dx/dt = ∂ H/∂ p = p , dp/dt = -∂ H/∂ x = V

When -V particle oscillates
between two turning points. On the _{o} < E < V_{o}phase plane (x, p) its orbit rotates
periodically (the blue oval curves) around elliptic fixed point O
(equilibrium position at x = π ).
For E > V orbits pass
over maxima of _{o}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 =
V separates these two regions. It goes out and come to
hyperbolic points _{o}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
_{ o}^{1/2}. |

dθ/dt = ∂H(I)/∂I = ω(I) = const

Under a periodic perturbation

dI/dt = -∂ H/∂ θ = -iε∑

where

I

After substitution

I

At a resonance

dθ/dt = ω(I) + ε dV

Corresponding Hamiltonian is

For small

Thus independently on

Nonresonant terms lead to chaotic dynamics near homoclinic orbits. To see this we can plot Poincare sections of the flow at the moments

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