Bifurcation Sweep Results

Bifurcation Sweep Results#

Experiment 1: Single Element and Coupled Pair#

First experiment: driven Stribeck oscillator with amplitude sweep across the stick-slip bifurcation boundary. Three configurations tested.

1. Single oscillator (drive at ω_d = 2ω₀)#

Clear bifurcation in RMS amplitude at A ≈ 0.5–0.6. Below threshold, the oscillator responds linearly at the drive frequency. Above, it enters stick-slip oscillation.

Subharmonic presence: The ω₀ = ω_d/2 channel grows with amplitude but remains 3–4 orders of magnitude below the fundamental. The Stribeck nonlinearity generates subharmonics, but does not efficiently convert energy into them in a single-oscillator configuration.

2. Coupled pair at ω_d = ω₀ (baseline)#

Regime

A range

η (transfer efficiency)

Linear (below bifurcation)

≤ 0.30

~1.00

Post-bifurcation

≥ 0.50

~0.01–0.02

Near-perfect energy transfer at low amplitudes. Above bifurcation, TX enters large-amplitude stick-slip and decouples from the medium. The bifurcation destroys transfer efficiency at the fundamental.

3. Coupled pair at ω_d = 2ω₀ and 3ω₀#

The single-element medium passes through the drive frequency linearly — it does not convert energy into the subharmonic channel. This motivated the lattice experiment.

Experiment 2: Stribeck Lattice (confirmed)#

Chain of N oscillators coupled by Stribeck friction. Drive element 0 at ω_d = 2ω₀. Measure spectrum at element N-1.

The lattice converts frequency. The subharmonic dominates.#

Length sweep (A = 1.0, ω_d = 2ω₀):

N

η

P(ω₀)/P(ω_d)

Dominant channel

2

0.998

0.06

ω_d (fundamental)

3

0.128

1.03

ω₀ > ω_d (crossover)

4

0.088

1.43

ω₀ > ω_d

8

0.040

2.13

ω₀ > ω_d

16

0.019

2.71

ω₀ > ω_d

N = 3 is the critical chain length. At N = 2 the medium passes through ω_d. At N = 3 the subharmonic ω₀ equals the fundamental at the receiver. Every additional element increases the ω₀/ω_d ratio — the lattice progressively filters for the subharmonic.

Amplitude sweep (N = 8, ω_d = 2ω₀):#

A

η

ω₀/ω_d at RX

0.05

0.862

0.06

0.50

0.996

0.06

0.80

0.051

0.97

1.00

0.040

2.13

5.00

0.028

58.89

Bifurcation threshold at A ≈ 0.8. Below it: linear passthrough at ω_d with high η. Above it: the lattice acts as a frequency converter. The subharmonic channel grows stronger with increasing amplitude while the fundamental saturates.

Spatial spectrum (N = 8, A = 1.0):#

Element 0 (TX):  P(ω_d) = 3.61e+02   P(ω₀) = 4.00e-01   ratio = 0.001
Element 1:       P(ω_d) = 1.93e-01   P(ω₀) = 3.92e-01   ratio = 2.03
Element 7 (RX):  P(ω_d) = 1.85e-01   P(ω₀) = 3.95e-01   ratio = 2.13

The conversion happens at the first contact. Element 0 is ω_d dominant (ratio 0.001). Element 1 is already ω₀ dominant (ratio 2.03). The remaining elements maintain this ratio with slight progressive filtering.

The ω_d component drops by 3 orders of magnitude at the first contact (362 → 0.19). The ω₀ component holds steady (0.40 → 0.39). This is the differential attenuation: the high-frequency mode dissipates in the slip regime while the subharmonic propagates in the stick regime.

Spatial spectrum (N = 16, A = 2.0):#

Same pattern at higher amplitude. First contact converts, ratio stabilizes at ~11–13 along the chain. The ω_d component attenuates slowly from 0.036 to 0.030 while ω₀ holds at ~0.39.

Key Results#

  1. The Stribeck lattice is a frequency converter. Energy injected at ω_d exits at ω_d/2, with the subharmonic dominating by 2–60× depending on drive amplitude.

  2. N = 3 is the critical chain length. Below 3 elements, the medium passes through the drive frequency. At 3, the subharmonic crosses over. This is the minimum spatial extent for the bifurcation cascade.

  3. Conversion happens at one contact, propagation is the rest. The first Stribeck contact does the frequency conversion. Subsequent contacts filter and propagate the subharmonic. This is consistent with a single bifurcation event followed by coherent stick-regime transport.

  4. Two regimes, one lattice:

    • Below bifurcation: linear passthrough, high η, ω_d dominant

    • Above bifurcation: frequency conversion, lower η, ω₀ dominant

    The “dual regime” is real. The lattice operates in both simultaneously at different amplitude scales — or transitions sharply between them.

  5. The stick regime is the efficient transport channel. P(ω₀) ≈ 0.39 propagates across 16 elements with negligible attenuation. P(ω_d) attenuates continuously. The subharmonic sits in the stick regime (low relative velocity → strong coupling → coherent transfer).

Connection to Tesla#

Tesla drove the Earth-ionosphere cavity at high frequency and high power. This simulation suggests the opposite approach:

  • Drive at 2× or 3× the target frequency

  • Let the medium’s Stribeck nonlinearity convert to the subharmonic

  • The subharmonic propagates coherently through the stick regime

  • Receivers tuned to ω₀ extract energy from the subharmonic channel

The medium’s friction is not the obstacle — it is the frequency converter. You need at least 3 coupling stages (N ≥ 3 in the lattice) for the conversion to activate.

The copper wire was chosen because it eliminates the medium’s nonlinearity — providing linear, frequency-preserving transport. But that linearization also eliminates the frequency conversion mechanism that would make the medium itself useful for long-range coherent transfer.