Encyclopedia of Systems Biology

2013 Edition
| Editors: Werner Dubitzky, Olaf Wolkenhauer, Kwang-Hyun Cho, Hiroki Yokota

Dynamical Systems Theory, Delay Differential Equations

  • Patrick NelsonEmail author
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-9863-7_272

Synonyms

Definition

Delay differential equations (DDE) are equations whose solution depends on not just a single initial condition at time, t = t 0, but also on the past history of the system. DDEs can be classified as retarded or neutral and continuous or discrete. In general, a discrete delay differential equation can be written as
$$\eqalign{ \frac{{{d^n}y}}{{d{t^n}}} =\ f( {t,\;{a_0}{{(t)}}y(t),\;a_1{{(t)}}y'(t), \ldots a{{(t)}_{{n - 1}}}{y^{{n - 1}}}(t),} \cr {b_0{{(t)}}y(t - \tau ) \ldots b_{{n - 1}}}{{(t)}{y^{{n - 1}}}(t - \tau )}) }$$
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References

  1. Asl FM, Ulsoy AG (2003) Analysis of a system of linear differential equations. ASME J Dyn Syst Meas Control 125:215–223CrossRefGoogle Scholar
  2. Bellman R, Cooke K (1963) Delay differential equations. Academic, New York/LondonGoogle Scholar
  3. Chebotarev NG, Meiman NN (1949) The Routh-Hurwitz problem for polynomials and entire functions. Trudy Mat Inst Steklov 26Google Scholar
  4. Corless RM et al (1996) On Lambert’s W function. Adv Comput Math 5:329–359CrossRefGoogle Scholar
  5. El’sgol’ts LE, Norkin SB (1973) An introduction to the theory and application of differential equations with deviating arguments. Academic, New YorkGoogle Scholar
  6. Forde J, Nelson P (2000) Applications of Sturm sequences to bifurcation analysis of delay differential equation models. J Math Anal Appl 300:273–284CrossRefGoogle Scholar
  7. Hale JK (1977) Theory of functional differential equations. Spring, New YorkCrossRefGoogle Scholar
  8. Krall AM (1965) Stability criteria for feedback systems with a time lag. SIAM J Control 2:160–170Google Scholar
  9. Kuang Y (1993) Delay differential equations with applications to population biology. Academic, BostonGoogle Scholar
  10. Pontryagin LS (1955) On the zeros of some elementary transcendental functions. Am Math Soc Transl 2:95–110Google Scholar
  11. Stepan G (1989) Retarded dynamical systems: stability and characteristic functions. Longman Scientific and Technical, Burnt MillGoogle Scholar
  12. Yi S, Ulsoy AG, Nelson PW (2007) Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter. Math Biosci Eng 4:1–12CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.CCMB 2017 Palmer CommonsUniversity of MichiganAnn ArborUSA