Functional analysis framework for two-dimensional fractional dynamical systems with applications to Lorenz and Euler systems

This paper develops a functional analysis framework for a class of two-dimensional Erdélyi–Kober (EK) fractional dynamical systems defined in locally compact Hausdorff spaces under the compact-open topology. The considered model incorporates delay and nonlinear interactions through EK fractional operators, which allow the description of dynamical systems with memory effects. By constructing appropriate operators in a Banach space setting, we analyze their continuity, boundedness, and the Lipschitz properties. Using classical results from functional analysis and operator theory, sufficient conditions are derived to establish the solvability and Hyers–Ulam stability of the proposed fractional system. The theoretical results demonstrate that the considered model remains stable under small perturbations of the system parameters. To illustrate the applicability of the developed framework, the results are applied to two significant fractional models: the EK fractional Lorenz system describing chaotic dynamics, and the two-dimensional fractional Euler system arising in fluid mechanics. These applications confirm the effectiveness of the proposed functional analytic approach for studying nonlinear fractional dynamical systems with memory and delay effects.