Generalized isocline method of plotting phase-plane trajectories

By considering simultaneously the
N-x
(or the
N-\dot{x}
), where
N=d\dot{x}/dx
and the
\dot{x}-x
planes, second order non-linear autonomous systems (not easily amendable to the existing methods) described by the differential equation
\ddot{x}=F(x, \dot{x})
can be studied. Changes in system behavior due to changes in initial conditions, nonlinear dampling and restoring forces can easily be studied. Simple and general as it is, the method presented here is believed to be novel.