复杂系统中涌现形成机理的讨论
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摘要
复杂性科学兴起于20世纪70至80年代,一般认为它是以复杂性和复杂系统为研究对象的学科。尽管目前复杂性科学还处于起始阶段,它已令人惊奇地被称为“21世纪的新科学”。
涌现是复杂系统的根本特征,通常认为它指的是复杂系统在自组织过程中新结构、新属性的出现。按照目前公认的看法,复杂性科学就是研究关于涌现的科学。复杂系统的涌现现象是由混沌边缘来完成的,换句话说这种由无序到有序过程是由混沌边缘来完成的。而混沌控制是产生混沌边缘的一种可能机制,但绝不是唯一机制,所有其他机制以及有关问题在目前看来还未解决。最近二十年,关于涌现的研究非常多,但到目前为止,还没有一个大家公认的关于涌现的定义。目前关于涌现方面的研究,多数是基于实验观察或者数值模拟的描述性工作,无法提供更多的洞察。这就使得人们在确认某个现象是否具有涌现特性上可能发生难以定夺的争论;另外一个方面,由于缺乏真正的理论深度,也难以获得更深刻的理解。
为了更深刻地理解涌现这个概念及从理论上探索复杂系统中涌现形成的机制,在本文中,我们利用对复杂系统已经有的认识,提出一条用理论与计算相结合的方法通过对复杂系统中各种同步行为的深入分析来研究复杂系统的中涌现形成机制的新思路。
首先,在一般的复杂系统的行为研究中,最令人感兴趣的是其行为的动态变化过程。这种动态变化过程在数学模型中往往与斑图有关。这样,在复杂系统的理论研究中,涌现常常表现为物理空间的斑图形成。所谓斑图,指的是复杂系统在演化过程中,常常会在物理空间上出现某种几何上具有某些规律性的模式的演化,即复杂系统各基本单元之间呈现出某种特定的关联作用。这样,按照我们提出的研究新思路,复杂系统中的各种同步关系就与系统中各式各样的斑图的涌现密切相关。我们相信通过讨论复杂系统基本单元之间的各种同步(特别是广义同步)对于理解斑图的形成以及涌现都是非常重要的,同步理论发展使我们看到了研究涌现形成机理的前景。正是出于这种考虑,我们首先研究两个系统之间实现广义同步的方法。受广泛作用于生物系统等复杂系统中的S形函数的启发,在本文第二章中,我们将采用一种新的强有力的控制方法以实现两个不同维数的(超)混沌系统的广义同步。
其次,从更深层次理解混沌边缘与无序到有序的转化关系是关于复杂系统中涌现的研究的关键点之一。在本文第三章中,我们通过同步化两个双向耦合的非恒同混沌系统以实现系统中无序和有序之间的转化。由于用来实现同步的结构适应过程中含有自由参数,因而同步化系统能涌现各种动力学行为。
最后,回到一般复杂系统中,由于复杂系统中涌现往往表现为各式各样的斑图形成,从动力系统的观点看,斑图就意味着复杂系统的物理空间不同部分之间在时间上存在某种特定的同步关系,这样一来,复杂系统中的各种同步关系就与系统中各式各样的斑图的涌现密切相关。在各种同步之中,由完全同步产生的斑图显然是平凡的,而广义同步虽具有广泛的意义,但目前由于数学处理的困难,我们还不能解决。为此在本文第四章中,我们选取滞后同步作为突破口,以格子动力系统作为模型来研究滞后同步下网格动力系统中斑图的形成,并首次用分析方法讨论了同步和斑图之间的关系,阐释了我们提出的关于斑图形成的同步观点。
作为一个新的研究方向,我们的工作仅开了一个头,还有许多工作可以继续下去。比较重要的一点是如何把第二、第三章的方法用到第四章讨论的一般复杂系统中去,以求得建立更为完善的讨论涌现形成机制的方法。
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.
涌现是复杂系统的根本特征,通常认为它指的是复杂系统在自组织过程中新结构、新属性的出现。按照目前公认的看法,复杂性科学就是研究关于涌现的科学。复杂系统的涌现现象是由混沌边缘来完成的,换句话说这种由无序到有序过程是由混沌边缘来完成的。而混沌控制是产生混沌边缘的一种可能机制,但绝不是唯一机制,所有其他机制以及有关问题在目前看来还未解决。最近二十年,关于涌现的研究非常多,但到目前为止,还没有一个大家公认的关于涌现的定义。目前关于涌现方面的研究,多数是基于实验观察或者数值模拟的描述性工作,无法提供更多的洞察。这就使得人们在确认某个现象是否具有涌现特性上可能发生难以定夺的争论;另外一个方面,由于缺乏真正的理论深度,也难以获得更深刻的理解。
为了更深刻地理解涌现这个概念及从理论上探索复杂系统中涌现形成的机制,在本文中,我们利用对复杂系统已经有的认识,提出一条用理论与计算相结合的方法通过对复杂系统中各种同步行为的深入分析来研究复杂系统的中涌现形成机制的新思路。
首先,在一般的复杂系统的行为研究中,最令人感兴趣的是其行为的动态变化过程。这种动态变化过程在数学模型中往往与斑图有关。这样,在复杂系统的理论研究中,涌现常常表现为物理空间的斑图形成。所谓斑图,指的是复杂系统在演化过程中,常常会在物理空间上出现某种几何上具有某些规律性的模式的演化,即复杂系统各基本单元之间呈现出某种特定的关联作用。这样,按照我们提出的研究新思路,复杂系统中的各种同步关系就与系统中各式各样的斑图的涌现密切相关。我们相信通过讨论复杂系统基本单元之间的各种同步(特别是广义同步)对于理解斑图的形成以及涌现都是非常重要的,同步理论发展使我们看到了研究涌现形成机理的前景。正是出于这种考虑,我们首先研究两个系统之间实现广义同步的方法。受广泛作用于生物系统等复杂系统中的S形函数的启发,在本文第二章中,我们将采用一种新的强有力的控制方法以实现两个不同维数的(超)混沌系统的广义同步。
其次,从更深层次理解混沌边缘与无序到有序的转化关系是关于复杂系统中涌现的研究的关键点之一。在本文第三章中,我们通过同步化两个双向耦合的非恒同混沌系统以实现系统中无序和有序之间的转化。由于用来实现同步的结构适应过程中含有自由参数,因而同步化系统能涌现各种动力学行为。
最后,回到一般复杂系统中,由于复杂系统中涌现往往表现为各式各样的斑图形成,从动力系统的观点看,斑图就意味着复杂系统的物理空间不同部分之间在时间上存在某种特定的同步关系,这样一来,复杂系统中的各种同步关系就与系统中各式各样的斑图的涌现密切相关。在各种同步之中,由完全同步产生的斑图显然是平凡的,而广义同步虽具有广泛的意义,但目前由于数学处理的困难,我们还不能解决。为此在本文第四章中,我们选取滞后同步作为突破口,以格子动力系统作为模型来研究滞后同步下网格动力系统中斑图的形成,并首次用分析方法讨论了同步和斑图之间的关系,阐释了我们提出的关于斑图形成的同步观点。
作为一个新的研究方向,我们的工作仅开了一个头,还有许多工作可以继续下去。比较重要的一点是如何把第二、第三章的方法用到第四章讨论的一般复杂系统中去,以求得建立更为完善的讨论涌现形成机制的方法。
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.
引文
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