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.


Let us consider an example from population biology.



Discussion (to be read after doing the numerical experiments)


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