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In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb {Z}^q$ -systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb {Z}^q$ -system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb {Z}^q$ -system has directional discrete spectrum along q linearly independent directions if and only if it has discrete spectrum.

In this paper, the parameter identification of gene regulatory network with time-varying delay is studied. Firstly, we introduce the differential equation model of gene regulatory network with unknown parameters and time delay. Secondly, for the unknown parameters in the time-varying model, a corresponding system with adaptive parameters and adaptive controller is introduced, and the parameter identification problem of the original model is transformed into the synchronization problem of the two systems. Thirdly, we design an effective adaptive controller and an adaptive law for parameters and construct a Lyapunov functional. Then we give a strict theoretical proof that the adaptive parameters can converge to unknown parameters by Barbalat's lemma. Finally, a numerical example is given to verify the validity of the theoretical results.

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