Math Insight

Equilibria in discrete dynamical systems

Math 1241, Fall 2020
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Due date: Oct. 2, 2020, 11:59 p.m.
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Total points: 3
  1. An equilibrium point of a dynamical system is a constant solution. If the initial condition $x_0$ of a discrete dynamical system is set to be the equilibrium value, then $x_1$, $x_2$, $x_3$, etc., will all be identical to that initial condition. (Of course, we could use a different variable, like $z$, instead of $x$.)

    Consider the dynamical system: \begin{align*} z_{t+1} - z_{t} &= 3(z_t-1)(z_t+5)\\ z_0 &= a \end{align*}

    1. Let the initial condition be $a=1$. With this initial condition calculate:
      $z_1 =$
      , $z_2 =$
      , $z_3 =$
      , $z_4 =$
      , $z_5 =$

      Given these calculations, we can conclude that $z_t=1$
      an equilibrium.

      Need help?
    2. Let the initial condition be $a=2$. With this initial condition calculate:
      $z_1 =$

      Given this calculation, we can conclude that $z_t=2$
      an equilibrium.

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    3. Let the initial condition be $a=-5$. With this initial condition calculate:
      $z_1 =$

      Given this calculation, we can conclude that $z_t=-5$
      an equilibrium.

      Need help?

  2. Consider the dynamical system: \begin{align*} y_{n+1} &= \frac{1}{8}(y_n-3)(y_n+3)\\ y_0 &= b \end{align*}
    1. Let the initial condition be $b=3$. With this initial condition calculate:
      $y_1 =$

      Given this calculation, we can conclude that $y_n=3$
      an equilibrium.

      Need help?
    2. Let the initial condition be $b=-3$. With this initial condition calculate:
      $y_1 =$

      Given this calculation, we can conclude that $y_n=-3$
      an equilibrium.

    3. Let the initial condition be $b=9$. With this initial condition calculate:
      $y_1 =$

      Given this calculation, we can conclude that $y_n=9$
      an equilibrium.

      Need help?
    4. Let the initial condition be $b=-1$. With this initial condition calculate:
      $y_1 =$

      Given this calculation, we can conclude that $y_n=-1$
      an equilibrium.

  3. The above approach of trying different initial conditions can verify whether or not a particular value is an equilibrium, but it's not easy to find equilibria that way. Instead, we can calculate the equilibria analytically by observing that we need to find a value of $x_n$ that leads to $x_{n+1}$ being same value. Therefore, to find equilibria, we can set $x_n=x_{n+1}$ to the same value, for example to the variable $E$. Then, we solve for $E$, which will gives us the value of the equilibria.

    Calculate the equilibrium point(s) for each of the following dynamical systems. (For brevity, we don't show the initial conditions, as we don't need to worry about initial conditions to calculate equilibria.) If there are more than one equilibrium for a given dynamical system, separate the values by commas.

    1. $x_{n+1} -x_n = \frac{1}{2}x_n$

      How many equilibria are there?
      .
      The equilibria are $E =$
      .

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    2. $x_{n+1} = \frac{1}{2}x_n$

      How many equilibria are there?
      .
      The equilibria are $x_n =$
      .
      (We might refer to the equilibria by $E$, to emphasize they are an equilibrium, or by the state variable $x_n$, to emphasize they are a particular solution to the dynamical system.)

      Need help?
    3. $x_{t+1} = 2 x_t -10$

      How many equilibria are there?
      .
      The equilibria are $E =$
      .

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    4. $y_{t+1} -y_t = 2y_t(1-y_t)$

      How many equilibria are there?
      .
      The equilibria are $y_t =$
      .
      (Remember, if there are more than one equilibrium, separate answers by commas.)

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    5. $z_{n+1} = 2z_n(1-z_n)$

      How many equilibria are there?
      .
      The equilibria are $E =$
      .

      Need help?
    6. Here's the dynamical system from the first problem. This time calculate the equilibria analytically. $$z_{t+1} - z_{t} = 3(z_t-1)(z_t+5)$$

      How many equilibria are there?
      .
      The equilibria are $E =$
      .

      Need help?
    7. Here's the dynamical system from the second problem. This time calculate the equilibria analytically. $$ y_{n+1} = \frac{1}{8}(y_n-3)(y_n+3)$$

      How many equilibria are there?
      .
      The equilibria are $E =$
      .

      Need help?

  4. If a dynamical system has parameters, then the equilibria could depend on those parameters. Consider the dynamical system \begin{align*} y_{n+1} - y_n &= y_n + b\\ y_0 &= c \end{align*} with two parameters $b$ and $c$.

    1. If you set $b=5$ and $c=3$, the dynamical system becomes:
      $y_{n+1} - y_n =$

      $y_0 = $
      .
      (Online, enter $y_n$ as y_n.)

      In this case, how many equilibria are there?

      What are the equilibria? $E =$

    2. If you leave $b$ and $c$ as parameters, then the number of equilibria and their values might depend on $b$ and/or $c$.

      In this case, how many equilibria are there?

      What are the equilibria? $E =$

      Need help?
    3. Does the number of equilibria depend on $b$?

      Does the number of equilibria depend on $c$?

      Do the values of the equilibria depend on $b$?

      Do the values of the equilibria depend on $c$?

  5. It's possible that not only the values of the equilibria, but also the number of equilibria could change depending on the values of the parameters. Consider the dynamical system: \begin{align*} q_{t+1}-q_t &= q_t^2 + \gamma\\ q_0 &= \eta. \end{align*} (The parameters are $\gamma$, the Greek letter gamma, and $\eta$, the Greek letter eta.)
    1. If you set $\gamma=-4$, how many equilibria are there?

