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The integrodifferential system with aftereffect (“heredity” or “prehistory”) dx/dt=Ax+-t R(t,s)x(s,)ds, is considered; here $\epsilon$ is a positive small parameter, $A$ is a constant $n\times n$ matrix, $R(t,s)$ is the kernel of this system exponentially decreasing in norm as $t\to \infty$. It is proved, if matrix $A$ and kernel $R(t,s)$ satisfy some restrictions and $\epsilon$ does not exceed some bound ${\epsilon }_{*}$, then the $n$-dimensional set of the so-called principal two-sided solutions $\tilde{x}(t,\epsilon )$ approximates in asymptotic sense the infinite-dimensional set of solutions...

In this paper we present an iterative algorithm for the construction of Mathieu functions of any order $N$ in the form of Fourier series (practically, polynomials), and also the corresponding Quick-BASIC program for realization of this algorithm with numerical values of the parameter.