Fractional Stability

Matignon's theorem and regional Jacobians for fractional systems.

Fractional-order local stability is not determined only by the sign of the real part of each Jacobian eigenvalue. For a commensurate system of order $q$, the relevant test is the angular condition from Matignon’s theorem.

Matignon’s Theorem

An equilibrium of a commensurate fractional-order system with $0 < q \le 1$ is locally asymptotically stable if every Jacobian eigenvalue $\lambda_i$ satisfies

For the non-smooth Chua vector field, the regional Jacobian is evaluated only away from the switching surfaces $x=\pm1$.

Audited Chua Example

The library exposes the regional Jacobian used in the algebra validation. The Matignon margin can be calculated directly with numpy:

import numpy as np

from hidden_attractors.models import (
    chua_nonsmooth_parameters,
    equilibria_nonsmooth,
    jacobian_nonsmooth,
)

q = 0.9998
params = chua_nonsmooth_parameters()
eq = equilibria_nonsmooth(params)
threshold = q * np.pi / 2.0

for name in ("E0", "E+"):
    eig = np.linalg.eigvals(jacobian_nonsmooth(eq[name], params))
    margins = np.abs(np.angle(eig)) - threshold
    stable = bool(np.all(margins > 0.0))
    print(name, stable, margins)

For Danca’s non-smooth case at q=0.9998, this calculation gives:

The threshold is 1.570482167530 radians. Full cross-tool values and the transfer-sign normalization are recorded in Chua Non-Smooth.

Relation To Hiddenness

Local stability is necessary context, not a complete hiddenness test. An attractor is classified as hidden only when its basin does not intersect neighborhoods of any equilibrium under the stated numerical contract. Consequently, after algebra and local stability, the workflow still requires trajectory validation and equilibrium-neighborhood or basin sampling.