Center manifold analysis of a DDE model of gene expression

Communications in Nonlinear Science and Numerical Simulation

Volumn 13, Issue 6, 2008, Pages 1112-1120

  Verdugo, Anael ^a   Rand, Richard ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

Center manifold; DDE; Delay; Gene transcription; Hopf bifurcation

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; GENE EXPRESSION; PROTEINS;

CENTER MANIFOLD THEORY; DELAY DIFFERENTIAL EQUATION; LINDSTEDT'S METHOD;

ORDINARY DIFFERENTIAL EQUATIONS;

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; GENE EXPRESSION; ORDINARY DIFFERENTIAL EQUATIONS; PROTEINS;

EID: 36448975461     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2006.09.011     Document Type: Article

References (12)

1. Campbell S.A., Belair J., Ohira T., and Milton J. Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback. Chaos 5 (1995) 640-645
2. Carr J. Applications of centre manifold theory (1981), Springer-Verlag, New York
3. Guckenheimer J., and Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983), Springer-Verlag, New York
4. Hassard B.D., Kazarinoff N.D., and Wan Y.-H. Theory and applications of Hopf bifurcation (1981), Cambridge University Press, Cambridge
5. Kalmar-Nagy T., Stepan G., and Moon F.C. Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn 26 (2001) 121-142
6. Monk N.A.M. Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays. Curr Biol 13 (2003) 1409-1413
7. Namachchivaya N.S., and Beddini R. Spindle speed variation for the suppression of regenerative chatter. J Nonlinear Sci 13 (2003) 265-288
8. Rand RH. Lecture notes on nonlinear vibrations (version 52). Available from: http://www.tam.cornell.edu/randdocs/nlvibe52.pdf 2005.
9. Rand R.H., and Armbruster D. Perturbation methods, bifurcation theory and computer algebra (1987), Springer-Verlag, New York
10. Rand R, Verdugo A. Hopf bifurcation formula for first order differential-delay equations. Comm Nonlinear Sci Numer Simulat, in press, doi:10.1016/j.cnsns.2005.08.005.
11. Stepan G. Retarded dynamical systems: stability and characteristic functions (1989), Longman Scientific and Technical, Essex
12. Verdugo A, Rand R. Hopf bifurcation in a DDE model of gene expression. Comm Nonlin Sci Numer Simulat, in press, doi:10.1016/j.cnsns.2006.05.001.