# American Institute of Mathematical Sciences

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

## Polynomial and linearized normal forms for almost periodic differential systems

 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

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.
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:
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##### 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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