Synchronization of piecewise continuous systems of fractional order

This paper proves analytically that synchronization of a class of piecewise continuous fractional-order systems can be achieved. Since there are no dedicated numerical methods to integrate differential equations with discontinuous right-hand sides for fractional-order models, Filippov’s regularization (Filippov, Differential Equations with Discontinuous Right-Hand Sides, 1988) is applied, and Cellina’s Theorem (Aubin and Cellina, Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska, Set-valued Analysis, 1990) is used. It is proved that the corresponding initial value problem can be converted to a continuous problem of fractional-order systems, to which numerical methods can be applied. In this way, the synchronization problem is transformed into a standard problem for continuous fractional-order systems. Three examples are presented: the Sprott’s system, Chen’s system, and Shimizu–Morioka’s system.

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