\(\mathcal H\)-stability of linear \(\theta\)-method with general variable stepsize for system of pantograph equations with two delay terms
The authors study the two-dimensional linear pantograph equation \(\dot x(t)= a_1 x(t)+b_1 y(q_1t)\), \(\dot y(t)=a_2 y(t)+b_2 x(q_2t)\) for \(0<q_1\leq q_2<1\). The authors first give sufficient conditions on the complex system parameters \(a_j\), \(b_j\) (\(j=1,2\)) such that the origin is globally asymptotically stable independent of \(q_1\) and \(q_2\). Then, they prove that the linear \(\theta\)-method with variable stepsize is \({\mathcal H}\)-stable if \(1/2<\theta<1\). \({\mathcal H}\)-stability means that all numerical solutions tend to zero if the origin is asymptotically stable regardless of \(q_1\), \(q_2\), and the sequence of stepsizes.
scientific article; zbMATH DE number 2118011