American Institute of Mathematical Sciences

Advanced
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

Vincent Naudot 1, and  Jiazhong Yang 2,

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.
Keywords: Tagged monomial function, Logarithmic Mourtada Type function., Normal form, Strongly 1-resonant condition, Tagged function.
Mathematics Subject Classification: Primary: 37E30; Secondary: 37G0.
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]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[2]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018
[3]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[4]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019
[5]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020
[6]

Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021070
[7]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[8]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079
[9]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[10]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[11]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[12]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027
[13]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[14]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[15]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051
[16]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[17]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1
[18]

Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907
[19]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[20]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences