Math Insight

Elementary discrete dynamical systems problems, part 2

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  1. For the discrete dynamical system \begin{align*} x_{ t+1 } - x_t &= 0.4x_t^2 -2.16x_t +1.472\\ x_0 &= 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.

  2. Consider the dynamical system \begin{align*} x_{ n+1} - x_n &= b x_n\left(1-\frac{ x_n }{ 8000 }\right) \quad \text{for $n=0,1,2,3, \ldots$} \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 discrete dynamical system \begin{align*} s_{ n+1 } - s_n &= 0.07 s_{n}^{3} - 0.21 s_{n}^{2} - 0.28 s_{n}\\ s_0 &= -7, \end{align*} determine the equilibria and their stability analytically.

  4. For the following discrete dynamical system \begin{align*} s_{ n+1 } &= h(s_n)\\ s_0 &= 4, \end{align*} where $h(s) = 0.09 s^{3} + 0.27 s^{2} + 0.64 s$, the equilibria are $E=-4$, $E=0$, and $E=1$. For each equilibrium, determine the stability.

  5. For the following discrete dynamical system \begin{align*} y_{ n+1 } &= 3.7y_n\left(1 -\frac{ y_n }{ 5 }\right) \\ y_0 &= 2.5, \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.

  6. Consider the dynamical system \begin{align*} z_{ t+1} - z_t &= 2.2 z_t\left(1-\frac{ z_t }{ K }\right) \quad \text{for $t=0,1,2,3, \ldots$} \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 $K$?

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Math 201, Spring 19