Asymptotic and numerical stability of systems of neutral differential equations with many delays

We are concerned with the asymptotic stability of a system of linear neutral differential equations with many delays in the form y ′ ( t ) = L y ( t ) + ∑ i = 1 d M i y ( t − τ i ) + ∑ i = 1 d N i y ′ ( t − τ i ) , where L , M i , N i ∈ C N × N ( i = 1 , 2 , … , d ) are constant complex matrices, τ i > 0 ( i = 1 , 2 , … , d ) are constant delays and y ( t ) = ( y 1 ( t ) , y 2 ( t ) … y N ( t ) ) T is an unknown vector-valued function for t > 0 . We first establish a new result for the distribution of the roots of its characteristic function, next we obtain a sufficient condition for its asymptotic stability and then we investigate the corresponding numerical stability of linear multistep methods applied to such systems. One numerical example is given to testify our numerical analysis.