| Boolean network provides a simple and effective model for describing gene regulation networks,neural networks and artificial intelligence systems,which has become a hot issue in recent years.As a natural generalization of Boolean network,multi-valued logical network has attracted the attention of many scholars because of its simple model structure,which can be used for the modeling and dynamic characteristic analysis of large logical system models.System synchronization not only explain many natural phenomena,but also have many potential applications.The research on Boolean network synchronization has been extended to be a special case of logical network,and many important research results have been obtained.Based on the semi-tensor product of matrices theory and the existing research works,this paper studies the synchronization of switched multi-valued logical networks,the inner synchronization of multi-valued logical networks,multi-valued logical networks with time delays and the synchronization of multi-valued logical networks under asynchronous update mechanism by using algebraic methods.The main contents and innovations of this paper are as follows:(1)The synchronization problem of switched multi-valued logical networks is studied.Switched network is a more complex and practical system.In the real process of gene network regulation,the dynamic model of the network is usually determined by different switched modes.This kind of system model can be described by switched logical network.This paper studies the synchronization of switched multi-valued logical networks under drive-response configuration.The drive system is multi-valued logical network and the response system is switched multi-valued logical network.The sufficient and necessary conditions for the synchronization of switched multi-valued logical networks are obtained by using the semitensor product of matrices theory.At the same time,a numerical simulation example is given to further verify the correctness of the results.(2)The inner synchronization problems of multi-valued logical networks and multi-valued logical networks with time delays are studied.This paper analyzes the dynamic model of inner synchronization of multi-valued logical networks without time-delays,gives the conditions for each node of the network to realize inner synchronization,and shows a numerical simulation example.The phenomenon of time delays is common in complex systems.For example,in the gene regulatory networks,the time delays caused by the slow process of transcription,translation and translocation.Based on the inner synchronization model of multi-valued logical network,the necessary and sufficient conditions for the inner synchronization of multi-valued logical network with time delays are obtained by using the semi-tensor product of matrices,and a numerical simulation example of multi-valued logical network with time delays meeting the inner synchronization conditions is shown.(3)The synchronization and inner synchronization problems of multi-valued logical networks under asynchronous update mechanism are studied.Synchronous update means that all nodes of the multi-valued logical network update at the same time,while asynchronous update means that any node of the multi-valued logical network may or may not update at a certain time.With the change of time,asynchronous logical network presents many states.The multi-valued logical network under asynchronous update mechanism is more complex and has a wider range of applications.All possible update states of asynchronous multi-valued logical network are analyzed.The asynchronous multi-valued logical network is transformed into its corresponding discrete-time system by using the semi-tensor product of matrices theory.The sufficient and necessary conditions for inner synchronization and synchronization of asynchronous multi-valued logical network are proved by algebraic method.Moreover,the correctness of the conclusions is verified by two numerical simulation examples.The main research methods are as follows:(1)A multi-valued logical network synchronization model is proposed.With the semi-tensor product of matrices theory,the dynamic system of multi-valued logical network is transformed into its standard discrete-time system,the logical relationship of multi-valued logical network synchronization is analyzed,and the corresponding general generation relationship expression is deduced for different synchronization models.For different multi-valued logical networks,the conditions of inner synchronization and synchronization are proved by algebraic method.Furthermore,numerical simulation examples are used to verify the relevant conclusions of multi-valued logical network synchronization.(2)The conclusions of this paper are verified by relevant numerical simulation examples.A lot of the semi-tensor product of matrices calculation and data analysis are involved in the synchronous example analysis of multi-valued logical network.Matlab,which has powerful data calculation,matrix operation and drawing function,is chosen as a simulation tool. |
