Robust Local Stability and Bifurcation Control of Cubic-Intuitionistic-Fuzzy Fractional Jerk Dynamics with Nonlocal Physics-Informed Learning

Fractional nonlinear systems combine the long memory dynamics and high parameter sensitivity. This brings difficulties for control design as well as accurate numerical approximation when unknown coefficients exist. In this paper, a computational method is proposed for a commensurate Caputo fractional jerk system with damping, stiffness, cubic-restoring, and fractional-order parameters modeled by cubic-intuitionistic-fuzzy (CIF) coefficient projections. Membership, nonmembership, and embedded point components are transmitted individually without defuzzification. We choose a full-state feedback law via an exhaustive restricted gain search at each vertex of the CIF outer support. The controller must have a Matignon angular margin of at least 25° and a worst eigenvalue real part no more than -0.25. For the chosen gain K = (1.1,0.6,0.8), the worst outer-support angular margin is 25.8757° with a maximum real part of -0.25210. The Matignon margin is expanded to 47.5089°, the pitchfork threshold is moved from μ = 0 to μ = k 1 = 1.1 outside the whole uncertain range 0.25 ≤ μ ≤ 0.65, and the origin is shifted from unstable to stable at the nominal position via feedback. Caputo trajectories are obtained by means of an Adams–Bashforth–Moulton predictor–correct–evaluate approach. Observed orders from a step-refinement research range from 1.94 to 2.24. The maximal terminal state norm is 0.04087 across 60 outer-support samples. The integrated state energy is reduced by 17.44 times with the nominal controller, and the control energy is 0.33934. A label-free fractional physics-informed neural network is trained using a dense L1 Caputo memory operator and precise initial-condition embedding too. It reproduces the dominating displacement with R 2 =0.9425, but lesser accuracy for the third small-amplitude condition reveals a present constraint of global physics-only training. The paradigm therefore unifies interpretable fractional stability, uncertainty propagation, resilient control, convergent temporal integration, and transparent physics-informed approximation for nonlinear systems with memory. In this research, robustness refers to the vertextested local fractional stability at the origin, rather than a theorem about global nonlinear stabilization or continuous-box sector stability.