Dynamical Systems and Stability Theory for Fractional Differential Equations: A Theorem-Level Map of the Frontier

Fractional differential equations (FDEs) are now used across control, anomalous transport, viscoelasticity, biological dynamics, networks, and stochastic systems. Yet the literature is often organized by application or operator rather than by theorem strength, making it difficult to distinguish mature theory from genuinely open dynamical questions. This review develops a theorem-level map of stability and dynamical-systems results for fractional differential equations, emphasizing exact versus sufficient criteria, state-space architecture, global dynamics, robustness, bifurcation semantics, persistence and extinction, variable/distributed order, and stochastic memory. Double/Multiple Allee dynamics is used as a cross-cutting test case. The principal future direction identified is the geometry of physically reachable memory-state fibers under present-state observation.