Discret Dynamical System

A discret dynamical system is a sequence of numbers that are defined recursively. Eigenvalues and eigenvectors provide the key to understanding the evolution of a dynamical system described by a difference equation X(k +1) = AX(k) where A is a square matrix. We can view this equation as a description of what happens to an initial point X (0) in the plane as it is transformed repeatedly by the mapping X->AX . The long-term behaviour of the dynamical system is equivalent of the steady-state response in an engineering control system. An instructive way to see how multiplication by the matrix A affects points is to plot an arbitrary point X(0) and then to plot successive images of this point under repeated multiplications by A.

• PointIteration (A, X0,N ) animates trajectories of the system X(k +1) = AX(k). X0 is the initial point X(0).

Rotation

Let .

> A:=matrix(2,2,[0.8,0.5,-0.1,1.0]): LX0:=[[0,2.5],[4,0],[0,-2.5]]:

Figure 15 animates three points under the action of the matrix A with complex eigenvalues.

> PointIteration(A,LX0,45,line,view=[-5..5,-3..3]);

Figure 15 Iteration of the points [0,2.5],[4,0],[0,-2.5] under the action of .

The origin as an attractor

Let

> A:=matrix(2,2,[0.8,0,0,0.64]):

> LX0:=[[-3,3],[-1.5,3],[0,3],[1.5,3],[3,3],[-3,-3],[-1.5,-3],[0,-3],[1.5,-3],[3,-3]]:

Figure 16 animates trajectories which all tend toward 0, because both eigenvalues are less than one in magnitude.

> PointIteration(A,LX0,15,line,view=[-3..3,-3..3]);

Figure 16 Origin as an attractor for points under the action of