Chaos Preservation Demo
Interactive demonstration showing 99.4% suppression of chaotic divergence using Normalized Nonlinear Composition (NNC) operators on the Lorenz attractor.
Honest Outcome: Neutral Performance (2025-12-13)
Previous claim: 772x improvement in chaos preservation was based on an incorrect Lyapunov exponent algorithm (step-size method).
Corrected result: Using the proper Benettin algorithm with Gram-Schmidt orthonormalization, meta-calculus achieves neutral performance - matching RK4 baseline exactly (100 time units) with optimal k=0.3 parameter.
This demonstrates honest bounds on the framework's capabilities. Chaotic systems resist scheme modifications due to inherent exponential error growth.
Understanding Lyapunov Exponents and Chaos
Chaotic systems are hypersensitive to initial conditions - tiny changes grow exponentially over time. The Lyapunov exponent measures this growth rate. For the Lorenz attractor, the largest Lyapunov exponent is approximately 0.906, meaning nearby trajectories diverge at rate e^(0.906t).
In numerical simulations, preserving chaotic dynamics is challenging because numerical errors also grow exponentially. After enough time, your simulation diverges from the true trajectory, even though it stays on an attractor - just not the one you started on.
Positive Lyapunov Exponent
Lambda > 0: Trajectories diverge exponentially. This is chaos. Small errors become large.
Negative Lyapunov Exponent
Lambda < 0: Trajectories converge. Stable fixed points. Errors decay over time.
Corrected Key Result: Neutral Performance
RK4 Baseline
100.0
time units before divergence
Meta-0.3 (Best)
100.0
matches RK4 exactly
Improvement
0%
neutral outcome
Using the corrected Benettin algorithm for Lyapunov exponent computation, we find that NNC operators with k=0.3 achieve the same attractor preservation time as classical RK4 integration (100 time units).
Honest conclusion: Meta-calculus neither helps nor hurts chaotic dynamics. The exponential sensitivity inherent in chaos dominates any scheme-level modifications. This represents a realistic boundary of the framework's applicability.
Interactive Chaos Preservation Analysis
Explore how different integration schemes preserve the Lorenz attractor geometry. The visualizer shows divergence time, energy drift, and Lyapunov error across classical (RK4) and meta-calculus schemes with varying blend parameters (k).
Technical Details
Lorenz System
The Lorenz attractor is defined by three coupled ODEs:
dy/dt = x * (rho - z) - y
dz/dt = x * y - beta * z
Standard parameters: sigma=10, rho=28, beta=8/3. Known largest Lyapunov exponent: 0.906.
Benettin Algorithm (Corrected)
Proper Lyapunov exponent computation requires tracking tangent vector evolution with periodic Gram-Schmidt orthonormalization:
- Evolve trajectory x(t) and tangent vector delta(t) simultaneously
- Every N steps: compute growth factor |delta(t)|
- Renormalize: delta(t) = delta(t) / |delta(t)|
- Accumulate: log(growth factors)
- Lyapunov = (1/T) * sum(log growth)
Previous step-size method was incorrect and produced inflated improvement claims.
NNC Operators
Normalized Nonlinear Composition blends classical derivatives with bigeometric derivatives using parameter k:
For chaos: k=0.3 matches RK4. Higher k values (0.5, 0.1) cause early divergence. Lower k approaches classical behavior.
Divergence Criterion
Trajectories diverge when spatial distance exceeds threshold:
Divergence time measures how long the scheme preserves attractor geometry relative to reference trajectory. Both RK4 and meta-0.3 achieve 100 time units.
Experimental Methodology
Paper P07 tested chaos preservation across 6 integration schemes:
- RK4: Classical Runge-Kutta baseline (100.0 time units)
- Meta-0.3: NNC with k=0.3 (100.0 time units, matches baseline)
- Meta-0.5: NNC with k=0.5 (2.6 time units, early divergence)
- Meta-0.1: NNC with k=0.1 (0.8 time units)
- Bigeometric: Pure D_BG (0.2 time units)
- Adaptive: Dynamic k selection (0.1 time units)
Data source: public/chaos_corrected.json
Simulation script: simulations/chaos_preservation_moo.py
Implications for Meta-Calculus Framework
What Chaos Teaches Us
Chaotic systems represent a boundary case for scheme-invariance. The exponential sensitivity to perturbations dominates any potential benefits from alternative derivative operators.
This is an honest limit: meta-calculus is neutral for chaos, not harmful but not helpful.
Where Meta-Calculus Excels
The framework shows strong performance in:
- Singular problems (CASCADE k=-1.0)
- Oscillatory systems (symplectic preservation)
- Energy-conserving dynamics
- Gravitational wave simulations
Related Experiments and Results
Explore related demonstrations and validation results: