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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 = p
).
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 2p
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 Dp = 4 V o1/2. |
[1] R.Z.Sagdeev,
D.A.Usikov and G.M.Zaslavsky
Nonlinear Physics: From the Pendulum to Turbulence and Chaos (1988)