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In this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits blow-up solutions, no matter how small the length of the delay is. In the non-delay system every solution approaches to a stable unit circle in the plane, thus time delay induces blow-up of solutions, which we call “delay-induced blow-up” phenomenon. Furthermore, it is shown that the system has a family of infinitely many periodic solutions, while the non-delay system has only one stable limit cycle. The system studied in this paper is an example that arbitrary small delay can be responsible for a drastic change of the dynamics. We show numerical examples to illustrate our theoretical results.

We construct here an embedded Dormand–Prince pair of explicit methods of orders 6 and 4 for systems of ordinary differential equations with special structure, namely with two parts, in which the right-hand sides are dependent only on the unknown functions from the other group. The number of stages is six, which is fewer than for general explicit Runge–Kutta methods. The comparison to Dormand–Prince method of the same computation cost is made showing the higher accuracy of the suggested method.

We consider here numerical methods for systems of retarded functional differential equations of two equations in which the right-hand sides are cross-dependent of the unknown functions, i.e. the derivatives of unknowns don’t depend on the same unknowns. It is shown that using the special structure of the system one can construct functional continuous methods of Runge–Kutta type with fewer stages than it is necessary in case of general Runge–Kutta functional continuous methods. Order conditions and example methods of orders three and four are presented. Test problems are solved, demonstrating the declared convergence order of the new methods.

An explicit embedded method of the Dormand-Prince type designed for integrating systems of ordinary differential equations of special form is examined. A family of economical fifth-order numerical schemes for integrating systems of structurally separated ordinary differential equations is constructed.

An explicit Runge-Kutta type method for systems of ordinary differential equations with special structure is considered. For partitioned systems a family of explicit methods of order six with just six stages is constructed, which makes them more efficient than classic Runge-Kutta methods of order six. It is shown that second order differential equations, which right-hand side doesn't depend on the first derivative, can be rewritten as the considered partitioned systems. Direct application of the constructed methods to them generates two different families of Runge-Kutta-Nystrom methods. The comparison of constructed methods with known methods of order six is held.

An explicit embedded pair of methods for systems of second order ordinary equations with special structure is considered. Two-parametric families of methods of orders four and three with automatic step-size control are constructed. The numeric comparison to known embedded Runge-Kutta pairs of the same order is held.

When solving delay differential equations (DDEs) with state-dependent delays the problem of breaking point detection is important. Points where the solution is not smooth enough to provide the order of the method must be included into the computational mesh, otherwise a reduction in the order of the solution will result. The problem, however is to detect and compute such points efficiently. Breaking points arise every time a delay falls on a previous breaking point (either of the calculated solution or in the history function). In the case of retarded DDEs the new breaking point is (at least) one order smoother than the previous breaking point that gave rise to it. For fixed or time-dependent delays the breaking points can be precomputed independent of the solution, but for state-dependent delays the positions of the breaking points depend on the computed solution. If a breaking point is detected and the step-size is changed in order to incorporate the point into the mesh, then the new step-size generates a new solution and the breaking point moves. Consequently, breaking point detection is traditionally performed iteratively, and is computationally expensive. The same breaking point can also be detected multiple times. In the current work we propose a fast non-iterative method for finding breaking points with sufficient precision to preserve the order of up to third or fourth order methods. Our method makes use of analytic continuation of the solution across breaking points (including possible breaking points in the initial history function), and we explain how we handle this carefully to attain the desired order. Test results are presented for Explicit Functional Continuous Runge–Kutta methods, showing that they retain their order of convergence when the solutions have breaking points.