M090 · Hamiltonian chaos / exact maps

Kicked Rotor & KAM Transition

Does keeping phase-space area fixed protect a kicked rotor from chaos? No. The kicked rotor keeps phase-space area exactly, and chaos still appears. Above a critical kick strength, the smooth curves that used to fence the motion in break apart. Nearby starts then separate quickly and the rotor can wander far in momentum, while some islands of regular motion survive.

Kick a free rotor and explore invariant curves, resonance islands, chaotic transport and regular accelerator islands. Drag the starting point in phase space, then compare the wrapped torus picture with the unwrapped momentum, the separation of neighboring trajectories, tangent growth and a check that local area is preserved.

Reviewed kick $\text{Pending}$

Actual kick strength $\text{Pending}$

Angle before next kick $\text{Pending}$

Unwrapped momentum $\text{Pending}$

Displayed wrapped momentum $\text{Pending}$

Next impulse $\text{Pending}$

Free kinetic energy $\text{Pending}$

Cylinder pair distance $\text{Pending}$

Finite-time growth per kick $\text{Pending}$

Valid tangent iterations $\text{Pending}$

Initial local area determinant $\text{Pending}$

Analytic vs finite derivative $\text{Pending}$

Single-map inverse defect $\text{Pending}$

Full atlas final RMS momentum drift $\text{Pending}$

Kicks per full atlas orbit $\text{Pending}$

Zero-momentum half-turn trace $\text{Pending}$

Watch the motion and compare the cases

Does keeping phase-space area fixed protect a kicked rotor from chaos?

Kick strength

K

1.20\,

Initial angle fraction

\theta_0/(2\pi)

0.14\,

Initial scaled momentum

p_0/(2\pi)

0.31\,

Initial angle perturbation exponent

\log_{10}\epsilon

-6.00\,

Requested kicks

N

1500.00\,

Seeded atlas orbits

N_{\rm atlas}

12.00\,

Reviewed collision or kick

n/N

0.00\,\%

Drag the orange ring on the torus section to set the initial angle and scaled momentum, or use the matching sliders from the keyboard. The rod shows a schematic kick followed by free rotation, and each sample is taken just before a kick. The grey atlas orbits run for at most 1500 kicks. Their final RMS drift is a statistic of a finite ensemble, not a diffusion coefficient. Energy can change because the kicks do work. The growth rate uses the cylinder metric and an analytic tangent vector that is renormalized after every kick. Two points that look close on the torus can still be far apart in momentum, because the torus wraps momentum around.

The rotor is classical, with ideal delta-function kicks, unit moment of inertia and unit time between kicks. The exact standard map uses a positive sine impulse. Only the display wraps momentum; the dynamics keep the full momentum on the cylinder. A finite section cannot prove that every invariant curve survives or breaks. The near-critical case is a reference neighborhood, not a computed transition threshold. Damping, finite pulse width and quantum localization are not modeled.

Start with the essentials

Focus question: Does keeping phase-space area fixed protect a kicked rotor from chaos?
One-sentence intuition: Area conservation, the stability of one point and statistics from a finite run answer three different questions.

Core mathematical model

An impulse followed by a drift

p_{n+1}=p_n+K\sin\theta_n,\qquad \theta_{n+1}=\theta_n+p_{n+1}\pmod{2\pi}

The plus sign in front of the sine comes from a cosine kick potential, with unit moment of inertia and unit time between kicks. Internally, angle and momentum are stored divided by two pi. Each sample is taken just before the next kick.

Area preservation is exact

DT=\begin{pmatrix}1+K\cos\theta&1\\K\cos\theta&1\end{pmatrix},\qquad \det DT=1

The kick and the drift are both canonical shears. The analytic tangent matrix is checked against central differences, and the small floating-point mismatch adds no physical dissipation. Exact area preservation coexists with regular islands and with trajectories that are sensitive to where they start.

Keep the full momentum on the cylinder

\widetilde p_n=(p_n+\pi)\bmod 2\pi-\pi,\qquad \Delta p_n=p_n-p_0

Only the torus section wraps momentum, and only for display. The dynamics, kinetic energy, distance between nearby orbits and transport all use the unwrapped momentum. A regular accelerator island can travel through many momentum cells and still stay locally stable.

Two fixed points, two stability tests

\operatorname{tr}DT(0,0)=2+K,\qquad \operatorname{tr}DT(\pi,0)=2-K

A matrix with determinant one is elliptic when its trace lies strictly between minus two and two. The fixed point at a half turn is elliptic for positive kick strengths below four; at the endpoints it is marginal and needs a nonlinear analysis. The fixed point at zero angle is hyperbolic for every positive strength. The stability of one point does not tell you how the whole map behaves.

Finite separation can level off while tangent growth continues

\lambda_N=\frac1N\sum_{j=0}^{N-1}\log\|DT(z_j)\widehat{\delta z}_j\|,\qquad d_N=\|z_N-z_N^{\rm near}\|_{\rm cylinder}

The tangent vector starts along the angle direction and is rescaled after every kick. The nearby orbit is never rescaled. The growth plot shows the base-ten logarithm of two gains: the finite separation and the accumulated tangent stretch. Before you read meaning into the finite-time exponent, extend the record and try a different initial perturbation.

Work and transport measure different things

E_n=\frac{p_n^2}{2},\qquad \sigma_p(N)=\sqrt{\frac1M\sum_{j=1}^{M}(p_N^{(j)}-p_0^{(j)})^2}

Each instantaneous kick exchanges energy with the rotor. The RMS momentum drift of the atlas orbits uses its own final kick count, shown separately, and fixed deterministic seeds. It is measured from each orbit's own starting momentum, not from the ensemble mean, and it is not a measured diffusion coefficient. A finite sample cannot give a universal threshold for the destruction of KAM curves.

Common difficulties

Preserving area does not mean regular motion

Typical misconceptionAn area-preserving map cannot be chaotic.

Better mental modelA map can stretch a small patch strongly in one direction and squeeze it in another. The area stays the same, yet nearby points separate.

Wrapping momentum only changes the picture

Typical misconceptionIf an orbit stays inside the wrapped plot, its physical momentum must stay bounded.

Better mental modelThe wrapping is for display only. A regular accelerator island looks compact on the torus while its unwrapped momentum keeps climbing.

A short-time exponent proves nothing by itself

Typical misconceptionAny positive finite-time exponent proves the orbit is chaotic in the long run.

Better mental modelA passing shear or a single unstable point can also make nearby orbits separate for a while. Extend the record and compare the structure of the section.

Energy is supplied by the kicks

Typical misconceptionIf the energy changes, the integration must be wrong.

Better mental modelThe kicks do work on the rotor, so the exact map changes its energy. To check the numerics, test the map's derivative and its inverse instead.

Run the experiment

01
Find regular structures
Compare the Free integrable rotor, Weak-kick invariant curves and Stable resonance island presets.
What to observe: Invariant curves and islands give the section its structure.
02
Change record length
Compare Near-critical neighborhood with Strong-kick sensitivity, then press Double record length.
What to observe: What a finite record shows changes as it grows longer. Islands that survive can sit next to a sea of sensitive orbits.
03
Reveal hidden transport
Choose Regular accelerator island.
What to observe: On the torus the orbit stays in a small region, but its unwrapped momentum grows almost linearly.
04
Test an unstable fixed point
Choose Unstable fixed point, then drag the orange initial ring.
What to observe: An orbit sitting exactly on the fixed point never moves, yet its tangent exponent is positive and nearby trajectories move away.