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 a1. The amplitudes of motion (Subscript[A, x]), (Subscript[A, y]) obey (((2 (Subscript[A, x])))^2)+((Subscript[A, y])^2)= constant.
Notice that there is no driving harmonic.

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’ can also be moved with the crosshair that’s at origin.The x,y cursor, and the black track are plotted in x,y space. The blue track is in x,x’ space. The gold 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. Away from resonance, there are still two stable fixed points but they are not symmetric about x=0.
Turn colour to zero if you want only x,y.

Manipulate[Module[{sol=NDSolve[{x''[θ]+(nux^2) x[θ]+a1 ((y[θ])^2)==0,
y''[θ]+(nuy^2) y[θ]+2 a1 x[θ] y[θ]==0,x'[0]==p[[1]],y'[0]==p[[2]],
x[0]==q[[1]],y[0]==q[[2]]},{x[θ],y[θ],x'[θ],y'[θ]},{θ,0,2 π T}]},ParametricPlot[{Evaluate[{x[θ],x'[θ]}/.sol],Evaluate[{y'[θ],y[θ]}/.sol],Evaluate[{x[θ],y[θ]}/.sol]},{θ,0,2 π T},PlotRange->{{-1,1},{-1,1}},ImageSize->{600,600},PlotPoints->600,LabelStyle->{FontFamily->"Times New Roman",24},PlotStyle->{RGBColor[1-.75 colsc,1-.5 colsc,1],RGBColor[1,1-0.5 colsc,1-0.75 colsc],RGBColor[0,0.,0.]},PlotLegends->{"x,x' phase space","y',y phase space","x,y realspace"},PlotLabel->"Walkinshaw Resonance (perturbing field ind. of θ)"]],{{p,{.0,.0}},Locator},{{q,{.4,.56}},Locator},{{colsc,1,"colour"},.0,1.,.1},{{T,6,"turns"},.01,70,1},{{nux,1.22,"(Subscript[ν, x])"},0,2,.01},
{{nuy,0.61,"(Subscript[ν, y])"},0,2,.01},
{{a1,.1,"strength"},-0,.5,.01},AppearanceElements->All]