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) = AxThis 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.