American Institute of Mathematical Sciences

Advanced
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]

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

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

[1]

Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9
[2]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010
[3]

Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time - periodic systems without a small parameter. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 753-782. doi: 10.3934/dcds.2006.14.753
[4]

Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857
[5]

Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[6]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301
[7]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[8]

Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268
[9]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[10]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[11]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[12]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[13]

Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575
[14]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604
[15]

Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1241-1255. doi: 10.3934/dcdsb.2019218
[16]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172
[17]

P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure & Applied Analysis, 2004, 3 (2) : 161-173. doi: 10.3934/cpaa.2004.3.161
[18]

Massimo Tarallo. Fredholm's alternative for a class of almost periodic linear systems. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2301-2313. doi: 10.3934/dcds.2012.32.2301
[19]

Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030
[20]

Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (123)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences