Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 24, pp. 1-32.

Title: Nonlinear perturbations of piecewise-linear systems with
       delay and applications to gene regulatory networks
        
Authors: Valeriya Tafintseva (Norwegian Univ. of Life Sciences, Norway)
         Arcady Ponosov  (Norwegian Univ. of Life Sciences, Norway)
        
Abstract:
 We study piecewise-linear delay differential systems which describe
 gene regulatory networks with Boolean interactions.
 Under very general assumptions put on the regulatory functions
 it is shown how to construct the limit dynamics of the systems by
 applying singular perturbation analysis.
 The obtained results are compared with those based on the multilinear
 representation of the regulatory functions usually considered in the
 literature. It is shown that sliding modes may be quite different
 in the multilinear and general case.
 Polynomial representations of the systems are proposed to describe
 generic cases of the dynamics. The results presented in this paper
 may give the insight into the theory of gene regulatory networks.

Submitted November 14, 2012. Published January 27, 2013.
Math Subject Classifications: 34K07, 34K26, 34K60.
Key Words: Delay differential equations; singular perturbation analysis;
           polynomial representation; gene regulatory networks.