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

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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

Abstract
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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 and 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.
[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.
[3]	I. U. Bronstein and A. Ya. Kopanskii, "Smooth Invariant Manifolds and Normal Forms," World Scientific, River Edge, NJ, 1994. doi: 10.1142/9789812798749.
[4]	A. D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer-Verlag, Berlin, 1989.
[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.
[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.
[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.
[8]	V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory and Dynamical Systems, 16 (1996), 1071-1086.
[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.
[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.
[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.
[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.
[13]	S. Sternberg, The structure of local homeomorphisms, III, Amer. J. Math., 81 (1959), 578-604. doi: 10.2307/2372915.
[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.

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.
[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.
[3]	I. U. Bronstein and A. Ya. Kopanskii, "Smooth Invariant Manifolds and Normal Forms," World Scientific, River Edge, NJ, 1994. doi: 10.1142/9789812798749.
[4]	A. D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer-Verlag, Berlin, 1989.
[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.
[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.
[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.
[8]	V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory and Dynamical Systems, 16 (1996), 1071-1086.
[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.
[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.
[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.
[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.
[13]	S. Sternberg, The structure of local homeomorphisms, III, Amer. J. Math., 81 (1959), 578-604. doi: 10.2307/2372915.
[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.

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