This is Walkinshaw coupling resonance (Subscript[ν, x])=2(Subscript[ν, y])
(or, the swinging spring, or, the elastic pendulum)

author: Rick Baartman, TRIUMF, 2020

The Hamiltonian contains the coupling term (xy^2) , of strength azero. The amplitudes of motion Ax, Ay obey (((2 Ax))^2)+(Ay^2)= constant.

At startup, immediately “Evaluate Notebook”. After that, you can manipulate the parameters.

Plot explainer: The point with crosshairs can be moved with the mouse. It is the initial x,y coordinates. Initial x’,y’ are zero.This cursor, and the blue track are plotted in x,y space. The gold track is in x,x’ space. The green track is in y’,y space (note: it’s not y,y’ but y’,y, to make it easier to identify with x,y space).
Exactly on resonance, stable fixed point are when initial y/x =+ or -(Sqrt[8]), giving a parabolic track in (y,x). The unstable fixed point is when initial y=0.

Manipulate[Module[{sol=NDSolve[{x''[t]+(nux^2) x[t]+azero ((y[t])^2)==0,
y''[t]+(nuy^2) y[t]+2 azero x[t] y[t]==0,x'[0]==0.,y'[0]==0.,
x[0]==p[[1]],y[0]==p[[2]]},{x[t],y[t],x'[t],y'[t]},{t,0,2 π T}]},ParametricPlot[{Evaluate[{x[t],y[t]}/.sol],Evaluate[{x[t],x'[t]}/.sol],Evaluate[{y'[t],y[t]}/.sol]},{t,0,2 π T},PlotRange->{{-1,1},{-1,1}},ImageSize->{600,600},PlotPoints->600]],{{p,{.08,.8}},Locator},{{T,70,"turns"},.1,70,1},{{nux,1.4,"(Subscript[ν, x])"},0,2,.01},
{{nuy,0.7,"(Subscript[ν, y])"},0,2,.01},
{{azero,.18,"strength"},-0,1,.01},AppearanceElements->All]