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In this paper, we introduce the notion of recurrence entropy in the contextof nonlinear control systems. A set is said to be ($\tau$-)recurrent if everytrajectory that starts in the set returns to it (within at most $\tau$ units oftime). Recurrence entropy quantifies the complexity of making a set$\tau$-recurrent measured by the average rate of growth, as time increases, ofthe number of control signals required to achieve this goal. Our analysisreveals that, compared to invariance, recurrence is quantitatively lesscomplex, meaning that the recurrence entropy of a set is no larger than, andoften strictly smaller than, the invariance entropy. Our results further offerinsights into the minimum data rate required for achieving recurrence. We alsopresent an algorithm for achieving recurrence asymptotically.

Recurrence of Nonlinear Control Systems: Entropy and Bit Rates