Nikolis, Athanassios; Seimenis, Ioannis Solving dynamical systems with cubic trigonometric splines. (English) Zbl 1071.65174 Appl. Math. E-Notes 5, 116-123 (2005). The aim of the paper is the numerical solving using cubic trigonometric splines, of special nonlinear dynamical systems. The first part represents an introduction for the dynamical system and the trigonometric basic splines which will be used in the sequel. The second part is devoted to the presentation of the method and to obtain the condition which ensures the existence of the numerical solution, for \(2\times 2\) systems. The third part concerns the derivation of the error estimates in the one-dimensional case. The authors obtain that the convergence order of the spline approximations of the solution is cubic. The fourth part contains the numerical results which are obtained for \(2\times 2\) and \(3\times 3\) systems using the presented method in comparison with those obtained with the method of rational Fourier series approximations. Thus the accuracy of the results shows the efficiency of the trigonometric splines in approximating periodical functions. Reviewer: R. Militaru (Craiova) Cited in 5 Documents MSC: 65P10 Numerical methods for Hamiltonian systems including symplectic integrators 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems 65L70 Error bounds for numerical methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations Keywords:initial value problems; numerical results; error estimates; rational Fourier series approximations; nonlinear dynamical systems; comparison of methods; convergence PDFBibTeX XMLCite \textit{A. Nikolis} and \textit{I. Seimenis}, Appl. Math. E-Notes 5, 116--123 (2005; Zbl 1071.65174) Full Text: EuDML