Perturbed Hybrid Pantograph Systems with Deformable Derivatives: Well-Posedness, Stability, Numerical Sensitivity, and a Delay-Feedback Toy Example

We study a perturbed coupled system of generalized hybrid pantograph equations involving the deformable derivative of Zulfeqarr–Ujlayan–Ahuja. A central point of the revision is made explicit: for classically differentiable functions this derivative is local and satisfies Dτu=(1−τ)u+τu′. Therefore, in the present differentiable setting the memory or aftereffect is produced by the proportional pantograph delays, while the deformable order τ supplies an order-dependent local relaxation/drift term. After rewriting the system as an equivalent integral equation on X=C(I,R2), we establish invariant-ball conditions, existence and uniqueness within invariant balls, generalized Ulam–Hyers stability, and Lipschitz continuous dependence on the perturbation amplitude ε. The assumptions and constants are stated so that the restrictive roles of the Lipschitz bounds, the interval length, and |ε| are transparent. We then provide numerical parameter sensitivity diagrams for illustrative pantograph systems and include step-size refinement checks and performance indices. The numerical and plasma-inspired sections are deliberately framed as exploratory delay-feedback examples rather than as first-principles plasma models or rigorous bifurcation theory.