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, Dynamical Systems, 1 (1988), 1-148, Springer-Verlag, Berlin.  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), 19-22.  Google Scholar

[3]

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

[4]

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

[5]

K.-T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math., 85 (1963), 693-722. 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), 1-43. 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), 3509-3522. 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), 1071-1086.  Google Scholar

[9]

V. Naudot and J. Yang, Linearization of families of germs of hyperbolic vector fields, Dynamical Systems, 23 (2008), 467-489. 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), 187-219.  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), 543-550. 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), 623-631. doi: 10.2307/2372774.  Google Scholar

[13]

S. Sternberg, The structure of local homeomorphisms, III, Amer. J. Math., 81 (1959), 578-604. doi: 10.2307/2372915.  Google Scholar

[14]

J. Yang, Polynomial normal forms for vector fields on $R^3$, Duke Math. J., 106 (2001), 1-18. 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, Dynamical Systems, 1 (1988), 1-148, Springer-Verlag, Berlin.  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), 19-22.  Google Scholar

[3]

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

[4]

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

[5]

K.-T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math., 85 (1963), 693-722. 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), 1-43. 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), 3509-3522. 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), 1071-1086.  Google Scholar

[9]

V. Naudot and J. Yang, Linearization of families of germs of hyperbolic vector fields, Dynamical Systems, 23 (2008), 467-489. 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), 187-219.  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), 543-550. 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), 623-631. doi: 10.2307/2372774.  Google Scholar

[13]

S. Sternberg, The structure of local homeomorphisms, III, Amer. J. Math., 81 (1959), 578-604. doi: 10.2307/2372915.  Google Scholar

[14]

J. Yang, Polynomial normal forms for vector fields on $R^3$, Duke Math. J., 106 (2001), 1-18. doi: 10.1215/S0012-7094-01-10611-X.  Google Scholar

[1]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $ H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004
[2]

Qi Wang. Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function. Communications on Pure & Applied Analysis, 2015, 14 (2) : 383-396. doi: 10.3934/cpaa.2015.14.383
[3]

Peter Frolkovič, Karol Mikula, Jozef Urbán. Distance function and extension in normal direction for implicitly defined interfaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 871-880. doi: 10.3934/dcdss.2015.8.871
[4]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[5]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[6]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413
[7]

H. N. Mhaskar, T. Poggio. Function approximation by deep networks. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4085-4095. doi: 10.3934/cpaa.2020181
[8]

Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669
[9]

Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.

[10]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741
[11]

Steve Hofmann, Dorina Mitrea, Marius Mitrea, Andrew J. Morris. Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets. Electronic Research Announcements, 2014, 21: 8-18. doi: 10.3934/era.2014.21.8
[12]

Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403
[13]

Feng Ma, Jiansheng Shu, Yaxiong Li, Jian Wu. The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1173-1185. doi: 10.3934/jimo.2020016
[14]

Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51
[15]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475
[16]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
[17]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[18]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
[19]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
[20]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences