# Inner synchronization of controlled multi-valued logical networks

## TL;DR

This paper explores how to synchronize the states of multi-valued logical networks using a driving node and provides theoretical conditions for achieving synchronization.

## Contribution

The paper introduces a novel synchronization model and derives necessary and sufficient conditions for inner synchronization using the semi-tensor product.

## Key findings

- A necessary and sufficient condition for synchronization is derived when all nodes match the driving node's state.
- A more general synchronization model is proposed where nodes synchronize among themselves, not necessarily with the driving node.
- Simulation examples confirm the validity of the synchronization criteria.

## Abstract

This paper studies the inner synchronization problem of multi-valued logical networks by introducing an external single node, which is also called the driving node. Firstly, for the case where the synchronized state of all network nodes equals the state of the driving node, the inner synchronization model of the controlled multi-valued logical network is established, and a necessary and sufficient condition is obtained using the semi-tensor product. Secondly, based on the above synchronization model, a more general model of inner synchronization is discussed, where all nodes of the multi-valued logical network synchronize with each other, but not necessarily to the same as state of the driving node; inner synchronization criteria are established through theoretical proof. Finally, two simulation examples are given to verify the validity of the conclusions.