International Journal For Multidisciplinary Research

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Eigenvalue-based Stability of Nonlinear Caputo-type Fractional Differential Systems: a Spectral and Operator-theoretic Study

Author(s) Mr. Butta shivaprasad, Dr. Rachna Khandelwal
Country India
Abstract Abstract
This paper develops a mathematically qualified stability framework for commensurate Caputo systems in which spectral geometry, nonlinear remainder estimates and fractional resolvent bounds are treated as distinct but connected ingredients. For 0 < α < 1, the autonomous problem ᴄD₀₊ᵅx(t) = Ax(t) + f(x(t)), x(0) = x₀, is examined first in ℂⁿ and then in an abstract Banach-space setting. In finite dimensions, the sharp linear criterion σA⊂ λ ∈ ℂ 0: argλ>απ2 is obtained from the asymptotics of the Mittag–Leffler function. A nonlinear linearisation result is then stated under the precise requirement f0=0 andfxx→ 0 as x→0, together with local well-posedness. The paper distinguishes asymptotic stability from stronger Mittag–Leffler estimates and explains why a spectral inclusion alone is insufficient for an unbounded operator unless compatible sectorial resolvent bounds are available. A two-dimensional cubic system is analysed in detail, and a parameter-dependent example shows how fractional order can change the stability classification even though the matrix is fixed. The resulting framework supplies a reproducible sequence—equilibrium translation, Fréchet linearisation, spectral-sector test, remainder control and numerical verification—that can support doctoral research in fractional dynamics. The contribution is theoretical and interpretive: no unreported experimental data are claimed.
Keywords: Caputo derivative; fractional-order system; Mittag–Leffler function; linearisation; spectral sector; sectorial operator; asymptotic stability.
Keywords Keywords: Caputo derivative; fractional-order system; Mittag–Leffler function; linearisation; spectral sector; sectorial operator; asymptotic stability
Field Engineering
Published In Volume 8, Issue 4, July-August 2026
Published On 2026-08-18
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CrossRef DOI prefix of IJFMR is 10.36948/ijfmr

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