When -Vo < E < Vo particle oscillates
between two turning points. On the phase plane (x, p) its orbit rotates
periodically (the blue oval curves) around elliptic fixed point O
(equilibrium position at x = π ).
For E > Vo orbits pass
over maxima of V(x) and go to infinity. The motion is unbounded (as
since V(x) is periodic, therefore we can consider motion on
a cylinder and join the 0 and 2π
points). The red separatrix (or homoclinic orbit) at E =
Vo separates these two regions. It goes out and come to
hyperbolic points X (unstable equilibrium positions at x =
2pn ). Dynamics near homoclinic orbits
is very sensitive to perturbations because a small force can throw orbit
over (or under) maximum of V(x) and change qualitatively its motion.
The "width" of the homoclinic orbit is
max Δp = 4 V o1/2.