Eigenvalue Based Local Stability Analysis of Nonlinear Caputo Type Fractional Systems and the Effects of Fractional Order and System Parameters"

Abstract: The location of a Jacobian spectrum relative to a fractional stability sector determines whether linearisation can certify local asymptotic stability of a commensurate nonlinear Caputo system. A binary classification, however, gives little indication of how close the model lies to losing that certificate. This paper develops an explicit angular margin and a complementary matrix-perturbation margin for systems of order between zero and one. The known strict-sector stability theorem is used as the analytical foundation. A two-dimensional nonlinear family with Jacobian eigenvalues μ ± iω gives a closed-form critical fractional order and exact parameter sensitivities. A second family with a triangular, nonnormal Jacobian shows why unchanged eigenvalues do not guarantee small finite-time amplification. A disk-distance calculation then translates a diagonalizable matrix perturbation bound into a sufficient certificate for persistence of the strict sector condition. All numerical values are calculated from stated formulas rather than fitted to trajectories. The contribution is a transparent reporting framework that distinguishes order sensitivity, parameter sensitivity and eigenvector conditioning. The paper does not infer nonlinear bifurcations or global attraction solely from an eigenvalue crossing. Keywords: nonlinear fractional system; Jacobian spectrum; local stability; fractional order; angular margin; nonnormal matrix; perturbation certificate. Keywords: nonlinear fractional system; Jacobian spectrum; local stability; fractional order; angular margin; nonnormal matrix; perturbation certificate.