3.—A Note on Quadratic Systems

Abstract

Synopsis‘Quadratic system’ is here used in the sense introduced by W. A. Coppel for a pair of simultaneous Riccati equations with constant coefficients. The problem is to study the occurrence of movable singularities for solutions of a systemunder suitable restrictions on the linesThe aim of the restrictions is to obtain systems where the trajectories have four finite and three infinite topological singularities which form specified configurations. For the movable (function theoretical) singularities the infinitary (topological) singularities are decisive. They are the endpoints of the coordinate axes and of the line F: L2–L1 = 0. A detailed qualitative discussion is given for the trajectories. The system forces various properties of monotony on the integral curves.The qualitative discussion is completed by quantitive data in the form of series expansions for the solutions. Here the value of the parameteris critical.If q is not an integer, then there is one and only one solution whose trajectory is associated with the upper end of the line F which has a simple pole at t = 0 (or any other preassigned value). All other trajectories in this family belong to solutions with expansions in terms of non-logarithmic psi-series of the formfor a singularity at t = 0. Here – 1 ≦ m, 0 ≦ n and c is an arbitrary constant.On the other hand, if q is a positive integer, then we have logarithmic psi-series of the formwhere Pn(s) and Qn(s) are polynomials in s of degree ≦ [n/q].Solutions associated with the ends of the axes have non-logarithmic psi-series. One coordinate becomes infinite like 1/t at zero, the other goes to zero like tλ where λ = –b1/a1 for the x-axis and –a2/b2 for the y-axis. The psi-series have exponents of the form j + Kλ, where j and k are integers, –1 ≦ j, 0 ≦ k.The results are used to construct new convergence proofs for the solutions of Emden's equation at a movable singularity.