Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions |
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Authors: | Eva Kaslik Seenith Sivasundaram |
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Affiliation: | a Institute e-Austria Timisoara, Bd. V. Parvan nr. 4, room 045B, 300223, Timisoara, Romaniab Department of Mathematics and Computer Science, West University of Timisoara, Bd. V. Parvan nr. 4, 300223, Romaniac Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA |
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Abstract: | Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann-Liouville and Grunwald-Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system. |
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Keywords: | Periodic solution Non-existence Fractional-order derivative Mellin transform Caputo Riemann-Liouville Grunwald-Letnikov Neural network |
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