Appearance of nonlinear resonances and homoclinic orbits in Hamiltonian
dynamics is motivated following Chirikov and Zaslavsky.
Hamiltonian and equations of motion for nonlinear pendulum are (see e.g. )
H(p,x) = p2/2 + V(x)
= p2/2 + Vocos x = E = const
¶H/¶p = p ,
-¶H/¶ x =
Vo sin x .
For canonical action-angle variables (I, q)
Hamiltonian H(I) depends on I only and equations of motion are
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.
-¶H/¶q = 0 ,
I = const ,
const . (*)
For a N-dimentional integrable dynamical system one can find
equation of motion in the form (*) with
Ik , qk vectors.
In this case all orbits are situated on N-dimensional tori.
Under a periodic perturbation V(I, q, t) =
V(I, q, t + T)
H(I, q, t) =
Ho(I) + e
V(I, q, t) = Ho(I) +
Vnm(I) einq -
dI/dt = -¶H
n Vnm(I) einq -
imWt , (**)
where W = 2p /T.
For small e we will search solution
as a series
I = I o + e
I 1 + ... ,
q = q o +
e q 1 + ... ,
I o = const,
q o =
w(I o)t .
After substitution I o,
q o into (**) we get
dI 1/dt =
-iSn,m n Vnm(I o)
exp[i(nw(I o) -
mW )t] ,
I 1 =
-Sn,m n Vnm(I o)
/[nw(I o) - mW ]
ei[nw(I o) -
mW ]t + const .
At a resonance nw(I) -
mW = dnm ~ 0 ,
I 1 and q 1
contain divergent terms with small denominators
~ Vnm /dnm.
In nonlinear systems frequency w(I) is
different for different I , therefore we can get resonanses at any
external frequency W for some I, n, m .
Single resonance approximation.
To avoid this divergence for a single resonance we can solve (**) taking
into account only secular terms with nw -
mW ~ 0
dI/dt = enVnm
sin(nq - mW t +
w(I) + e dVnm/dI
cos(nq - mW t +
Corresponding Hamiltonian is
H = Ho(I) + e
Vnm(I) cos(y) ,
y = nq -
mW t + j .
For small DI = I - Io
it turns  into the nonlinear pendulum Hamiltonian
y) = nw'
DI 2/2 +
neV cos( y ) ,
w' = dw/dI .
Thus independently on w(I), V(I)
dynamics near resonance is described approximately by this universal
Hamiltonian. Resonant terms lead to appearence of homoclinic orbits
on the (y, DI)
phase plane (similar to Fig.1) with the width
max DI = 4
Nonresonant terms lead to chaotic dynamics near homoclinic orbits.
To see this we can plot Poincare sections of the flow at the moments
t = n T . One model example is the Standard map
below. You see resonances, invariant circles and chaos near homoclinic orbits.
 R.Z.Sagdeev, D.A.Usikov and G.M.Zaslavsky
Nonlinear Physics: From the Pendulum to Turbulence and Chaos (1988)
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updated 26 August 2003