# Updating function discrete time dynamical system

These physical models constitute a side of dynamical systems which may be used as a quantitative tool to analyze the environment around us.On the other hand some dynamical systems may involve more simplifications and approximations and thus do not carry with them the same numerical accuracy or prediction of exact values.Some examples of state variables may include the population of a colony, the density of a chemical in a solution, the amount of money in a bank account, the position of a particle, temperature or anything that can be represented by a number or set of numbers.In each of these examples the state of the system may be represented by a number.The simplest type of dynamical system describes the evolution of a state variable which changes at a constant rate. On our birthday our age makes a jump from one integer value to the next.If we measured our age as any positive number the change would not be noticeable, but it is a bit awkward to say things like "I am approximately 26.997 years old." Instead we say "tomorrow I will be 27 years old." Another point worth addressing is the use of relevant units.But, exact solutions for such complex systems are often too difficult to solve or too complicated to understand.Instead the system of differential equations which model say the Belousov-Zhabotinky reaction may still shed interesting information by utilizing simple dynamical system techniques without the need of an exact solution. Although our age is continuous our birthday is a discrete event.

This defines the state variable which in this case is a representation of a persons age.

The right hand side of the above equation is the updating function for a person aging at a constant rate.

So far we have defined what it is that is changing, the state variable, and how it changes, the updating function.

Now that we have a complete dynamical system which may be written as \[ \left\{ \begin a_ & = & a_n 1 \ a_0 & = & 0\ \end \right.

\] we would like to write down a solution to the system.

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