Math Insight

Elementary discrete dynamical system problems

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  1. Consider the dynamical system \begin{align*} W_{ t+1} -W_t &= \delta W_t \quad \text{for $t=0,1,2,3, \ldots$ } \end{align*} where $\delta$ is a parameter, and $\delta \ge -1$.

    1. Solve the dynamical system to give a formula for $W_t$ in terms of the initial condition $W_0$, the parameter $\delta$, and the number $t$.
    2. Find the equilibria. Hint: treat the case $\delta=0$ separately from the case $\delta \ne 0$.
    3. For each equilibrium, determine for which values of $\delta$ the equilibrium is stable. Hint: treat the case $\delta=0$ separately from the case $\delta \ne 0$.

  2. Consider the discrete dynamical system \[ \left\{ \begin{array}{r c l} w_{ t+1} & = & f(w_t)\\ w_0 & = & 5 \end{array} \right. \] where the function $f$ is graphed along with the diagonal $w_{ t+1} = w_t$, below.

    1. Find the equilibria of the system. Label points corresponding to the equilibria on the graph above and give estimates of their numerical values.

    2. Use cobwebbing to determine the stability of each equilibrium.
    3. For any stable equilibrium, determine the interval around the equilibrium for which initial conditions $w_0$ lead to solutions $w_{ t }$ that approach the equilibrium as the time $t$ increases. In other words, find the largest interval around the equilibrium that is part of the basin of attraction.

  3. Consider the dynamical system \begin{align*} P_{ t +1}-P_t = \frac{(P_t -\lambda)(P_t-\alpha)(P_t-\mu)}{ \xi }, \quad \text{for $t=0,1,2,3, \ldots$ } \end{align*} where $\lambda$, $\alpha$, $\mu$, and $\xi$ are parameters, with $\xi \ne 0$. Find all equilibria.

  4. Consider the discrete dynamical system \[ \left\{ \begin{array}{r c l} x_{ t+1} & = & x_t\left(2.0-x_t\right) \\ x_0 & = & 1.2\\ \end{array} \right. \]

    1. Calculate the equilibria analytically.

      $E=$________

    2. Compute the next three points of the solution.

      $x_0=$________

      $x_1=$________

      $x_2=$________

      $x_3=$________

    3. On the graph below, label the equilibria and cobweb the dynamical system for four steps.

  5. Consider the dynamical system \begin{align*} P_{ n+1} - P_n &= \frac{ P_n(P_n -2)}{ 4 } \quad \text{for $n=0,1,2,3, \ldots$ .} \end{align*}

    1. Find the equilibria.
      1. If $P_0 = -0.01$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
      2. If $P_0= 0.01$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
      3. Based on the above calculations, what can you infer about the stability of one of the equilibria?
      1. If $P_0 = 1.99$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
      2. If $P_0= 2.01$, calculate $P_1$, $P_2$, $P_3$, and $P_4$.
      3. Based on the above calculations, what can you infer about the stability of one of the equilibria?

  6. Solve the dynamical system \begin{align*} r_{ t+1} - r_t &= 0.9 r_t\\ r_0 &= 72.6. \end{align*}

  7. What is an equilibrium of a dynamical system? What does it mean for an equilibrium to be stable? To be unstable?

  8. Consider the dynamical system \begin{align*} p_{ t+1} - p_t &= \frac{ a p_t +k }{ c }, \quad \text{for $t=0,1,2,3, \ldots$}\\ p_0 &= m \end{align*} where $a$, $k$, $c$, and $m$ are parameters, and $c \ne 0$. Find all equilibria.