The paper analyzes dynamical growth models predicting cyclic development of investigated economic factors. The research deals with an optimal control problem based on the economic growth model with the production function of the Cobb-Douglas type. Following the Pontryagin maximum principle, we derive the Hamiltonian system and conduct its qualitative analysis, which reveals conditions for the cyclic behavior of the optimal solutions around the isolate steady state. Numerical experiments illustrate the obtained results by demonstrating phase portraits corresponding to steady states of the focal type. © 2023 American Institute of Physics Inc.. All rights reserved.