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Vector Dynamical Systems: Linear Case

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.
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Example

Let us consider an example from population biology.

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Last modified: March 27, 1995

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