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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

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 & Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345
References:
[1]

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[2]

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[3]

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A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer-Verlag, 1974.  Google Scholar

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J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.  Google Scholar

[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.  Google Scholar

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W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236. doi: 10.1017/S014338570400094X.  Google Scholar

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W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988. doi: 10.1002/cpa.20083.  Google Scholar

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H. Poincaré, Thesis, 1879, also Oeuvres I, Gauthier Villars, Paris, (1928), 59-129. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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S. Stenberg, Finite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., 10 (1961), 451-474.  Google Scholar

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F. Takens, Normal forms for certain singularities of vector fields, An. Inst. Fourier., 23 (1973), 163-195. doi: 10.5802/aif.467.  Google Scholar

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A. Vanderbauwhede, Center manifolds and their basic properties. An introduction, Delft Progr. Rep., 12 (1988), 57-78.  Google Scholar

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.  Google Scholar

[2]

D. Bainov and P. Simenov, Integral Inequalities and Applications, Kluwer academic publishers, 1992. doi: 10.1007/978-94-015-8034-2.  Google Scholar

[3]

Yn. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Math., 702, Springer-Verlag, 1979.  Google Scholar

[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.  Google Scholar

[5]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[6]

H. Dulac, Solution d'un systeme d'equations differentielles dans le voisinage de valeurs singulieres, Bull. Soc. Math. Fr., 40 (1912), 324-383.  Google Scholar

[7]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer-Verlag, 1974.  Google Scholar

[8]

J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.  Google Scholar

[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.  Google Scholar

[10]

W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236. doi: 10.1017/S014338570400094X.  Google Scholar

[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.  Google Scholar

[12]

H. Poincaré, Thesis, 1879, also Oeuvres I, Gauthier Villars, Paris, (1928), 59-129. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Stenberg, Finite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., 10 (1961), 451-474.  Google Scholar

[16]

F. Takens, Normal forms for certain singularities of vector fields, An. Inst. Fourier., 23 (1973), 163-195. doi: 10.5802/aif.467.  Google Scholar

[17]

A. Vanderbauwhede, Center manifolds and their basic properties. An introduction, Delft Progr. Rep., 12 (1988), 57-78.  Google Scholar

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