Polynomial and linearized normal forms for almost periodic differential systems

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January 2016, 36(1): 345-360. doi: 10.3934/dcds.2016.36.345

Polynomial and linearized normal forms for almost periodic differential systems

Weigu Li ^1, , Jaume Llibre ^2, and Hao Wu ^3,

1.	School of Mathematics, Peking University, Beijing 100871
2.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
3.	Department of Mathematics, Southeast University, Nanjing 210096, Jiangsu, China

Received October 2013 Revised May 2014 Published June 2015

Abstract
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Under a Creative Commons license

For almost periodic differential systems $\dot x= \varepsilon f(x,t,\varepsilon)$ with $x\in \mathbb{C}^n$, $t\in \mathbb{R}$ and $\varepsilon>0$ small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system $\dot x= \varepsilon \lim_{T \to \infty} \frac {1} {T} \int_0^T f(x,t,0) dt$, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non--resonant.

Keywords: averaging method, linearization., normal form, Almost periodic differential systems.

Mathematics Subject Classification: Primary: 34C29, 34C2.

Citation: Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345

References:

[1]	L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	D. Bainov and P. Simenov, Integral Inequalities and Applications, Kluwer academic publishers, 1992. doi: 10.1007/978-94-015-8034-2.
[3]	Yn. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Math., 702, Springer-Verlag, 1979.
[4]	K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Am. J. Math., 85 (1963), 693-722. doi: 10.2307/2373115.
[5]	W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629. Springer-Verlag, Berlin-New York, 1978.
[6]	H. Dulac, Solution d'un systeme d'equations differentielles dans le voisinage de valeurs singulieres, Bull. Soc. Math. Fr., 40 (1912), 324-383.
[7]	A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer-Verlag, 1974.
[8]	J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.
[9]	Yu. S. Il'yashenko and S. Yu. Yakovenko, Finitely-smooth normal forms of local families of diffeomorphisms and vector fields, Russian Math Sur., 46 (1991), 1-43. doi: 10.1070/RM1991v046n01ABEH002733.
[10]	W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236. doi: 10.1017/S014338570400094X.
[11]	W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988. doi: 10.1002/cpa.20083.
[12]	H. Poincaré, Thesis, 1879, also Oeuvres I, Gauthier Villars, Paris, (1928), 59-129.
[13]	R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Diff. Equ., 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.
[14]	S. Siegmund, Normal forms for nonautonomous difference equations. Advances in difference equations, IV, Comput. Math. Appl., 45 (2003), 1059-1073. doi: 10.1016/S0898-1221(03)00085-3.
[15]	S. Stenberg, Finite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., 10 (1961), 451-474.
[16]	F. Takens, Normal forms for certain singularities of vector fields, An. Inst. Fourier., 23 (1973), 163-195. doi: 10.5802/aif.467.
[17]	A. Vanderbauwhede, Center manifolds and their basic properties. An introduction, Delft Progr. Rep., 12 (1988), 57-78.

show all references

References:

[1]	L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	D. Bainov and P. Simenov, Integral Inequalities and Applications, Kluwer academic publishers, 1992. doi: 10.1007/978-94-015-8034-2.
[3]	Yn. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Math., 702, Springer-Verlag, 1979.
[4]	K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Am. J. Math., 85 (1963), 693-722. doi: 10.2307/2373115.
[5]	W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629. Springer-Verlag, Berlin-New York, 1978.
[6]	H. Dulac, Solution d'un systeme d'equations differentielles dans le voisinage de valeurs singulieres, Bull. Soc. Math. Fr., 40 (1912), 324-383.
[7]	A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer-Verlag, 1974.
[8]	J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.
[9]	Yu. S. Il'yashenko and S. Yu. Yakovenko, Finitely-smooth normal forms of local families of diffeomorphisms and vector fields, Russian Math Sur., 46 (1991), 1-43. doi: 10.1070/RM1991v046n01ABEH002733.
[10]	W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236. doi: 10.1017/S014338570400094X.
[11]	W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988. doi: 10.1002/cpa.20083.
[12]	H. Poincaré, Thesis, 1879, also Oeuvres I, Gauthier Villars, Paris, (1928), 59-129.
[13]	R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Diff. Equ., 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.
[14]	S. Siegmund, Normal forms for nonautonomous difference equations. Advances in difference equations, IV, Comput. Math. Appl., 45 (2003), 1059-1073. doi: 10.1016/S0898-1221(03)00085-3.
[15]	S. Stenberg, Finite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., 10 (1961), 451-474.
[16]	F. Takens, Normal forms for certain singularities of vector fields, An. Inst. Fourier., 23 (1973), 163-195. doi: 10.5802/aif.467.
[17]	A. Vanderbauwhede, Center manifolds and their basic properties. An introduction, Delft Progr. Rep., 12 (1988), 57-78.

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