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学者姓名：任志华
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Abstract ：
In this paper, we investigate parameterized difference systems, which can yield normal forms of vector fields and diffeomorphisms together. Using it, the normal forms of periodic difference and differential systems can be achieved uniformly.
Keyword ：
Normal form Normal form Differential system Differential system Poincare domain Poincare domain Difference system Difference system
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| GB/T 7714 | Ren, Zhihua , Liu, Meijia , Wu, Hao . Normal Forms for Difference and Differential Systems [J]. | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2022 , 21 (2) . |
| MLA | Ren, Zhihua 等. "Normal Forms for Difference and Differential Systems" . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 21 . 2 (2022) . |
| APA | Ren, Zhihua , Liu, Meijia , Wu, Hao . Normal Forms for Difference and Differential Systems . | QUALITATIVE THEORY OF DYNAMICAL SYSTEMS , 2022 , 21 (2) . |
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Abstract ：
We establish Poincare ' type theorems of normal forms for quasi periodic systems near the invariant torus based on the exponential dichotomy spectra of linear parts at the normal direction.
Keyword ：
Normal form Normal form quasi-periodic system quasi-periodic system Poincare? theorem Poincare? theorem Key words and phrases Key words and phrases
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| GB/T 7714 | Ren, Zhihua , Wang, Tian , Wu, Hao . COMMENTS ON POINCAR?E THEOREM FOR QUASI-PERIODIC SYSTEMS [J]. | COMMUNICATIONS ON PURE AND APPLIED ANALYSIS , 2022 . |
| MLA | Ren, Zhihua 等. "COMMENTS ON POINCAR?E THEOREM FOR QUASI-PERIODIC SYSTEMS" . | COMMUNICATIONS ON PURE AND APPLIED ANALYSIS (2022) . |
| APA | Ren, Zhihua , Wang, Tian , Wu, Hao . COMMENTS ON POINCAR?E THEOREM FOR QUASI-PERIODIC SYSTEMS . | COMMUNICATIONS ON PURE AND APPLIED ANALYSIS , 2022 . |
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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 ：
periodic difference systems periodic difference systems analytic linearization analytic linearization Siegel's theorem Siegel's theorem Normal form Normal form
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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 normal form normal form nonstrongly 1-resonant nonstrongly 1-resonant
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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 ：
Resonance Resonance Conjugacy Conjugacy Smooth normal form Smooth normal form Poincare type map Poincare type map
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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 ：
Normal form Normal form Strongly 1-resonant Strongly 1-resonant Diffeomorphism Diffeomorphism
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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 Resonance Resonance Diffeomorphisms Diffeomorphisms
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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 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 ：
Poincare type diffeomorphisms Poincare type diffeomorphisms hyperbolic diffeomorphisms hyperbolic diffeomorphisms smooth conjugacy smooth conjugacy 0-resonant diffeomorphisms 0-resonant diffeomorphisms
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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 ：
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 ：
smooth conjugacy smooth conjugacy moduli-free normal form moduli-free normal form hyperbolic fixed point hyperbolic fixed point
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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 ：
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 ：
Eigenvalues and eigenfunctions Eigenvalues and eigenfunctions Nonlinear dynamical systems Nonlinear dynamical systems Linearization Linearization Dynamical systems 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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