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.