Pseudo closed trajectories in the family of trajectories defined by a system of differential equations
Some of these curves are considered in a rather incidental way in the writings of Poincare.** However, the full concept of pseudo closed trajectories does not seem to have been discussed explicitly heretofore. We assume that all of the variables in the equations (1) are real, that the functions X and Y are continuous in an open connected region R in the xy-plane, that these functions satisfy Lipschitz conditions locally throughout R, and that the real curves X = 0 and Y = 0, if they exist, have only simple intersections. We assume that these points of intersection have no point of condensation in the region R. We also assume that if the functions X and Y are not linear, they satisfy conditions which are sufficient to insure that the topological properties of the family of trajectories in a small neighborhood of any singular point (point of intersection of the curves X = 0, Y = 0) other than a center are the same as in the case obtained by replacing X and Y by their linear approximations. The last assumption is satisfied if X and Y are, for instance, of class C2 in R. In brief, our assumptions are simply the ones that are usually employed in discussions of the family of trajectories. The necessary information about the implications of the assumptions is readily available in the literature.f Since the notion of a trajectory is a familiar one, we have used the term so far without any explanations. However, in order to avoid the danger of future ambiguities, it will be well now to define the term explicitly. Let x — <p(t), y — ip(t) be a solution of the system of equations (1). It is understood that the solution is defined over a certain maximum (finite or infinite) open interval J(,) on the i-axis. As t describes 7(<> once in the sense of increasing t, the point (x, y) = (<p(t), $(t)) describes a certain curve T in R. We call T a trajectory.