Chirikov Standard Map Kick and Twist

Mathematical Definition

The map iterates in two stages: a kick changes the momentum p, and a twist changes the angle θ. Given a kick strength K, the next state is given by:

The angle θ is taken modulo 2π, meaning the map operates on a cylinder. The momentum p remains unwrapped. Area preservation is ensured because the Jacobian determinant is 1. The fixed points are at (0, 0) and (π, 0). The origin is hyperbolic for K > 0, while (π, 0) is elliptic for 0 < K < 4.

Worked Example

Let's calculate the first iteration with K = 1, starting at θ₀ = π/2 and p₀ = 0.

1. Calculate the kick: Δp = K sin(θ₀) = 1 × sin(π/2) = 1.
2. Update momentum: p₁ = p₀ + Δp = 0 + 1 = 1.
3. Update angle: θ₁ = (θ₀ + p₁) mod 2π = (π/2 + 1) mod 2π &approx; (1.5707963267948966 + 1) = 2.5707963267948966.
4. The next kick would be K sin(θ₁) &approx; 0.5403023058681398.
5. The Jacobian components at the updated state are J₁₁ = 1 + K cos(θ₁) &approx; 0.1585290151921035 and J₂₁ = K cos(θ₁) &approx; -0.8414709848078965.

Experiment

Interact with the Chirikov Standard Map Kick and Twist to see how modifying the kick strength K or the initial position shapes the resulting orbit. Notice how small K yields relatively orderly bounded curves for some inputs, whereas large K can induce widespread stochastic spreading.

1. Set Kick strength to 1 and Initial angle to pi/2.
2. Set Kicks to 5. Observe how the momentum rapidly updates along the cylinder representation.
3. Increase Kick strength to 3 to observe stronger stretching and chaotic diffusion around the origin.

Limitations

Since this numerical trace is finite, one cannot infer global chaos or calculate a robust Lyapunov exponent. The finite phase portrait does not perfectly capture long-term structural stability of the standard map without exhaustive trajectory lengths.

References

The standard map has been extensively studied to understand resonance overlap and macroscopic chaotic diffusion limits in simplified 2D phase geometries.

• Chirikov standard map (Scholarpedia). This article on Scholarpedia supports the equations and analysis of canonical chaotic dynamics of the Chirikov standard map.