth normal form Smooth normal form
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| GB/T 7714 | Ren, Zhihua , Xia, Li , Yang, Jiazhong . On classification of the Poincare type maps on R-3 [J]. | BULLETIN DES SCIENCES MATHEMATIQUES , 2015 , 139 (5) : 582-598 . |
| MLA | Ren, Zhihua 等. "On classification of the Poincare type maps on R-3" . | BULLETIN DES SCIENCES MATHEMATIQUES 139 . 5 (2015) : 582-598 . |
| APA | Ren, Zhihua , Xia, Li , Yang, Jiazhong . On classification of the Poincare type maps on R-3 . | BULLETIN DES SCIENCES MATHEMATIQUES , 2015 , 139 (5) , 582-598 . |
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Abstract ：
In this paper we study polynomial normal forms and smooth (C-infinity) classification of one kind of diffeomorphisms which have infinite many resonant relations but are finitely determined. We derive a complete list of normal forms of all such germs with arbitrary degeneracy of their nonlinear parts. (C) 2013 Elsevier Inc. All rights reserved.
Keyword ：
Diffeomorphism Diffeomorphism Normal form Normal form Strongly 1-resonant Strongly 1-resonant
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| GB/T 7714 | Ren, Zhihua , Yan, Shuo , Yang, Jiazhong . Finite determinacy and polynomial normal forms for diffeomorphisms near a strongly 1-resonant fixed point [J]. | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2014 , 413 (1) : 476-481 . |
| MLA | Ren, Zhihua 等. "Finite determinacy and polynomial normal forms for diffeomorphisms near a strongly 1-resonant fixed point" . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 413 . 1 (2014) : 476-481 . |
| APA | Ren, Zhihua , Yan, Shuo , Yang, Jiazhong . Finite determinacy and polynomial normal forms for diffeomorphisms near a strongly 1-resonant fixed point . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2014 , 413 (1) , 476-481 . |
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Abstract ：
We prove that on R-n, under the given algebraic conditions imposed on eigenvalues of their linear approximation operators, any two hyperbolic diffeomorphism germs with generic nonlinear parts are C-1 conjugate to each other if and only if their linear parts are similar. (C) 2012 Elsevier Masson SAS. All rights reserved.
Keyword ：
C-1 conjugacy C-1 conjugacy Diffeomorphisms Diffeomorphisms Resonance Resonance
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| GB/T 7714 | Ren, Zhihua , Peng, Zhaoxia . On smooth conjugacy of hyperbolic diffeomorphism germs [J]. | BULLETIN DES SCIENCES MATHEMATIQUES , 2013 , 137 (5) : 584-588 . |
| MLA | Ren, Zhihua 等. "On smooth conjugacy of hyperbolic diffeomorphism germs" . | BULLETIN DES SCIENCES MATHEMATIQUES 137 . 5 (2013) : 584-588 . |
| APA | Ren, Zhihua , Peng, Zhaoxia . On smooth conjugacy of hyperbolic diffeomorphism germs . | BULLETIN DES SCIENCES MATHEMATIQUES , 2013 , 137 (5) , 584-588 . |
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Abstract ：
In this paper, we study the smooth classifications of germs of diffeomorphisms near a hyperbolic fixed point based on the smooth moduli-free polynomial normal forms of the corresponding diffeomorphisms and give the following result. On R-n, n <= 5, with two kinds of exceptions, any two hyperbolic germs of diffeomorphisms with generic nonlinear parts are at least C-1 conjugated if and only if their linear parts are similar.
Keyword ：
hyperbolic fixed point hyperbolic fixed point moduli-free normal form moduli-free normal form smooth conjugacy smooth conjugacy
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| GB/T 7714 | Ren, Zhihua , Peng, Zhaoxia . Smooth moduli-free normal forms of hyperbolic germs of diffeomorphisms [J]. | DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL , 2013 , 28 (2) : 251-262 . |
| MLA | Ren, Zhihua 等. "Smooth moduli-free normal forms of hyperbolic germs of diffeomorphisms" . | DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL 28 . 2 (2013) : 251-262 . |
| APA | Ren, Zhihua , Peng, Zhaoxia . Smooth moduli-free normal forms of hyperbolic germs of diffeomorphisms . | DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL , 2013 , 28 (2) , 251-262 . |
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Abstract ：
In this note, we study smooth classification of all germs of 0-resonant diffeomorphisms on R-3 having generic nonlinear parts. We prove that, for the Poincare type diffeomorphisms, except for one germ, any two such germs are at least C-3 conjugated if and only if their linear parts are similar; and for the non-Poincare ones, any such two germs can be C-infinity conjugated provided that they have similar linear parts.
Keyword ：
0-resonant diffeomorphisms 0-resonant diffeomorphisms hyperbolic diffeomorphisms hyperbolic diffeomorphisms Poincare type diffeomorphisms Poincare type diffeomorphisms smooth conjugacy smooth conjugacy
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| GB/T 7714 | Ren, Zhihua , Yang, Jiazhong , Yu, Fuwang . ON CONJUGATING EQUIVALENCE OF 0-RESONANT DIFFEOMORPHISMS ON R-3 [J]. | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2013 , 23 (6) . |
| MLA | Ren, Zhihua 等. "ON CONJUGATING EQUIVALENCE OF 0-RESONANT DIFFEOMORPHISMS ON R-3" . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 23 . 6 (2013) . |
| APA | Ren, Zhihua , Yang, Jiazhong , Yu, Fuwang . ON CONJUGATING EQUIVALENCE OF 0-RESONANT DIFFEOMORPHISMS ON R-3 . | INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS , 2013 , 23 (6) . |
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Abstract ：
To discuss the linearization of nonlinear dynamical systems, some formal maps with the so-called weak-resonant nonlinear parts were studied by using classical normal form theory. The conclusion is that in such cases the maps are linearizable near their fixed points, regardless of resonant relations the eigenvalues of the maps admit.
Keyword ：
Dynamical systems Dynamical systems Eigenvalues and eigenfunctions Eigenvalues and eigenfunctions Linearization Linearization Nonlinear dynamical systems Nonlinear dynamical systems
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| GB/T 7714 | Ren, Zhi-Hua , Liu, Yan-Hua , Yan, Shuo . Linearization of some formal maps on Cn [J]. | Journal of Beijing University of Technology , 2013 , 39 (1) : 153-156 . |
| MLA | Ren, Zhi-Hua 等. "Linearization of some formal maps on Cn" . | Journal of Beijing University of Technology 39 . 1 (2013) : 153-156 . |
| APA | Ren, Zhi-Hua , Liu, Yan-Hua , Yan, Shuo . Linearization of some formal maps on Cn . | Journal of Beijing University of Technology , 2013 , 39 (1) , 153-156 . |
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