A system is known as a model in which each part is related with each other with some relationship between them. And thus, a complex system is a system which is made up of many interrelated parts which are mutually related and are really complex to understand or describe. The field of complex and analytical dynamic studies has undergone vigorous growth in two nearly short periods. The work of initial studies revolves around the iterations near a fixed point. Afterwards on variety of other dynamic and complex systems too work began like, rational maps, higher degree polynomials, mero-morphic systems etc.

Complex Systems Theory

The theory behind the complex system is that all the parts of this system are interconnected with each other as if they have been interwoven. Thus, to understand about the nature of these complex systems, we must know about the behavior of its every part and how they work together to complete the entire system.
Complex system theory includes the theory in which huge and large number of units are organized together into such aggregations in which we can generate a pattern, for storing some information and can even get engaged in some collective decision making. The equations of complex systems are generally derive from information theory, statistical physics and non-linear dynamics, and looking organized but these are considered fundamentally complex because of unpredictable behaviors of natural systems.
In recent times, many different communities of science and mathematics have given a combined definition to those systems which are complex. According to this definition a complex system can be described in brief the phenomena, aggregates, problems, structure or organisms which is sharing common theme on some criterion:
1)    That these systems are complicated or intricate inherently.
2)    These systems can be completely deterministic very rarely.
3)    The mathematical models of such systems are mostly of the non linear type.
4)    These systems are pre disposed to some unexpected and unpredictable outcomes which are also termed as the so-called emergent behaviors.
In the field of mathematics, the largest contribution made by the study of complex systems was in the form of the deterministic systems with the discovery of chaos. This is a property used in some dynamical systems which is strongly related to its non linear characteristic. For some scientist and researchers, complex systems mean a structure having a lot of variations. But there are some scientist and researchers too, who say that a complex system is one having large number of interacting components, in which every part is inter connected and inter related with a unique or various relation. It is commonly said that the moment where the causality of a system breaks down, the complexity of a system starts. And the analysis for describing such complex systems generally need the requirement of differential equations that are non linear. Hence, the knowledge of differential calculus is a pre requisite for understanding the modeling of complex systems as they uses the concept of non linear differential equations.

The word complex is an adjective in general sense that describes a component or a system which is difficult to verify as well understand by just the design or its function or both. In these cases the complexity is determined by the factors like number of components, number of conditional branches and their intricacies, the types of data structures involved, degree of nesting, etc.

The complexity theory may also indicate the large number of units that can organize themselves into aggregations which can generate a pattern, store some information and can even get engaged in some collective decision making.

Examples of Complex Systems

We can find various examples of Complex systems which are being used not only in the filed of mathematics, sciences, etc. but in our real world day to day life which are discussed below:

Some of the examples of complex systems from our real world system are as follows:

1) The Governments of different Countries as every government has many responsibilities like taxation, transportation, military etc and each of these responsibilities or functions and are in itself complex in nature. Hence, this is a very simple example of a complex system which we come across in our daily life.
2) The second example is the different kind of families. As every family (nuclear, extended etc.) is made up of some individuals who are related to some other individual by a relation, and has to make a link with the outside environment. Hence, every family also constitutes an example of a complex system.
3) The ecosystem of the earth and its subsystem ecosystems like oceans, rain forest, desert, weather etc. Note that each sub system eco system is in itself a complex system.
4) Any corporation or company is also an example of a complex system as it is made up of many inter related sub complex systems.

Some of the examples of complex systems used in the field of mathematics are listed below:
1) Complex systems used in Graph Theory in the form of complex networks:

Let $G$ be a graph having $V$ as the set of vertices and $E$ as the set of directed or undirected edges.
Let $k$i represents the number of connected edges from the vertex $v$i, then $C$i, the clustering coefficient of a vertex $vi$ is given by the following formula:
$C_{i}$ = $\frac{2\left | \left \{ e_{jk} \right \} \right |}{k_{i}(k_{1}-1)}$:$v_{j},v_{k}\epsilon N_{i}, e_{jk}\epsilon E$
The following is an example of the complex network, which is known as small world network as shown below:

2) Complex systems used in probability:
If the probability of heads is equal to $\frac{1}{2}$ and the coin is a fair coin then the unpredictability or the complexity will be considered equal to 1. Similarly, if the probability of heads is equal to 1 and the coin is a perfectly biased coin then the unpredictability or the complexity will be considered equal to 0.
In natural patterns of landforms, complexity is a manifestation of two main characteristics.
1) One of the natural form of patterns is from the processes that are not linear at all, that is the ones that can modify the properties of our environment in which they are operating or which are strongly coupled.
2) The other one is in the systems which are open and can be driven from neutral or equilibrium just by the exchange of momentum, energy, information or material across their boundaries.

Understanding Complex Systems

To understand about any complex system we must know the difference between a simple and a complex system. Some examples of simple systems are a pendulum, a spinning wheel, an oscillator, an orbiting planet. We must understand that every complex system have properties which are universal in nature, while the systems which are simple follows only one function that has to performed at one time only. Thus, on comparing simple versus complex systems, we can understand about complex system thoroughly.

Complex Systems can be better understood by looking at the following features, which every complex system has:
1) The property of Non Linearity: This is a pre requisite for a system to be a complex system and is a must for every complex system. A system is said to be linear if we can combine any of the two solutions to obtain another, and can also multiply any of the solution by some factor to obtain another. When this principle of superposition is not applicable then the system is said to be non linear. Non linearity in the some equations may also have a consequence that even small changes in the initial values or conditions can make a big change in different results.
2) The feature of Feedback: It is also a very important condition for complex and dynamic systems. When any inter connected part of a system interact with some other part, in some way, the later part receives some feedback. The later time of the feedback completely depends as on how the interaction with them is taking place. But only the existence of feedback is not enough for a system to be complex. The most abstract way to represent the prevalence of the feedback in a system that is complex is given by the casual graphs’ theory. A chain of some casual arrows will not indicate any feedback while a graph having loops of some casual arrows will depict feedback. In some places, feedback is often used by a control system. Feedback is also been used for error correction in the filed of statistics and reliability theory.

3) Spontaneous Order: Related notions in a system include organization, determinism, symmetry, periodicity and pattern. The most confusing issue of complex systems is how the order in them is related to the content of information of its states and dynamics as the information is processed. With complexity, the total number of order is also incompatible.

4) Robustness and Lack of Central Control: The order in a system that is complex is robust due to the fact of being distributed non centrally. It is said to be stable only under perturbations of the system. A system that is centrally controlled is vulnerable to malfunctions in a few of the key components. The order of a system can be maintained by utilizing some error correction mechanism.

5) Emergence: Emergence is a quantity that is calculated by understanding the collective behavior of all the parts of a system and hence, it is also one of the important properties of a system to be complex as well.
Thus, the main aim to understand the field of complex systems is to known about its universal properties or features which are discussed above.
The various difficulties that are faced in the field of complex systems are:
a) Unless and until it is not clear as what is meant by the structure and variations originally of a system, the structure’s information is not proving itself to be informative or helpful.
b) One have to choose between the conflation of complexity which have a number of components or the conflation of science of complexity with non linear and chaotic dynamics, or conflation of a system which is complex and have different histories as possible in a hand and a fully subjective answer to our questions.
c) It takes us to a territory that is more interesting.
d) A central idea of non linearity is introduced.
e) What many means in a complex system in accordance to “many components” also matters to the complexity of the system.
f) Informative characterization of a system is difficult and thus idea to argue over it is introduced.