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The function adds squared deviation from a target Ω*, a weighted squared coherence shortfall, and a linear free-energy term. Nonnegativity requires suitable coefficients and F≥0. Calling it a Lyapunov function additionally requires showing that it decreases along the actual coupled dynamics and is positive relative to the intended equilibrium.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| V | Candidate stability objective |
| Omega_star | Target state |
| lambda_ | Coherence penalty |
| mu | Free-energy weight |
How it works
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Assumptions & limits
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Original context
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}
}
/**
* @brief Lyapunov stability analysis
*/
namespace stability {
// V(Ω) = (Ω - Ω*)² + λ·(1-C)² + μ·F
inline double lyapunov_function(double omega, double coherence, double free_energy) {
double term1 = (omega - TARGET_OMEGA) * (omega - TARGET_OMEGA);
double term2 = LYAPUNOV_LAMBDA * (1.0 - coherence) * (1.0 - coherence);
double term3 = LYAPUNOV_MU * free_energy;
return term1 + term2 + term3;
}
// dV/dt = (V(t) - V(t-1))/dt
inline double lyapunov_derivative(double v_curr, double v_prev, double dt = 0.1) {
return (v_curr - v_prev) / dt;Exact original notation
V(Ω) = (Ω - Ω*)² + λ·(1-C)² + μ·F