Discussion On Mechanism For Emergence In Complex Systems

It is commonly believed that the complexity science is a scientific discipline studying complexity and complex systems,the rise of which is beginning from 1970s to 1980s.Although the complexity science is still at a beginner level,it has been miraculously called 'the science of 21st century'.Emergence is the essential feature of a complex system,commonly believed, which refers to the appearance of new structures and new properties during the self-organization in a complex system.And the science of complexity is a kind of science about emergence in essential on the basis of the current classic view.Emergence in a complex system is accomplished through edge of chaos:in other words,the process from disorder to order is fulfilled through edge of chaos.Chaos control is a potential mechanism producing edge of chaos but is absolutely not the only one,and all the rest of mechanisms and relative problems are sill unresolved.In the last couple of decades there has been massive researches on emergence,but up to now, three is no a generally accepted definition concerning emergence.Most of the present studies on emergence are descriptive work basing on observations and simulations,which can not provide more insight.The case makes the contention hard to decide come into being with regard to determining whether a phenomenon possesses characteristics of emergence.On the other hand,the deeper understanding on emergence is hard to obtain due to lack of genuine theoretical depth.In order to understand emergence more deeply and theoretically explore mechanism for emergence in complex systems,utilizing available knowledge about complex systems,in this paper,we propose a new idea to study mechanism for emergence in complex systems by integration of theoretical and calculational methods through further in-depth study on synchronization of all kinds in complex systems. Firstly,in the research on behaviors of a complex system,what is the most interesting is exploration of its dynamic variation,which frequently is related to patterns in a mathematical model.As a result,in the theoretical study of complex systems,emergence often presents as pattern formation in physical space.So-called pattern refers to evolution of the modes with some geometrical rules appearing in physical space during the evolution in a complex system.In fact.emergence frequently expresses as patterns. whose typical display is some kind of geometrical modes in time and space. Therefore.according to our new idea studying mechanism for emergence, synchronization of all sorts in a complex system is relevant to emergence of various patterns in this system.We are of the opinion that in order to understand pattern formation and emergence,it is very important to discuss various types of synchronization(especially generalized synchronization) among basic units in a complex system,and the development of theory about synchronization opens up vistas of research on mechanism for emergence.Just with this in mind,we first study approaches for realizing generalized synchronization between two systems.Inspired by sigmoidal functions which acts extensively in biological systems,in the second chapter of this paper,we provide a new.systematic and powerful control approach to implement generalized synchronization between two chaotic(hyperchaotic) systems with different dimensions.Secondly,further in-depth comprehension of the relationship between edge of chaos and conversion of disorder to order is one of the key points for studying emergence in a complex system.In the third chapter of this paper,we realize changes between disorder and order by synchronizing two nonidentical chaotic systems with bidirectional coupling.Due to there being free parameters in the process of structural adaptation,the synchronized system is able to adjust to changes in the environment,that is,possesses characteristics of emergence.Lastly,emergence of pattern formation,for example,is often the rep- resentation form of emergence.In fact,emergence is a dynamic process of pattern formation occurring over time from the point of dynamical systems, and the mechanism which evokes pattern formation is various synchronized relations between the basic units of complex systems.Among various types of synchronization,the pattern resulting from complete synchronization is too commonplace,and there is no mature mathematical method to deal with generalized synchronization,which have extensive significance.Therefore, in the fourth chapter of this paper,we study pattern formation in lattices dynamics under lag synchronization,which is chosen as a breakthrough of our idea about emergence,and discuss the relationship between synchronization and pattern in analytical methods for the first time,which will illustrate our opinion interpreting pattern formation through synchronization.As a new research aspect,our work is only a beginning and there is a lot to do in this area.What is more important is how to utilize the approaches presented in Chapter 2 and Chapter 3 to discuss the emergence in general complex systems,which can make us establish a more perfect method of discussing mechanism for emergence in complex systems.