Math Insight

Higher dimensional discrete dynamical systems

Math 2241, Spring 2016
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Due date: Feb. 10, 2016, 11:59 p.m.
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Total points: 3
  1. So far, we have studied only two-dimensional discrete dynamical systems. For higher-dimensional systems, the ideas are the same. Let's take a look at an example: \begin{eqnarray*} x_{n+1}&=&1.1 x_{n} + 1.3 y_{n} + 0.8 z_{n}\\ y_{n+1}&=&0.7 x_{n} + 0.3 y_{n} + 0.8 z_{n}\\ z_{n+1}&=&0.4 x_{n} + 1.2 y_{n} + 0.7 z_{n}\\ x_0&=&49\\ y_0&=&32\\ z_0&=&43 \end{eqnarray*} Suppose we want to determine the long-term behavior of the system.

    1. The first step is to find the eigenvalues and eigenvectors of the corresponding matrix, which is $$A=\left[\begin{matrix}1.1 & 1.3 & 0.8\\0.7 & 0.3 & 0.8\\0.4 & 1.2 & 0.7\end{matrix}\right]$$ Finding the eigenvalues by hand is difficult for larger matrices, so we will have a computer program do it for us. There are many programs that can do this. We'll show you how to compute the eigenvalues and eigenvectors using Wolfram Alpha ® and R ©.

      For Wolfram Alpha, start by going to www.wolframalpha.com. Enter eigenvalues { { 1.1, 1.3, 0.8 }, { 0.7, 0.3, 0.8 }, { 0.4, 1.2, 0.7 } }. In general, each row is entered in curly brackets.

      For R, open the R console and start by creating the matrix. Enter A=matrix(c(1.1, 1.3, 0.8, 0.7, 0.3, 0.8, 0.4, 1.2, 0.7),3,3,byrow=TRUE) . You may wish to verify that the matrix is correct by just typing A on the next line. The c() command specificies the elements of the matrix, while the next two numbers specifies the size of the matrix. Next, enter eigen(A) to compute the eigenvalues and eigenvectors. Include at least five significant digits in your answer.

      Enter the eigenvectors and corresponding eigenvalues below:
      $\lambda_1=$
      , $\vc{v}_1=$

      $\lambda_2=$
      , $\vc{v}_2=$

      $\lambda_3=$
      , $\vc{v}_3=$

    2. The long-term behavior for a three-dimensional system is similar as for a two-dimensional system. Except for special initial conditions, the long-term behavior will be determined by the largest (in magnitude) eigenvalue and its eigenvector.

      What is the magnitude of the largest eigenvalue?
      Since that eigenvalues magnitude is
      , the solution will be
      (unless the initial condition is a linear combination of just the other two eigenvectors, which is unlikely).

    3. We don't have a three-dimensional applet for you to visualize how multiplication by $A$ transforms vectors. But, the principles you learned in two dimensions work for three and higher dimensions.

      If the initial condition is a scalar multiple of an eigenvector (which is unlikely), what happens to the direction of the vector of state variables?

      If the initial condition is not a scalar multiple of an eigenvector, what happens to the direction of the vector of state variables?

      Now suppose we want to know what the ratio of $x_n$ to $y_n$ approaches. We'll ignore the possibility that our initial condition is an eigenvector, because that is an extremely unlikely situation. The ratio that is approached is the same as the ratio of $x$ to $y$ in the eigenvector. What is this ratio?

      If after a long time, $z_n = 1000$, what would you expect for values of $x_n$ and $y_n$?
      $x_n \approx $
      , $y_n \approx$

      Need help?

