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__Polynomial chaos expansion and stability analysis of uncertain DDEs__

R. Vermiglio, A. Ern and O.L. Maitre

**Abstract**

Many problems in science and engineering are modeled with systems of delay differential equations (DDEs). Since they are infinite dimensional dynamical systems, the stability analysis requires effective numerical methods. Beside the mathematical model and the numerical method, another relevant aspect is the specification of the data. The uncertainty in the model constants are often modeled as random processes.
The Polynomial Chaos (PC) expansion provides a representation of random variables, vectors and processes with respect to a basis set of functionals, usually orthogonal polynomials. It is based on the original Wiener's theory of homogeneous chaos [2] and it has been successfully applied in different fields for uncertainty quantification e.g. [1].
We propose to apply PC expansion to quantify the resulting uncertainty in the numerical stability analysis of equilibria of DDEs with random parameters. We introduce some basic ideas and we present preliminary results.

**Bibliography**

[1] O. Le Ma\^ itre and O. Knio, Spectral Methods for Uncertainty Quantification
with Applications to Computational Fluid Dynamics, Springer-Verlag Ed. (2010)

[2] N. Wiener, The homogeneous chaos, Amer. J. Math., 60 (1938), pp. 897-936.