Figure 2

(A) Semiclassical closed trajectory in the plane xOy generated from the initial conditions \((x,y)=(10,-20)\) and \((p_x,p_y)=(-5,20)\). The corresponding dimensionless amplitudes of the elastic field are \(\beta _1=\frac{\pi }{4}\) and \(\beta _2=-10\) which lie in the region of stability as for Fig. 1. The loop closes after a period \(T=6\). The orange dot represents its initial and final positions. (B) Broken loop of inverted free evolution. It takes the initial conditions \((x,y)=(0,0)\) and \((p_x,p_y)=(-5,20)\) with the amplitudes \(\beta _1=-11.86\) and \(\beta _2=-0.4\) which lie exactly in the separatrix between the stability and instability regions. For any operation that is realised with amplitudes of this kind, it will take a period of \(T=2\) to complete the control operation. The orange (red) dot represents its initial (final) position.