Math Insight

Elementary discrete dynamical systems problems, part 2

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  1. For the following discrete dynamical system \begin{align*} x_{ t+1 } &= f(x_t)\\ x_0 &= -5, \end{align*} where $f(x) = 0.02 x^{3} + 0.04 x^{2} + 0.94 x$, the equilibria are $E=-3$, $E=0$, and $E=1$. For each equilibrium, determine the stability.

  2. For the discrete dynamical system \begin{align*} z_{ t+1 } - z_t &= -0.3z_t^2 +1.2z_t -0.333\\ z_0 &= -1, \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.

  3. For the discrete dynamical system \begin{align*} z_{ n+1 } - z_n &= - 0.07 z_{n}^{3} - 0.42 z_{n}^{2} - 0.56 z_{n}\\ z_0 &= 7, \end{align*} determine the equilibria and their stability analytically.

  4. For the following discrete dynamical system \begin{align*} z_{ t+1 } &= g(z_t)\\ z_0 &= -4, \end{align*} where $g(z) = - 0.09 z^{3} + 0.09 z^{2} + 2.53 z + 1.35$, the equilibria are $E=-3$, $E=-1$, and $E=5$. For each equilibrium, determine the stability analytically. Then, on the below graph, sketch tangent lines to $g$ at each equilibrium and cobweb near each equilibrium to graphically verify your conclusions about stability.

  5. Consider the dynamical system \begin{align*} w_{ n+1} - w_n &= 1.7 w_n\left(1-\frac{ w_n }{ K }\right) \quad \text{for $n=0,1,2,3, \ldots$} \end{align*} where $K$ is a positive parameter.

    Find all equilibria and determine their stability.

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

  6. Consider the dynamical system \begin{align*} x_{ t+1} - x_t &= r x_t\left(1-\frac{ x_t }{ 8000 }\right) \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $r$ is a nonzero parameter.

    The system has two equilibria. What are they?

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