Consider the system $$\dot{x}=y, \qquad \dot{y}=xw^2-GM/{x^2}, \qquad \dot{z}=w, \qquad \dot{w}=-2yw/{x}$$ that represent a satellite in orbit where x= radius (distance from earth) y= radius velocity. z=angle and w=angle velocity.
i tried to solve it with matlab and ode45 with r=42200000 starting point. And i get what it seems to be a periodic solution, the radius oscillates , and the angle keep going, and this is for diverse starting point. At some point the radius start to diverge.
how i explained it is that in a certain area i have a periodic solution. after i'm to far away from earth so i don't have the gravity keeping me there.
is there a way to demonstrate/visualize those closed orbit? like a phase plane portrait or some kind of poincare maps?