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Abstract The Quantum Approximate Optimization Algorithm (QAOA) is a powerful tool in solving various combinatorial problems such as Maximum Satisfiability and Maximum Cut. Hard computational problems, however, require deep circuits that place high demands on classical variational parameter optimization. Ultimately, this has necessitated investigations into alternative methods for effective QAOA parameterizations. Here, we study a parameterization scheme based on classical chaotic recursive mapping, which enables significant reductions in the scaling of the variational parameter space. Through numerical investigations of hard Maximum Satisfiability problems, we demonstrate that the chaotic mapping can effectively match the performance of standard QAOA when subject to a limited number of classical optimization iterations and short-depth circuits. Insight into this behavior is elucidated through the lens of classical dynamical systems and used to inform hybridized schemes that leverage both standard and chaotic parameterizations. It is shown that these hybridized approaches can boost QAOA performance beyond that of the standard approach alone, especially for deep circuits. Through this study, we provide a new perspective that introduces a generalized framework for specifying performant, dynamical-map-based QAOA parameterizations.