December  2010, 3(4): 667-682. doi: 10.3934/dcdss.2010.3.667

Finite smooth normal forms and integrability of local families of vector fields

1. 

Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, 33431 Boca Raton, United States

2. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China

Received  March 2009 Revised  May 2010 Published  August 2010

In this paper we study a class of smooth vector fields which depend on small parameters and their eigenvalues may admit certain resonances. We shall derive the polynomial normal forms for such systems under $C^k$ conjugacy, where $k$ can be arbitrarily large. When the smoothness of normalization is less required, we can even reduce these systems to their quasi-linearizable normal forms under $C^{k_0}$ conjugacy, where $k_0$ is good enough to preserve certain qualitative properties of the original systems while the normal forms are as convenient as the linearized ones in applications. Concerning the normalization procedure, we prove that the transformation can be expressed in terms of Logarithmic Mourtada Type (LMT) functions, which makes both qualitative and quantitative analysis possible.
Citation: Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667
References:
[1]

V. I. Arnold and Yu. S. Ilyashenko, Ordinary differential equations,, Encyclopaedia of Math. Sci. 1, 1 (1988), 1.   Google Scholar

[2]

P. Bonckaert, V. Naudot and J. Yang, Linearization of germs of hyperbolic vector fields,, C. R. Math. Acad. Sci. Paris, 336 (2003), 19.   Google Scholar

[3]

I. U. Bronstein and A. Ya. Kopanskii, "Smooth Invariant Manifolds and Normal Forms,", World Scientific, (1994).  doi: 10.1142/9789812798749.  Google Scholar

[4]

A. D. Bruno, "Local Methods in Nonlinear Differential Equations,", Springer-Verlag, (1989).   Google Scholar

[5]

K.-T. Chen, Equivalence and decomposition of vector fields about an elementary critical point,, Amer. J. Math., 85 (1963), 693.  doi: 10.2307/2373115.  Google Scholar

[6]

Yu. S. Ilyashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields,, Russian Math. Surveys, 46 (1991), 1.  doi: 10.1070/RM1991v046n01ABEH002733.  Google Scholar

[7]

M. Martens, V. Naudot and J. Yang, A strange attractor with large entropy in the unfodling of a low resonant degenerate homoclinic orbit,, Intern. Journ. of Bifurcation & Chaos, 16 (2006), 3509.  doi: 10.1142/S0218127406016951.  Google Scholar

[8]

V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit,, Ergodic Theory and Dynamical Systems, 16 (1996), 1071.   Google Scholar

[9]

V. Naudot and J. Yang, Linearization of families of germs of hyperbolic vector fields,, Dynamical Systems, 23 (2008), 467.  doi: 10.1080/14689360802331162.  Google Scholar

[10]

V. S. Samovol, Linearization of systems of differential equations in a neighbourhood of invariant toroidal manifolds,, Proc. Moscow Math. Soc., 38 (1979), 187.   Google Scholar

[11]

V. S. Samovol, A necessary and sufficient condition of smooth linearization of an autonomous planar system in a neighborhood of a critical point,, Math. Notes, 46 (1989), 543.  doi: 10.1007/BF01159105.  Google Scholar

[12]

S. Sternberg, On the structure of local homeomorphisms of Euclidean $n$-space, II,, Amer. J. Math., 80 (1958), 623.  doi: 10.2307/2372774.  Google Scholar

[13]

S. Sternberg, The structure of local homeomorphisms, III,, Amer. J. Math., 81 (1959), 578.  doi: 10.2307/2372915.  Google Scholar

[14]

J. Yang, Polynomial normal forms for vector fields on $R^3$,, Duke Math. J., 106 (2001), 1.  doi: 10.1215/S0012-7094-01-10611-X.  Google Scholar

show all references

References:
[1]

V. I. Arnold and Yu. S. Ilyashenko, Ordinary differential equations,, Encyclopaedia of Math. Sci. 1, 1 (1988), 1.   Google Scholar

[2]

P. Bonckaert, V. Naudot and J. Yang, Linearization of germs of hyperbolic vector fields,, C. R. Math. Acad. Sci. Paris, 336 (2003), 19.   Google Scholar

[3]

I. U. Bronstein and A. Ya. Kopanskii, "Smooth Invariant Manifolds and Normal Forms,", World Scientific, (1994).  doi: 10.1142/9789812798749.  Google Scholar

[4]

A. D. Bruno, "Local Methods in Nonlinear Differential Equations,", Springer-Verlag, (1989).   Google Scholar

[5]

K.-T. Chen, Equivalence and decomposition of vector fields about an elementary critical point,, Amer. J. Math., 85 (1963), 693.  doi: 10.2307/2373115.  Google Scholar

[6]

Yu. S. Ilyashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields,, Russian Math. Surveys, 46 (1991), 1.  doi: 10.1070/RM1991v046n01ABEH002733.  Google Scholar

[7]

M. Martens, V. Naudot and J. Yang, A strange attractor with large entropy in the unfodling of a low resonant degenerate homoclinic orbit,, Intern. Journ. of Bifurcation & Chaos, 16 (2006), 3509.  doi: 10.1142/S0218127406016951.  Google Scholar

[8]

V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit,, Ergodic Theory and Dynamical Systems, 16 (1996), 1071.   Google Scholar

[9]

V. Naudot and J. Yang, Linearization of families of germs of hyperbolic vector fields,, Dynamical Systems, 23 (2008), 467.  doi: 10.1080/14689360802331162.  Google Scholar

[10]

V. S. Samovol, Linearization of systems of differential equations in a neighbourhood of invariant toroidal manifolds,, Proc. Moscow Math. Soc., 38 (1979), 187.   Google Scholar

[11]

V. S. Samovol, A necessary and sufficient condition of smooth linearization of an autonomous planar system in a neighborhood of a critical point,, Math. Notes, 46 (1989), 543.  doi: 10.1007/BF01159105.  Google Scholar

[12]

S. Sternberg, On the structure of local homeomorphisms of Euclidean $n$-space, II,, Amer. J. Math., 80 (1958), 623.  doi: 10.2307/2372774.  Google Scholar

[13]

S. Sternberg, The structure of local homeomorphisms, III,, Amer. J. Math., 81 (1959), 578.  doi: 10.2307/2372915.  Google Scholar

[14]

J. Yang, Polynomial normal forms for vector fields on $R^3$,, Duke Math. J., 106 (2001), 1.  doi: 10.1215/S0012-7094-01-10611-X.  Google Scholar

[1]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[2]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[3]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[4]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[5]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[6]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[7]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

[8]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386

[9]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[10]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[11]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[12]

Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383

[13]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347

[14]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[15]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[16]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020297

[17]

Jérôme Lohéac, Chaouki N. E. Boultifat, Philippe Chevrel, Mohamed Yagoubi. Exact noise cancellation for 1d-acoustic propagation systems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020055

[18]

Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054

[19]

Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327

[20]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]