Nonlinear resonance

Appearance of nonlinear resonances and homoclinic orbits in Hamiltonian dynamics is motivated following Chirikov and Zaslavsky.

Nonlinear pendulum

Hamiltonian and equations of motion for nonlinear pendulum are (see e.g. [1])
    H(p,x) = p2/2 + V(x) = p2/2 + Vocos x = E = const
    dx/dt = H/p = p ,     dp/dt = -H/ x = Vo sin x .
Oscillator 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.
For canonical action-angle variables (I, q) Hamiltonian H(I) depends on I only and equations of motion are
    dI/dt = -H/¶q = 0 ,     I = const ,
    dq/dt = H/I = w(I) = 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.

Resonances

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) + e Sn,m Vnm(I) einq - imWt ,
    dI/dt = -H /¶q = -ie Sn,m n Vnm(I) einq - imWt ,     (**)
    dq/dt = H/I .
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. Universal Hamiltonian

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 + j) ,
    dq/dt = w(I) + e dVnm/dI cos(nq - mW t + j) .
Corresponding Hamiltonian is
    H = Ho(I) + e Vnm(I) cos(y) ,   y = nq - mW t + j .
For small DI = I - Io it turns [1] into the nonlinear pendulum Hamiltonian
    H(DI, 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 (eV/|w'|)1/2 .
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.

[1] 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