By Prof. Edward R. Scheinerman, Mathematics
This article allows readers from quite a lot of backgrounds and with restricted technical must haves to discover the dynamical structures and arithmetic usually. The booklet is designed for readers who are looking to proceed exploring arithmetic past linear algebra, yet are usually not prepared for hugely summary fabric.
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Extra resources for Invitation to Dynamical Systems
We have x(k) = Ak x0 + Ak−1 + Ak−2 + · · · + A + I b. To simplify this, observe that Ak−1 + Ak−2 + · · · + A + I I − A = I − Ak , and so, provided I − A is invertible, we have x(k) = Ak x0 + (I − Ak )(I − A)−1 b. The case: all |λ| < 1. The case: some |λ| > 1. 9) The formula, of course, is valid provided I − A is invertible, which is equivalent to saying that 1 is not an eigenvalue of A. Now, what happens as k → ∞? If the absolute values of A’s eigenvalues are all less than 1 (hence I − A is invertible), then Ak tends to the zero matrix, hence ˜ = (I − A)−1 b.
We start with x(0) a good bit to the right of x ˜. Observe that x(1) is to the left of x ˜, but not nearly as far. Successive iterations take us to alternate sides of x ˜, but getting closer and closer—and ultimately converging—to x ˜. 3. In this case we have a > 1, so the line y = f (x) = ax+b is sloped steeply upward. We start x(0) just slightly greater than x ˜. Observe that x(1) is now to the right of x(0), and then x(2) is farther right, etc. 1. 3: Iterating f (x) = ax + b with a > 1. 4: Iterating f (x) = ax + b with a < −1.
Vn be the n linearly independent eigenvectors associated with λ1 , λ2 , . . , λn . Then we can write x0 = x(0) = c1 v1 + c2 v2 + · · · + cn vn , from which it follows that x(k) = c1 λk1 v1 + c2 λk2 v2 + · · · + cn λkn vn , from which we factor out λk1 to get x(k) = λk1 c1 v1 + c2 λk2 λkn v + · · · + c vn . 2 n λk1 λk1 Since |λ1 | > |λj | for all j > 1, the ratios λkj /λk1 all go to 0. Thus if c1 = 0, and for k large we have x(k) ≈ c1 λk1 v1 . Geometrically, this means that the state vector is heading to infinity, in essentially a straight line, in the direction of the eigenvector v1 .
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