The xn-x Plane for Analysis of Certain Second-Order Nonlinear Systems

Analysis of certain second-order nonlinear systems, not easily amenable to the phase-plane methods, and described by either of the following differential equations x
^n-2
+ f(x)x
²ⁿ
+g(x)x
ⁿ
+h(x)=0 +f(x)x-
ⁿ
+h(x)=0 n≫0 can be effected easily by drawing the entire portrait of trajectories on a new plane; that is, on one of the x
ⁿ
x planes. Simple equations are given to evaluate time from a trajectory on any of these n planes. Poincare's fundamental phase plane x-x is conceived of as the simplest case of the general x
ⁿ
x plane.