      What are the values of the equilibria? $E=$

      (If there are more than one equilibrium, separate answers by commas. If there are no equilibria, enter none.)
      Need help?
    2. If you set $\gamma=4$, how many equilibria are there?

      What are the values of the equilibria? $E=$

      (If there are more than one equilibrium, separate answers by commas. If there are no equilibria, enter none.)
    3. If you set $\gamma=0$, how many equilibria are there?

      What are the values of the equilibria? $E=$

      (If there are more than one equilibrium, separate answers by commas. If there are no equilibria, enter none.)
    4. Does the number of equilibria depend on $\gamma$?

      Does the number of equilibria depend on $\eta$?

      Do the values of the equilibria depend on $\gamma$?

      Do the values of the equilibria depend on $\eta$?

  6. For the dynamical system \begin{align*} m_{t+1}-m_t &= a(m_t-b)(m_t-c)(m_t-d)\\ m_0 &= k, \end{align*} for parameters $a$, $b$, $c$, $d$, and $k$, with $a \ne 0$, what are the equilibria?
    $E =$
    Need help?

  7. For the dynamical system \begin{align*} a_{t+1} &= p(a_t-q)(a_t-r)(a_t-s)(a_t-u) + a_t\\ a_0 &= v, \end{align*} for parameters $p$, $q$, $r$, $s$, $u$, and $v$, with $p \ne 0$, what are the equilibria?
    $E=$
    Need help?

  8. As we've seen already, linear dynamical systems are an important type of dynamical system, giving rise exponential growth or decay. We'll be coming back to them again and again. Here, let's explore how the equilibria depend on parameters when we add a constant term $b$, obtaining the dynamical system \begin{align*} z_{n+1} - z_n &= az_n + b\\ z_0 &= c \end{align*}

    1. If you set $a=-0.4$ and $b=2$, what are the equilibria?
      $E = $

    2. If you leave $a$ and $b$ as parameters, but require that $a \ne 0$, what are the equilibria? (They might depend on the values of $a$, $b$, and/or $c$.)
      $E=$

      Need help?

    3. As long as $a \ne 0$, does the number of equilibria depend on the parameter $a$?

      On $b$?

    4. One special case that we have avoided so far is when $a=0$. Is your above formula for the equilibrium valid for when $a=0$?
      Why or why not?

      If $a=0$ does the change in the state variable $z$ in each time step depend on the value of $z$?
      The change in $z$ at each time step is
      .

      What if, in addition to $a=0$, we also set $b=0$, then what is the change in $z$ at each time step?
      . In that case, if we start with the initial condition $c=7$, then $z_1= $
      , $z_2 =$
      , $z_3 =$
      . We can conclude that $z_t=7$
      an equilibrium.

      Was there anything special with the value $7$?
      If $a=0$ and $b=0$, can you find any other equilibria?
      In fact, if we start with
      initial condition, the value of $z_t$
      change with time step $t$. In this case, we can conclude that
      number is an equilibrium for the dynamical system. How many equilibria are there?
      (If for some reason, you need to enter the symbol $\infty$ online in an answer blank, you can type oo or the symbol ∞.)

      Need help?

    5. Let's see if anything is different when $a = 0$ but $b \ne 0$. Recall when $a=0$, the change in $z$ at each time step is
      . Now, if $b \ne 0$, we know that at every time step the value of $z$
      change. Can we find any initial condition $c$ for which $z$ stays at that value, i.e. $z_1=c$?
      . Therefore, if $a = 0$ and $b \ne 0$, the dynamical system has how many equilibria?

      Need help?
    6. Just to be repetitive, in this dynamical system written in
      form, the value of $a$ where things get weird is $a =$
      . For that value of $a$, there are either
      equilibria or there are an
      number of equilibria, depending on the value of $b$.
      Need help?

  9. Let's repeat this exercise using function iteration form rather than the difference form of the previous problem. We are going to be mean and use the same parameter value $a$, even though this $a$ has a different meaning than it did above. We'll write the dynamical system as \begin{align*} u_{n+1} &= au_n + b\\ u_0 &= c \end{align*} so it looks awfully similar to the last problem. But, as you'll see, this parameter $a$ acts differently.
    1. If you set $a=0.6$ and $b=2$, what are the equilibria?
      $E = $

      (As always, separate multiple answers by commas.)
    2. If you leave $a$ and $b$ as parameters, but require that $a \ne 1$, what are the equilibria? (They might depend on the values of $a$, $b$, and/or $c$.)
      $E=$
    3. As long as $a \ne 1$, does the number of equilibria depend on the parameter $a$?

      On $b$?
    4. One special case that we have avoided so far is when $a=1$. Is your above formula for the equilibrium valid for when $a=1$?
      Why or why not?

      If $a=1$, to calculate the value of the state variable $u$ at the next time step, you just need to add
      to its previous value.

      What if, in addition to $a=1$, we also set $b=0$? If we start with the initial condition $c=7$, then $u_1= $
      . We can conclude that $u_n=7$
      an equilibrium.

      Was there anything special with the value $7$?
      If $a=1$ and $b=0$, then
      number is an equilibrium for the dynamical system. How many equilibria are there?

    5. Let's see if anything is different when $a = 1$ but $b \ne 0$. Can we find any initial condition $c$ for which $u$ stays at that value, i.e. $u_1=c$?
      . Therefore, if $a = 1$ and $b \ne 0$, the dynamical system has how many equilibria?
      Need help?
    6. Just to be repetitive, in this dynamical system written in

      form, the value of $a$ where things get weird is $a =$
      . For that value of $a$, there are either
      equilibria or there are an
      number of equilibria, depending on the value of $b$.
      Need help?

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Math 1241, Fall 2020