On the nonexistence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical systems

Abstract

Using Fourier analysis as tool, we prove that the Caputo fractional-order derivative of the non-constant periodic function cannot be periodic, and thus we confirm the nonexistence of the non-constant exact periodic solutions in a class of the Caputo fractional-order initial-valued dynamical systems (including the autonomous case) from a different viewpoint. We also demonstrate with theoretical discussion and numerical examples that the non-constant long-time periodic solution might exist in the Caputo fractional-order systems. The investigation clarifies the difference and similarity of the periodic solutions between the Caputo fractional-order dynamical systems and the integer-order dynamical systems in a systematic way.