**Discrete Dynamical Systems -- Adapted from VLA Module**

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with(LinearAlgebra): |

A *discrete dynamical system* is a sequence of vectors related to one another by a square matrix *A* as follows:

, for *k* = 0, 1, 2, ...

If we think of *k* as representing time in some units, we can think of a dynamical system as describing the evolving behavior over time of the set of variables represented by the vectors . We refer to this behavior as the "dynamics" of the "system" of vectors .

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**Example 1:** To determine the long-term behavior of the vectors we'll examine the eigenvector decomposition:

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Ex1 := Matrix([[525/1000,15/100],[-1875/10000,975/1000]]); |

First we find the eigenvalues and associated eigenvectors of *A*:

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h,P := Eigenvectors(Ex1); |

Therefore we have the eigenvector decomposition

+

Since both eigenvalues are less than one in absolute value, then both and have limit 0. Now the vector tends toward the zero vector in the limit every , so both the x-coordinate population and the y-coordinate population tends to 0. We sometimes say that the points are "attracted to the origin." Furthermore, since , then as long as is not 0, then the points will be close to for large *k*, which tells us that approaches the origin along a line with direction vector ie y=(5/2) x.

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We plot the "trajectory" of the vectors , which is a plot of the points

Note: while we don't typically need the coefficients and , we could compute them as follows if we need them in applications:

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x0 := Vector([1.5,1]); |

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a := P^(-1).x0; |

(1.1) |

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**Example 2:** To determine the long-term behavior of the vectors we'll examine the eigenvector decomposition:

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Ex2 := Matrix([[9/10,-12/100],[-5/10,8/10]]); |

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h,P := Eigenvectors(Ex2); |

The eigenvector decomposition is , where the values of and depend on . Since the larger eigenvalue of *B* is 11/10, the points eventually move away from the origin and get arbitrarily far away. An associated eigenvector is , which tells us that as long as is nonzero, then eventually the trajectory follows the line through the origin y=-x.

However, note what happens below when the initial vector is an eigenvector associated with the smaller eigenvalue (ie =0). In this case we die off along the y= x line.

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x0 := Vector([.4,1]): |

**Example 3:** To determine the long-term behavior of the vectors we'll examine the eigenvector decomposition:

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Ex3 := Matrix([[9/10,2/10],[1/10,8/10]]); |

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h,P := Eigenvectors(Ex3); |

Thus .

The larger eigenvalue is 1 so, for most starting positions, the system tends towards the line y=1/2 x associated to the eigenvector . We can say a bit more than this as =1 and so

which is the line parallel to (since changes over time) and through the tip of ie the place on y=1/2 x given by the starting position and a parallel transport there. So eventually we hit that line and then stay fixed there forevermore (unless

=0 in which case we die off along the y=-x line). For most starting populations, in the limit the populations will achieve the ratio of 2 of population a for each 1 of population b.

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**Example 4:** Here is a system whose dynamics are more complicated:

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Ex4 := Matrix([[8/10,5/10],[-1/10,1]]); |

We sometimes say that the points are "attracted to the origin along a spiral." The spiral behavior is caused by the presence of complex eigenvalues.

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