Spotlight #58 – Chaos Divergence, Rheology & Signal Analysis

I. Measuring Chaos: The Lyapunov Exponent

What exactly diverges?

A deterministic system evolves from one state to the next by fixed rules, so two trajectories starting at the same point remain identical forever. The question is: how quickly do trajectories that start close by diverge? For a double pendulum, starting at (θ₁, θ₂) = (2.0, 1.0) and at (2.0 + 10⁻⁶, 1.0) — a difference of less than a millionth of a radian — produces completely different motions within a few tens of seconds.

The Lyapunov exponent λ quantifies this: on average, the separation δ(t) ≈ δ(0)·e^λt. If λ > 0, the system is chaotic. The larger λ, the faster two initially nearby states diverge and the shorter the horizon over which prediction is meaningful. A prediction error doubles every t_½ = ln 2 / λ seconds.

Ensemble approach and regression

Rather than renormalizing a single perturbation (the standard Benettin algorithm), the simulation takes a simpler approach: run N = 30 members simultaneously, each offset by Δθ₁(k) = k · Δθ, and track max|θ₁(k,t) − θ₁(0,t)| over time. Since the maximum separation grows exponentially (until saturation), log₁₀(separation) grows linearly. A linear regression through the middle 75% of the log-separation history gives a slope that, after conversion, estimates λ in bits per second.

Regular vs chaotic initial conditions

Not all initial conditions of the double pendulum are chaotic. With small angles (say both < 20°) the system behaves almost like two coupled linear pendulums — the ensemble fan stays narrow for minutes and the log-separation plot is nearly flat (λ ≈ 0). Increasing the initial angle pushes the system across an invisible boundary in phase space between Kolmogorov-Arnold-Moser (KAM) tori (regular behaviour) and chaotic sea. The "Crossing angles" preset (θ₁ = 2.5, θ₂ = 2.5 rad) reliably produces chaos within a few integration steps.

II. Viscoelastic Materials: Between Elastic and Viscous

Why do polymers bounce and flow?

Silly Putty bounces off the floor (elastic) but sags slowly under its own weight (viscous). This is not a property failure — it is exactly what viscoelasticity predicts. The key is the ratio of the observation timescale t_obs to the material relaxation time τ_R. The Deborah number De = τ_R / t_obs:

• De ≫ 1 (fast deformation, slow material): elastic-dominated behaviour = bouncing.
• De ≪ 1 (slow deformation, long time): viscous-dominated behaviour = sagging.
• De ≈ 1: mixed response where both energy storage and dissipation are significant.

The name honours the biblical prophetess Deborah who said "the mountains flowed before the Lord" (Judges 5:5) — rhetorically, all matter flows on a long enough timescale.

Three model geometries and their signatures

Each model is a circuit of springs (elastic elements, modulus G₀) and dashpots (viscous elements, viscosity η):

• Maxwell — Series. Under constant stress, strain increases without bound (flows). Under constant strain, stress decays fully to zero. Good for liquids: colognes, polymer solutions. G→0 at low frequency.
• Kelvin-Voigt — Parallel. Under constant stress, strain increases asymptotically to σ/G₀ (bounded creep). Stress does not relax to zero. Good for gels, rubber, biological tissues. G′ = const at all frequencies.
• Standard Linear Solid (SLS) — Maxwell arm parallel to a spring. Both bounded creep and genuine stress relaxation. G at long times = αG₀ where α ∈ (0,1). Best fit for most real polymers, cartilage, muscle.

Reading the dynamic spectrum G′(ω) and G″(ω)

In an oscillatory shear experiment with strain ε = ε₀ sin(ωt), the in-phase component of the stress is G′ (storage modulus, energy stored elastically per cycle) and out-of-phase is G″ (loss modulus, energy dissipated). The crossover frequency ω_c = 1/τ_R is where G′ = G″; below it viscous dominates, above it elastic dominates.

For the Maxwell model this crossover is a perfect crossing of two simple curves. Look at the frequency-sweep tab in the simulation: drag G₀ or η and watch ω_c shift left and right. Since τ_R = η/G₀, doubling viscosity at constant elasticity shifts the crossover to lower frequency — the material has a longer memory.

III. Seeing Signals in Time-Frequency Space

From "which frequencies" to "which frequencies at which time"

The Discrete Fourier Transform answers a global question: what frequencies are present in the entire signal? For a stationary sine wave this is exactly what you need. But for a chirp (frequency sweeping over time), or a music note that decays, or a drum hit with a sharp transient followed by room reverb, the DFT gives a smeared average that loses the temporal structure.

The Short-Time Fourier Transform answers the local question: what frequencies are present in each short window of time? By sliding a window of N samples across the signal, computing the DFT at each position, and stacking the results, we get the spectrogram: a 2D image with time on the horizontal axis, frequency on the vertical axis, and colour encoding log-magnitude. A chirp appears as a diagonal streak; a pure tone as a horizontal line; a drum hit as a vertical burst that quickly fades.

The uncertainty principle in signal processing

There is a fundamental trade-off — mathematically equivalent to Heisenberg's uncertainty principle — between time resolution and frequency resolution in the STFT:

Δt · Δf ≥ 1 / (4π)    (Gabor limit)

A short window gives good time resolution (you can see exactly when an impulse occurs) but poor frequency resolution (the spectrum of a short chirp is blurry). A long window gives sharp frequency resolution but blurs time. The simulation lets you change the window length N: try N = 128 vs. N = 1024 on the chirp signal and observe this trade-off directly in the spectrogram.

Window functions and spectral leakage

If a sinusoid does not complete an exact integer number of cycles within the DFT window, the abrupt truncation at the window edges creates a discontinuity that smears energy across all frequency bins — this is spectral leakage. Window functions taper the signal smoothly to zero at the edges, reducing leakage at the cost of a wider main lobe.

To see leakage live: select a pure Sine signal and zoom in on the DFT spectrum panel. With Rectangular windowing you'll see multiple secondary peaks neighbouring the main spike. Switch to Blackman: the secondary peaks collapse dramatically, leaving only the main spike and its immediate neighbours. Switch back to Rectangular: the leakage returns. This is the most direct demonstration of why audio engineers default to Hanning or Hamming windowing.

The AM signal dissected

Amplitude modulation (AM radio) multiplies a carrier cos(2πf_ct) by a slowly varying message signal 1 + m·cos(2πf_mt). Expanding:

x(t) = cos(2πfct) + (m/2)cos(2π(fc+fm)t) + (m/2)cos(2π(fc-fm)t)

The DFT should therefore show three peaks: the carrier at f_c and two sidebands at f_c ± f_m. Try the AM preset in the simulation: set f₀ = 400 Hz (carrier frequency), and you'll see the carrier peak flanked by two smaller sidebands, with the separation proportional to f_m = f₀/10 = 40 Hz.

Connections Across the Three Themes

All three Wave 61 simulations are about measuring hidden structure in dynamic systems. The Lyapunov exponent measures how fast information about initial conditions decays in a chaotic system. The complex modulus measures how energy is partitioned between storage and loss in a viscoelastic material. The Fourier transform measures the energy distribution across frequencies in a signal.

There is even a direct mathematical link: the complex modulus G*(ω) = G′(ω) + iG″(ω) that you compute in the viscoelastic simulation is related to the Fourier transform of the stress relaxation modulus G(t): G*(ω) = iω∫₀^∞ G(t) e^−iωt dt. Rheology and signal processing are, at some level, the same mathematics.