Math Insight

Quiz 3

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Total points: 8
  1. For the discrete dynamical system $z_{ t+1} = f(z_t)$, the graph of $f$ is shown below (thick green curve) along with the diagonal (thin black line).

    Determine the equilibria of the dynamical system. Enter the equilibria in increasing order, separated by commas. All equilibria are integers.

    Equilibria:

  2. For the dynamical system \begin{align*} a_{ t+1} -a_t &= q(a_t-d)(a_t-s)(a_t-r) \\ a_0 &= m, \end{align*} for parameters $q$, $d$, $s$, $r$, and $m$, with $q \ne 0$, what are the equilibria?

    Equilibria:
    If there is more than one equilibrium, separate answers by commas.

  3. For the discrete dynamical system $s_{ t+1} = f(s_t)$, the graph of $f$ is shown below (thick green curve) along with the diagonal (thin black line).

    Determine the equilibria of the dynamical system. Enter the equilibria in increasing order, separated by commas. All equilibria are integers.

    Equilibria:

    For each equilibrium, use cobwebbing to determine the stability of the equilibrium. Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

    Equilibria are stable:

    For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

  4. Find all equilibria to the discrete dynamical system below. If there are multiple equilibria, separate your answers by commas. \[ \left\{ \begin{array}{r c l} y_{ t +1} & = & y_t ^2 -3y_t + 4 \\ y_0 & = & 40\\ \end{array} \right. \]

    $E = $