The dependence of fractional dynamical systems on the differentiation order is commonly studied through sensitivity, monotonicity, or inverse order determination. In this paper, we investigated a complementary phenomenon: critical fractional orders at which the first-order sensitivity of a Mittag–Leffler response with respect to its order vanishes. We first considered the map
$ \alpha\longmapsto E_{\alpha}(\lambda), \qquad 0<\alpha<1, $
for sufficiently small nonzero $ \lambda $. A uniform $ C^1 $ perturbation argument shows that this map possesses a unique critical order $ \alpha_*(\lambda) $ in $ (0, 1) $, emerging from the distinguished value $ \alpha_0 $ determined by $ \psi(\alpha_0 \ + \ 1) = 0 $. The critical point is proved to be nondegenerate; it is a strict minimum for $ \lambda < 0 $ and a strict maximum for $ \lambda > 0 $. Consequently, the usual linear order sensitivity is lost at the critical point and is replaced by the quadratic local law
$ \left| E_{\alpha_*(\lambda)+h}(\lambda) - E_{\alpha_*(\lambda)}(\lambda) \right| \asymp |h|^2. $
We also derived an explicit first-order asymptotic expansion of the critical branch. The analysis was then extended to the dynamical map $ E_{\alpha}(\lambda t^\alpha) $ arising from a scalar Caputo evolution equation. An explicit unperturbed critical-order curve was obtained and shown to persist, on compact time intervals, as a unique nondegenerate critical surface for small $ |\lambda| $. High-precision numerical computations confirmed the critical-branch displacement, the minimum–maximum classification, and the predicted quadratic scaling. The results characterize a local loss of first-order order sensitivity that is distinct from the monotonicity regimes usually employed in fractional-order identification.
Citation: Areen Al-khateeb. Critical fractional orders and quadratic loss of sensitivity in Mittag–Leffler dynamics[J]. AIMS Mathematics, 2026, 11(11): 36244-36274. doi: 10.3934/math.20261404
The dependence of fractional dynamical systems on the differentiation order is commonly studied through sensitivity, monotonicity, or inverse order determination. In this paper, we investigated a complementary phenomenon: critical fractional orders at which the first-order sensitivity of a Mittag–Leffler response with respect to its order vanishes. We first considered the map
$ \alpha\longmapsto E_{\alpha}(\lambda), \qquad 0<\alpha<1, $
for sufficiently small nonzero $ \lambda $. A uniform $ C^1 $ perturbation argument shows that this map possesses a unique critical order $ \alpha_*(\lambda) $ in $ (0, 1) $, emerging from the distinguished value $ \alpha_0 $ determined by $ \psi(\alpha_0 \ + \ 1) = 0 $. The critical point is proved to be nondegenerate; it is a strict minimum for $ \lambda < 0 $ and a strict maximum for $ \lambda > 0 $. Consequently, the usual linear order sensitivity is lost at the critical point and is replaced by the quadratic local law
$ \left| E_{\alpha_*(\lambda)+h}(\lambda) - E_{\alpha_*(\lambda)}(\lambda) \right| \asymp |h|^2. $
We also derived an explicit first-order asymptotic expansion of the critical branch. The analysis was then extended to the dynamical map $ E_{\alpha}(\lambda t^\alpha) $ arising from a scalar Caputo evolution equation. An explicit unperturbed critical-order curve was obtained and shown to persist, on compact time intervals, as a unique nondegenerate critical surface for small $ |\lambda| $. High-precision numerical computations confirmed the critical-branch displacement, the minimum–maximum classification, and the predicted quadratic scaling. The results characterize a local loss of first-order order sensitivity that is distinct from the monotonicity regimes usually employed in fractional-order identification.
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Figures(4) / Tables(2)
Areen Al-khateeb. Critical fractional orders and quadratic loss of sensitivity in Mittag–Leffler dynamics[J]. AIMS Mathematics, 2026, 11(11): 36244-36274. doi: 10.3934/math.20261404
The unperturbed critical-order curve $ \alpha_0(t) = \psi^{-1}(\log t)-1 $
Local profiles of $ F_{\lambda, 1}(\alpha) $ near their critical orders for $ \lambda = -0.05 $ and $ \lambda = 0.05 $
Log–log verification of the quadratic critical-order scaling for $ \lambda = \pm0.05 $ at $ t = 1 $
Order sensitivity of the full Mittag–Leffler response and its approximations. (a) Small-argument regime $ \lambda = -0.05 $, $ t = 1 $: the zero of the complete sensitivity is compared with the stationary order of the first nonconstant truncation. (b) Large-argument relaxation regime $ \lambda = -1 $, $ t = 100 $: the full order sensitivity is compared with derivatives of the one- and two-term asymptotic expressions for interior orders. The curves illustrate approximation differences and do not imply a global absence theorem for the full response
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