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In this paper, we show that a fast switch is able to lead a complex dynamical system to being asymptotically stable, although this system is completely unstable in every switch duration and even the associated connection matrices are randomly selected. Importantly, we define some new exponents by which we can figure out the essential patterns that guarantee the stability of fast switching systems, and besides, their calculations need little computational cost. More interestingly, we show the efficiency of some random switches in inducing stability through a comparison of the systems with different switch connection matrices and switch durations, and we give a design method for obtaining higher efficient random switch rules. We also investigate the generalization of the obtained results to a more realistic case where the switch obeys some renewal process. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

The efficiency of a random and fast switch in complex dynamical systems