# Vector Dynamical Systems: Linear Case

## Introduction

We consider next dynamical systems in which the state is a vector, e.g.,

```   x = (x_1, x_2 )
```
In this case the vector is an ordered pair of scalars --- these are the components x_1 and x_2. Thus the state is determined by two measurements. It could, for example, represent the population levels of two species in a given region at a given time. Thus, for an n-species population model, the state is an n-tuple
```  x = (x_1, x_2, ... x_n),
```
i.e., a vector in n-space.

As for scalar systems the next state is determined by a "generating" function f:

```  next state = f( current state )
```
As before an initial state x_0 generates a sequence of future states { x_n } by the rule
```  x_{n+1} = f(x_n)
```
The notion of equilibrium, or rest state also makes sense. These are the fixed points p, i.e., the vectors which solve the equation
```   p = f(p)
```

For now we will consider dynamical systems in which the generating function is given by matrix multiplication

```  f(x) = Ax
```
This is a case in which there is a "simple" formula for the n-th state:
```  x_n = A^n x,
```
where A^n denotes the n-th power of the matrix A.

### Example

Let us consider an example from population biology.

## Bibliography

Back to syllabus
Back to Department of Mathematics, University of Utah