Engineering & Science
Chirikov Resonance-Overlap Criterion: Hamiltonian Chaos & Standard Map Parameter
Evaluate Boris Chirikov’s resonance-overlap parameter s = (Δω_1 + Δω_2) / Ω_12 and Chirikov-Taylor standard-map stochasticity threshold K_c ≈ 0.9716.
When adjacent nonlinear pendulum separatrices touch (s >= 1), global Hamiltonian chaos erupts!
Boris Chirikov (1979) formulated the foundational analytical criterion for the onset of widespread deterministic chaos in tokamak magnetic confinement, particle accelerators, and asteroid Kirkwood gaps: compute the unperturbed half-widths Δω_1 and Δω_2 of two neighboring nonlinear resonances separated in frequency by Ω_12 = |ω_1 - ω_2|. Their Chirikov overlap ratio is s = (Δω_1 + Δω_2) / Ω_12!
When s < 1, invariant KAM (Kolmogorov-Arnold-Moser) tori survive between the islands and confine particle trajectories; once s >= 1 (or in the Chirikov-Taylor standard map where primary island half-width 2*SQRT(K) separated by 2*π gives s_map = (2 / π) * SQRT(K), refined by Greene’s golden-mean KAM breakup to K_c = 0.9716), last confining KAM surfaces dissolve into global stochastic diffusion!
| Resonance Configuration | Chirikov Overlap Ratio s | Standard Map K vs K_c = 0.9716 | Phase-Space Transport |
|---|---|---|---|
| Δω_1 = 1.4, Δω_2 = 1.2, Ω_12 = 2.0 (K = 1.25) | s = 1.300 (Overlap!) | s_map = 0.7118 (K > 0.9716) | Global Stochastic Sea / Fast Loss |
| Δω_1 = 0.6, Δω_2 = 0.5, Ω_12 = 2.0 (K = 0.50) | s = 0.550 (Separated) | K < 0.9716 | Bounded KAM Tori Confinement |
| Invalid Ω_12 <= 0 or Negative Width | Error Guard | Error Guard | "Invalid Resonance Parameters" |
Two-Resonance Chirikov Separatrix Overlap Ratio s = (Δω_1 + Δω_2) / Ω_12
Computes the dimensionless separatrix sum-to-spacing ratio from half-widths B2, B3 (rad/s) and resonance spacing B4 (rad/s).
=IF(OR(B2 < 0, B3 < 0, B4 <= 0, B5 < 0), "Invalid Resonance Parameters", ROUND((B2 + B3) / B4, 3))Chirikov-Taylor Standard Map Primary Overlap s_map = (2 / π) * SQRT(K)
Evaluates the primary standard-map overlap parameter from stochasticity kick strength K in B5 (global KAM breakup occurs at K_c = 0.9716).
=IF(OR(B2 < 0, B3 < 0, B4 <= 0, B5 < 0), "Invalid Resonance Parameters", ROUND((2 / PI()) * SQRT(B5), 4))Try the example
Use the original values or change them to test your understanding.