We introduce a Caputo–generalized conformable fractional (Caputo–GCF) derivative that separates the fractional order $ \sigma $ from an independently tunable structural memory parameter $ \nu+\zeta $, and recovers the classical Caputo, Riemann–Liouville and (formally, in a limiting sense) Hadamard derivatives as special cases. Linearity, boundedness and continuity of the operator are established. Applying it to a delayed fractional-order neural network, we prove existence and uniqueness of the equilibrium point and derive two explicit finite-time stability criteria — a Hölder-based criterion for $ 0 < \sigma\le\tfrac12 $ and a Cauchy–Schwarz-based criterion for $ \tfrac12 < \sigma < 1 $ — that explicitly quantify the roles of $ \sigma $, $ \nu+\zeta $, the delay, and the coupling matrices. Numerical experiments confirm the theoretical bounds and show faster empirical settling and smaller final error than the classical Caputo and GCF operators on a representative example. Applications to anomaly detection (Controlled Anomalies Time-Series dataset), delayed UAV attitude stabilization, and secure neural dynamics under bounded cyber disturbances further illustrate the framework's applicability.
Citation: Abeer Aljohani. A new Caputo–generalized conformable fractional derivative with applications to finite-time stability and secure neural systems[J]. AIMS Mathematics, 2026, 11(10): 35578-35630. doi: 10.3934/math.20261380
We introduce a Caputo–generalized conformable fractional (Caputo–GCF) derivative that separates the fractional order $ \sigma $ from an independently tunable structural memory parameter $ \nu+\zeta $, and recovers the classical Caputo, Riemann–Liouville and (formally, in a limiting sense) Hadamard derivatives as special cases. Linearity, boundedness and continuity of the operator are established. Applying it to a delayed fractional-order neural network, we prove existence and uniqueness of the equilibrium point and derive two explicit finite-time stability criteria — a Hölder-based criterion for $ 0 < \sigma\le\tfrac12 $ and a Cauchy–Schwarz-based criterion for $ \tfrac12 < \sigma < 1 $ — that explicitly quantify the roles of $ \sigma $, $ \nu+\zeta $, the delay, and the coupling matrices. Numerical experiments confirm the theoretical bounds and show faster empirical settling and smaller final error than the classical Caputo and GCF operators on a representative example. Applications to anomaly detection (Controlled Anomalies Time-Series dataset), delayed UAV attitude stabilization, and secure neural dynamics under bounded cyber disturbances further illustrate the framework's applicability.
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Figures(9) / Tables(12)
Abeer Aljohani. A new Caputo–generalized conformable fractional derivative with applications to finite-time stability and secure neural systems[J]. AIMS Mathematics, 2026, 11(10): 35578-35630. doi: 10.3934/math.20261380
State trajectories of the three-dimensional Caputo–GCF delayed neural network in Example 2 for $\sigma = 0.9$, $\nu+\zeta = 1.5$, and $\tau = 0.35$
Norm evolution $\|x(t)-x^*\|_2$ for the Caputo–GCF delayed neural network in Example 2 with $\sigma = 0.9$, $\nu+\zeta = 1.5$, and $\tau = 0.35$
Effect of the structural memory parameter $\nu+\zeta$ on the reported stability bound for Example 2
Effect of the delay parameter $\tau$ on the estimated settling time for Example 2 with $\sigma = 0.9$ and $\nu+\zeta = 1.5$
Comparison between the classical Caputo derivative and the proposed Caputo–GCF derivative for Example 2 with $\sigma = 0.9$ and $\tau = 0.35$
Visualization of the Controlled Anomalies Time-Series dataset in the selected test region. The plot shows the primary observable $\texttt{cso1}$, its rolling mean, and the detected anomaly timestamps. The large deviation around the anomaly interval indicates a sudden disturbance in the observed signal, while the rolling mean shows the local trend of the time series
Fractional-memory surface computed from the CATS anomaly activity. The anomaly-intensity sequence $ \rho(t) $ is obtained from the anomaly activity of the dataset and convolved with the power-law memory kernel $ (t-k+1)^{-\sigma} $ for $ \sigma\in[0.55, 0.95] $. Smaller values of $ \sigma $ preserve anomaly effects for a longer time, while larger values of $ \sigma $ lead to faster memory decay
State trajectories under a bounded cyber disturbance (shaded interval)
Evolution of the security energy $ \|x(t)\|_2 $ showing finite-time recovery
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