Math Insight

Elementary discrete dynamical systems problems, part 2

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  1. For the following discrete dynamical system \begin{align*} w_{ n+1 } - w_n &= 0.06(w_n -4)(w_n -2)(w_n -3)\\ w_0 &= 9, \end{align*} determine the equilibria and their stability.

    To save calculations, you can use the fact that $$\left(w - 4\right) \left(w - 3\right) \left(w - 2\right)=w^{3} - 9 w^{2} + 26 w - 24.$$

  2. Consider the dynamical system \begin{align*} x_{ t+1} &= (b+1) x_t -\frac{ b x_t^2 }{ 10000 } \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $b$ is a nonzero parameter.

    The system has two equilibria. What are they?

    For each equilibrium, determine the range of $b$ for which the equilibrium is stable.

  3. For the following discrete dynamical system \begin{align*} y_{ n+1 } &= 3.4y_n\left(1 -\frac{ y_n }{ 7 }\right) \\ y_0 &= 4.2, \end{align*} find the equilibria and determine their stability analytically. Then, on the plot below, cobweb near each equilibrium to graphically verify your conclusions about stability.

  4. Consider the dynamical system \begin{align*} z_{ t+1} - z_t &= 1.4 z_t\left(1-\frac{ z_t }{ N }\right) \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $N$ is a positive parameter.

    Find all equilibria and determine their stability.

    Does the stability of any of the equilibria depend on the value of $N$?

  5. For the following discrete dynamical system \begin{align*} x_{ t+1 } &= h(x_t)\\ x_0 &= 10, \end{align*} where $h(x) = - 0.02 x^{3} - 0.06 x^{2} + 1.2 x$, the equilibria are $E=-5$, $E=0$, and $E=2$. For each equilibrium, determine the stability.

  6. For the discrete dynamical system \begin{align*} w_{ n+1 } - w_n &= 0.3w_n^2 -0.21w_n -4.182\\ w_0 &= -6, \end{align*} find the equilibria and determine their stability analytically. Then, on the plot below, cobweb near each equilibrium to graphically verify your conclusions about stability.

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Math 1241, Fall 2025