As the values of parameters in periodic systems vary, a nodal point appearing on a locus of period doubling bifurcation points crosses over a locus of turning points. We consider the nodal point lying just on the locus of turning points and consider its accurate location. To compute it, we consider an extended system which consists of an original equation and an additional equation. We present a result assuring that this extended system has an isolated solution containing the nodal point.

This paper is concerned with singular points and bifurcation points of quasiperiodic solutions to quasiperiodic differential systems. In fact, from the results of numerical experiments, we can observe such points in Duffing's equation with a quasiperiodic forcing term. The purpose of this paper is to propose a method for computing them. The essential idea of our method is to compute a quasiperiodic solution to an enlarged differential system which consists of an original quasiperiodic system and an additional differential system corresponding to the first variation equation of the original system. By making use of the Poincar mapping, a quasiperiodic solution to a quasiperiodic differential system is reduced to a periodic solution to a periodic difference system. Therefore, this periodic solution can be computed by using a finite trigonometric series.