The x¿n-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. Poincaré´s fundamental phase plane x¿¿x is conceived of as the simplest case of the general x¿ⁿ¿x plane.