Although the science of complexity is in fashion, there is almost no agreement on what the complex system is. In this article, I would like to suggest a working definition of the complex system: indeterminate determination.In order to understand the complex system, you must understand two concepts: complexity and system. Since system was defined at the previous article, I will explain complexity in this article.

## 1. Complexity is Indetermincy

For example, a series

(1) 2,4,6,8,10,12,14,16…

is simpler than

(2) 9,2,85,7,36,49,1,756…

Here simple means predictable. You can predict that the nth term of the series (1) is 2n. If the series (1), however, turns out to be

(1′) 2,4,6,8,10,12,14,16,31…

it suddenly gets unpredictable and complex.

## 2. The difference between complex and composite

When complexity should be so grasped, then the complex system can be conceived to be an indeterminate system that behaves unpredictably. In this respect, complex systems are similar to chaos, though the former is more comprehensive concept than the latter. It is a requisite for chaos to behave according to deterministic nonlinear rules, while this is not the case as to complex systems.

A complex system is usually defined as a system consisting of a large number of elements that are interconnected and interactive. Such a definition is misleading, as it makes the distinction between complex systems and composite systems ambiguous.

The distinction between complex/simple should be distinguished from that between composite/single. Whether a system is composite or single depends on whether elements constituting the system are plural or singular, while whether a system is simple or complex depends on whether the behavior of the system is determinate or not.

The logistic map f (x)=4x(1-x) is a single system containing only one variable; however it is also a complex system because applied to itself repeatedly it behaves indeterminately. On the other hand, a precision clock is a composite system composed of many parts; however, it is also a simple system because the behavior is determinate. If you do not distinguish complex/simple systems from composite/single systems, you can explain neither complex single systems nor simple composite systems.

## 3. Redefining complex systems

By the way, how do other scientists define the term? Some list the following requisites

- The number of agents constituting a model is medium.
- Agents have intellect.
- Agents interact with each other based on local information.

I will examine and interpret them from my point of view:

- Medium is a relative quantity. 100 million is large as a population and small as a number of molecules. Medium should be interpreted as a probability between 0 and 1; that is to say, the first requires that systems should be exposed to complex environment.
- Taken literally, the second requisite confines complex systems to human society or at most primatial society. This scope is too narrow. The word intellect derives from inter+legere, "choose among", so the second requires systems should have function of reducing the complexity they face.
- Local can be interpreted as partial. The third requires reducing complexity based on partial information should increase the complexity of systems themselves.

To sum up, complex systems are systems that are exposed to complex environment and, through reducing the complexity they face, increase the complexity of systems themselves. The expression "indeterminate determination" seems rather paradoxical but it is a special case following the second law of thermodynamics; decreasing some entropy increases more entropy.

## 4. Increasing entropy through reducing it

I gave an example a refrigerator to explain the second law of thermodynamics at the previous issue. A refrigerator is not a complex system because it can decrease interior temperature (entropy) determinately though it increases exterior entropy.

In the case of information systems, however, a system and its environment are not separated physically. That’s why information systems can be complex systems.

The logistic map f(x)=4x(1-x) as a function reduces the complexity of the next generation but this reduction makes its behavior unpredictable. A complex system similar to the logistic map is rumor circulation. Rumors usually get exaggerated. Suppose a message is applied to the same exaggeration transformation repetitively from member to member in a community. This is similar to the repetitive and self-referential application of a logistic map. Those members as information systems reduce complexity, which result in increase in information entropy. Rumors often spread panic, namely increase social entropy.

## 5. Complex systems and chaos

Most of complex systems are not single systems like logistic equations but composite systems including plural variables. For example, Lorenz equations that model the chaos of air movement:

- dx/dt=-10x+10y
- dy/dt=28x-y-xz
- dz/dt=8/3z+xy

consists of three variables. "dx/dt" means a variable x differentiated by time and shows how x changes temporally. Lorenz equation plots indeterminate locus known as Lorenz attractor in 3-dimensional space.

This locus crosses at no points, which means it behaves non-periodically for ever. Lorenz invented the terms "butterfly effect" to express the sensitive dependence on initial conditions of the Lorenz attractor.

At the Lorenz equations three variables are defined in terms of self and others. We usually decide our behaviors based on the state of others concerned and ourselves. So, you can compare Lorenz equations to social relationship. Suppose such an eternal triangle; a husband (x) is not interested in his wife (z) and charmed by a woman (y) in the neighborhood, while the woman feels hesitant in his wife’s presence though she has some interest in the husband.

Here we have a social version of Lorenz equations:

- The husband decides his behavior in reference to his own feelings and the woman in the neighborhood
- The woman in the neighborhood decides her behavior in reference to her own feelings and the husband, and his relationship to his wife.
- The wife decides her behavior in reference to her own feelings and his affair with the woman.

The bilateral relation between mutually loving man and woman is stable, while the eternal triangle "eternally" remains complex without converging on a stable state. As the story of the eternal triangle is complex and unpredictable, it can be a theme of dramas. Thanks to butterfly effects scenarists can get an unlimited number of stories with initial conditions changed.

It was Poincare’s *Le probleme de trois corps et les equations de la dynamique* that first raised the problem of chaos. He insisted in 1887 that the solar system be not a stable system because three or more bodies under the gravity law shows no periodicity, namely behave chaotically. To use Poincare’s terms the eternal triangle is a three-body problem in sociology.

Of course, in actual societies, others whose selection is concerned with mine are more than two. In a society as a complex system, every agent expects the expectation of the others: if that person acts so and so, I will act so and so. The reduction of complexity on a personal level increases the complexity of social systems. As we cannot formulate the rule of our behavior completely, social systems might not be considered to be chaos but they are at least complex systems.