  2. Let's look at a model of a monarch butterfly population. The model incorporates the four stages of the butterfly's life: egg, caterpillar, pupa (chrysalis), and butterfly. Each day, an egg hatches into a caterpillar with probability $0.08$ or stays an egg with probability $0.2$. A caterpiller will turn into a pupa with probability $0.06$ or stay a caterpillar with probability $0.47$. A pupa will become an adult butterfly with probability $0$ or stay a pupa with probability $0.58$. Lastly, an adult butterfly will survive to the next day with probability $0.73$. Additionally, adult butterflies lay eggs at an average rate of $100$ per day. This can be written as the following mathematical model: \begin{eqnarray*} e_{n+1}&=&0.2e_n+100b_n\\ c_{n+1}&=&0.08e_n+0.47c_n\\ p_{n+1}&=&0.06c_n+0.58p_n\\ b_{n+1}&=&0.1p_n+0.73b_n\\ e_0&=&250000\\ c_0&=&800\\ p_0&=&300\\ b_0&=&400 \end{eqnarray*} We want to predict if the monarch butterfly population will be able to sustain itself long-term or if they will eventually die off. We also want to know what fraction of the population will be adult butterflies in several years.

    1. Let's start by looking at the first week of butterfly populations. We'll use R to compute these values. First, create a matrix for the system and a matrix to represent the initial condition vector. The matrix corresponding to the butterfly model is $$A=\left[\begin{matrix}0.2 & 0 & 0 & 100\\0.08 & 0.47 & 0 & 0\\0 & 0.06 & 0.58 & 0\\0 & 0 & 0.1 & 0.73\end{matrix}\right]$$ Enter A=matrix(c(0.2, 0, 0, 100, 0.08, 0.47, 0, 0, 0, 0.06, 0.58,0, 0, 0, 0.1, 0.73),4,4,byrow=TRUE) . For the initial condition vector, enter b=c(250000, 800, 300, 400). You may wish to verify that the matrices are correct by entering just the name on the next line (A or b, respectively).

      Now, to find the next day's population, b=A%*%b . The %*% is the R command for matrix multiplication. Written this way, we are overwriting the value of the vector b with the next day's populations. To see the actual value, enter b on the next line. Repeat these two steps for each day, and enter the values you get below (include at least one decimal place).

      Write the state vectors below, with $\vc{x}_n=\begin{bmatrix} e_n \\ c_n \\ p_n \\ b_n \end{bmatrix}$.
      $\vc{x}_1=$

      $\vc{x}_2=$

      $\vc{x}_3=$

      $\vc{x}_4=$

      $\vc{x}_5=$

      $\vc{x}_6=$

      $\vc{x}_7=$

      Based on this first week, do you think the population will be larger or smaller in several years?
      Are you confident about your answer?

    2. To get a certain answer, we need to find the eigenvalues and eigenvectors. To find the eigenvalues and eigenvectors in R, use the command eigen(A) with the matrix you created in part a. Include at least five significant digits in your answer.

      Enter the eigenvectors and corresponding eigenvalues below.

      $\lambda_1=$

      $\vc{v}_1=$

      $\lambda_2=$

      $\vc{v}_2=$

      $\lambda_3=$

      $\vc{v}_3=$

      $\lambda_4=$

      $\vc{v}_4=$

    3. Some of the eigenvalues are complex, involving the imaginary number $i=\sqrt{-1}$, which adds oscillatory effects to the system. We are interested in which eigenvalue is largest in magnitude. List the magnitudes of the eigenvalues in increasing order:

      What are the long term prospects for the monarch butterfly population? Will it eventually grow larger, or will the butterflies gradually die off?

    4. It's good a thing we aren't relying on visualizing the effects of $A$ on the system. Since we have for state variables, we are dealing with four-dimensional vectors. Our visualizations don't work so well in four dimensions! Even so, we can easily work with higher dimensional vectors by just thinking of them as lists of numbers, in this case representing population sizes.

      Now let's consider the fraction of the population which are adults in several years. The fraction on day $n$ is given by $\frac{b_{n}}{b_{n} + c_{n} + e_{n} + p_{n}}$. Suppose the state variable is an eigenvector. What is the fraction on day $n+1$? Write the fraction in terms of e_n, c_n, p_n, b_n and lambda, where lambda denotes the corresponding eigenvalue.

      Estimate the fraction of adults in several years. Include at least five significant digits in your answer.

      Need help?

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Math 2241, Spring 2016