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Synchronization of piece-wise continuous systems of fractional order

The aim of this study is to prove analytically that synchronization of a piece-wise continuous class of systems of fractional order can be achieved. Based on our knowledge, there are no numerical methods to integrate differential equations with discontinuous right hand side of fractional order which model these systems. Therefore, via Filippov's regularization [1] and Cellina's Theorem [2,3], we prove that the initial value problem can be converted into a continuous problem of fractional-order, to which numerical methods for fractional orders apply. In this way, the synchronization problem transforms into a standard problem for continuous systems of fractional order. Three examples of fractional-order piece-wise systems are considered: Sprott system, Chen and Shimizu-Morioka system.