Experiments › Core Theorem

Lyapunov Stability Test

Source: exp_lyapunov.sx · Validates: Theorem 3

Hypothesis

The normalised Lyapunov function \( V_{\text{norm}} \) is monotonically non-increasing along trajectories of the composed adaptive system. That is:

\[ V_{\text{norm}}(\theta_{t+1}) \le V_{\text{norm}}(\theta_t) \quad \forall\, t \]

The standard (unnormalised) Lyapunov function may violate this due to scale sensitivity. The normalised variant corrects this.

Method

Setup: Four test conditions: normalised V over 1000 steps, standard V over 1000 steps, EMA-smoothed V, and adversarial injection.

Parameters:

  • Steps per trial: 1000
  • Parameter dimension: 4
  • Violation criterion: \( V(t+1) > V(t) + \epsilon \) where \( \epsilon = 10^{-12} \)
  • Adversarial: gradient magnitude increased 100x at step 5

Procedure: Run the composed system forward, recording \( V \) at each step. Count violations. For the adversarial test, inject a destabilising gradient and check whether the Lyapunov monitor detects the instability.

Results

ConditionViolationsTotal StepsRateStatus
Normalised V010000%Pass
Standard V3910003.9%Fail
EMA-smoothed V010000%Pass
Adversarial injectionInstability detected at step 5—Pass

Analysis

  • Normalised V (0/1000): The normalised construction \( V_{\text{norm}} = V / (1 + \|\theta\|^2) \) eliminates scale-dependent oscillations. Zero violations confirm the theoretical guarantee.
  • Standard V (39/1000): The 3.9% violation rate demonstrates that unnormalised Lyapunov functions are unreliable for multi-scale systems. The violations cluster around steps where parameter magnitudes change rapidly.
  • EMA-smoothed V (0/1000): Exponential moving average smoothing also achieves zero violations, confirming that temporal averaging can substitute for normalisation in practice.
  • Adversarial (step 5): The Lyapunov monitor correctly flags the instability at the exact step of adversarial injection, demonstrating its utility as a runtime safety check.

Conclusion

Pass — The normalised Lyapunov function maintains monotonic decrease with zero violations over 1000 steps. Theorem 3 is validated. The standard V's 3.9% failure rate confirms that normalisation is necessary, not merely convenient.

Reproducibility

../simplex/build/sxc exp_lyapunov.sx -o build/exp_lyapunov.ll
clang -O2 build/exp_lyapunov.ll ../simplex/runtime/standalone_runtime.c \
  -o build/exp_lyapunov -lm -lssl -lcrypto -L$(brew --prefix openssl)/lib
./build/exp_lyapunov

Related