# American Institute of Mathematical Sciences

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] 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 [4] 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 [5] 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 [6] 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 [7] Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263 [8] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264 [9] 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 [10] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [11] Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117 [12] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [13] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

2019 Impact Factor: 1.233