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Computing the characteristic roots for delay differential equations. (English) Zbl 1054.65079

The paper deals with the computation of rightmost characteristic roots of a system of linear delay differential equations (DDE) with multiple discrete and distributed delays as follows: \(y'(t)=\sum_{l=0}^kL_ly(t-\tau_l) +\int_{-\sigma_2}^{-\sigma_1}M(\theta )y(t+\theta ) \,d\theta \), \(t\geq 0\), where \(L_0,\dots,L_k\in C^{m\times m}\), \(\tau =\tau_k>\dots >\tau_l>\tau_0\), \(\sigma_2>\sigma_1\geq 0\) and \(M:[-\sigma_2, -\sigma_1]\rightarrow C^{m\times m}\) is a sufficiently smooth function.
The characteristic roots of this DDE constitute the spectrum of the infinitesimal generator \(A\) of the semigroup of solution operators. \(A\) is discretized by a suitable matrix and then its eigenvalues are computed. It avoids a complicated problem of the computation of roots of the characteristic equation of the DDE. The discretization scheme using the Runge-Kutta method is proposed. The convergence order of the computed approximate roots to the exact ones is proved for arbitrary meshes. Implementation issues lead to standard large and sparse eigenvalue problems. Numerical results for two different systems of DDEs are included.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators

Software:

DDE-BIFTOOL
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