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

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (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 (1979).   Google Scholar

[4]

K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point,, Am. J. Math., 85 (1963), 693.  doi: 10.2307/2373115.  Google Scholar

[5]

W. A. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics, (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.   Google Scholar

[7]

A. M. Fink, Almost Periodic Differential Equations,, Lecture Notes in Math., 377 (1974).   Google Scholar

[8]

J. K. Hale, Ordinary Differential Equations, Second edition,, Robert E. Krieger Publishing Co., (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.  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.  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.  doi: 10.1002/cpa.20083.  Google Scholar

[12]

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

[13]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Diff. Equ., 27 (1978), 320.  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.  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.   Google Scholar

[16]

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

[17]

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

show all references

References:
[1]

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (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 (1979).   Google Scholar

[4]

K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point,, Am. J. Math., 85 (1963), 693.  doi: 10.2307/2373115.  Google Scholar

[5]

W. A. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics, (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.   Google Scholar

[7]

A. M. Fink, Almost Periodic Differential Equations,, Lecture Notes in Math., 377 (1974).   Google Scholar

[8]

J. K. Hale, Ordinary Differential Equations, Second edition,, Robert E. Krieger Publishing Co., (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.  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.  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.  doi: 10.1002/cpa.20083.  Google Scholar

[12]

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

[13]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Diff. Equ., 27 (1978), 320.  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.  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.   Google Scholar

[16]

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

[17]

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

[1]

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

[2]

Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026

[3]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[4]

Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135

[5]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347

[6]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[7]

Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021002

[8]

Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025

[9]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176

[10]

Yongjie Wang, Nan Gao. Some properties for almost cellular algebras. Electronic Research Archive, 2021, 29 (1) : 1681-1689. doi: 10.3934/era.2020086

[11]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028

[12]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[13]

Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383

[14]

Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134

[15]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287

[16]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[17]

Aisling McGlinchey, Oliver Mason. Observations on the bias of nonnegative mechanisms for differential privacy. Foundations of Data Science, 2020, 2 (4) : 429-442. doi: 10.3934/fods.2020020

[18]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

[19]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[20]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (104)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]