M090 · Hamiltonian chaos / exact maps

Kicked Rotor & KAM Transition

Kick a free rotor and explore invariant curves, resonance islands, chaotic transport and regular accelerator islands. Drag an initial phase-space point; compare a torus atlas with unwrapped momentum, neighboring trajectories, tangent growth and a local area audit.

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 $\text{Pending}$

OBSERVE / COMPARE / VERIFY

Does preserving area prevent 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 in the torus section to edit the initial angle and scaled momentum, or use their keyboard-accessible sliders. The rod shows a schematic kick followed by free rotation; samples are taken before each kick. Grey atlas orbits use at most 1500 kicks, and their final RMS drift is a finite ensemble statistic, not a diffusion coefficient. Energy may change because the kicks do work. Growth uses the cylinder metric and an analytic tangent vector renormalized every kick; wrapped proximity on the torus can hide large momentum separation.

Classical ideal delta kicks, unit moment of inertia and unit kick interval. The exact standard map uses a positive sine impulse. Only the display wraps momentum; the dynamical state keeps its cylinder lift. Finite sections cannot prove survival or destruction of every invariant curve. The near-critical preset is a reference neighborhood, not a computed transition threshold. No damping, finite pulse width or quantum localization.

Start with the essentials

Focus question: Does preserving area prevent chaos?
One-sentence intuition: Conservation, local stability and a finite statistical picture answer 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}

This positive sine convention comes from a cosine kick potential with unit inertia and unit kick interval. The phase-space engine stores angle and momentum divided by two pi. Samples are 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

Kick and drift are canonical shears. The analytic tangent matrix is audited against central differences; finite floating-point discrepancy does not add physical dissipation. Exact area preservation coexists with regular islands and sensitive trajectories.

Retain the cylinder momentum

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

The torus display wraps momentum only for the section. Dynamics, kinetic energy, pair separation and transport retain unwrapped momentum. Regular accelerator islands can move through many momentum cells while remaining locally stable.

Two fixed points, two stability tests

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

A determinant-one matrix is elliptic when its trace lies strictly between minus two and two. The half-turn fixed point is elliptic for positive strengths below four; endpoints are marginal and need nonlinear analysis. The zero-angle fixed point is hyperbolic for every positive strength. Stability of one point never classifies the whole map.

Growth can saturate without tangent saturation

\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 begins along angle and is renormalized every kick. The nearby orbit is never renormalized. The growth plot compares base-ten logarithms of finite separation gain and accumulated tangent gain. Extend the record and vary the initial perturbation before interpreting the finite-time exponent.

Work and transport are different observations

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

Instantaneous kicks exchange energy with the rotor. The atlas RMS momentum drift uses its own stated final kick count and deterministic seeds; it is neither a variance centered on the ensemble mean nor a measured diffusion coefficient. Finite samples do not compute a universal KAM destruction threshold.

Common difficulties

Area does not imply regularity

Typical misconceptionAn area-preserving map cannot be chaotic.

Better mental modelSensitive stretching and contraction can preserve area together.

Momentum wrapping is a view

Typical misconceptionA wrapped orbit has bounded physical momentum.

Better mental modelRead the unwrapped transport panel and accelerator preset.

A short exponent is not a theorem

Typical misconceptionAny positive finite-time exponent proves an asymptotic chaotic orbit.

Better mental modelTransient shear or an unstable isolated point can grow; extend the record and compare structure.

Energy is supplied by the kicks

Typical misconceptionEnergy changes signal integration error.

Better mental modelThe exact map exchanges work with the drive. Audit its derivative and inverse instead.

Run the experiment

01
Find regular structures
Compare free, weak-kick and stable-island presets.
What to observe: Invariant curves and islands organize the section.
02
Change record length
Compare near-critical and strong-kick cases, then double the record.
What to observe: Finite evidence changes with time; surviving islands can coexist with a sensitive sea.
03
Reveal hidden transport
Select the regular accelerator island.
What to observe: The torus remains compact while lifted momentum grows nearly linearly.
04
Challenge a fixed point
Review the unstable fixed point and drag the initial ring.
What to observe: An exact stationary orbit can have a positive tangent exponent; nearby trajectories depart.