Lyapunov Stability Analysis

HardPython

Problem

Write a Lyapunov stability analysis for a first order differential equation.

Implement a numerical certificate for the scalar system x' = f(x), where f is represented by polynomial coefficients in ascending order. Use V(x) = 0.5 * (x - x_eq)^2 and evaluate V_dot = (x - x_eq) * f(x) at evenly spaced points on both sides of the equilibrium.

Return "asymptotically_stable" if every sampled derivative is less than -tolerance, "stable" if all are at most tolerance but at least one is nonnegative within tolerance, and "unstable" if any exceeds the tolerance.

Function signature: def classify_equilibrium(coefficients, x_eq, radius, samples, tolerance):

Input values are numeric; output is one of the three specified strings.

Constraints

1 <= len(coefficients) <= 100
-10^6 <= coefficients[i] <= 10^6
radius > 0
1 <= samples <= 100000
0 <= tolerance < 1
The supplied x_eq is an equilibrium of the polynomial, subject to the requested tolerance

Function Signature

def classify_equilibrium(coefficients, x_eq, radius, samples, tolerance):

Write a Lyapunov stability analysis for a first order differential equation.

Function signature: def classify_equilibrium(coefficients, x_eq, radius, samples, tolerance):

Input values are numeric; output is one of the three specified strings.