American Institute of Mathematical Sciences

Advanced
January  2001, 7(1): 155-176. doi: 10.3934/dcds.2001.7.155

Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II

Teresa Faria 1,

1. 

Departamento de Matemática, Faculdade de Ciências, and CMAF, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

Received  May 2000 Revised  August 2000 Published  November 2000

A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.
Keywords: Normal form theory, functional differential equations..
Mathematics Subject Classification: 34K30, 34K17, 34K18, 35K5.
Citation: Teresa Faria. Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 155-176. doi: 10.3934/dcds.2001.7.155
[1]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[2]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793
[3]

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
[4]

Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784
[5]

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
[6]

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
[7]

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
[8]

Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016
[9]

Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033
[10]

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
[11]

John A. D. Appleby, Denis D. Patterson. Subexponential growth rates in functional differential equations. Conference Publications, 2015, 2015 (special) : 56-65. doi: 10.3934/proc.2015.0056
[12]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[13]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[14]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[15]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362
[16]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[17]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283
[18]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[19]

R.S. Dahiya, A. Zafer. Oscillatory theorems of n-th order functional differential equations. Conference Publications, 2001, 2001 (Special) : 435-443. doi: 10.3934/proc.2001.2001.435
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences