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Errors encountered

Errors occurred in the following questions: 3, 4, 6

Elementary discrete dynamical systems problems, part 2

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  1. 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 $K$?

  2. 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.

  3. Error in question

    The following errors occurred when rendering the question:

    • Error in expression: Esort1_x02
      Invalid format for tuple: Esort1+[dx1,dx2][Abs(fprime_Esort1) < 1]
    • Error in expression: Esort2_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort2_x01
      Invalid format for tuple: Esort2-[dx1,dx2][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort1_x01
      Invalid format for tuple: Esort1-[dx1,dx2][Abs(fprime_Esort1) < 1]
    • Error in expression: Esort2_x02
      Invalid format for tuple: Esort2+[dx1,dx2][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort1_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort1) < 1]
    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.

  4. Error in question

    The following errors occurred when rendering the question:

    • Error in expression: Esort3_x02
      Invalid format for tuple: Esort3+[dx1,dx2][Abs(fprime_Esort3) < 1]
    • Error in expression: Esort1_x02
      Invalid format for tuple: Esort1+[dx1,dx2][Abs(fprime_Esort1) < 1]
    • Error in expression: Esort2_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort1_x01
      Invalid format for tuple: Esort1-[dx1,dx2][Abs(fprime_Esort1) < 1]
    • Error in expression: Esort3_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort3) < 1]
    • Error in expression: Esort2_x01
      Invalid format for tuple: Esort2-[dx1,dx2][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort2_x02
      Invalid format for tuple: Esort2+[dx1,dx2][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort3_x01
      Invalid format for tuple: Esort3-[dx1,dx2][Abs(fprime_Esort3) < 1]
    • Error in expression: Esort1_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort1) < 1]
    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*} 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.

  6. Error in question

    The following errors occurred when rendering the question:

    • Error in expression: Esort3_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort3) < 1]
    • Error in expression: Esort2_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort1_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort1) < 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.

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Math 201, Spring 2015