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Errors occurred in the following questions: 1

Discrete dynamical system stability practice

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  1. Error in question

    The following errors occurred when rendering the question:

    • Error in expression: Esort1_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort1) < 1]
    • Error in expression: Esort2_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort2) < 1]
    • Error in expression: Esort3_stability
      Invalid format for expression: [unstable,stable][Abs(fprime_Esort3) < 1]
    For the following discrete dynamical system \begin{align*} w_{ n+1 } &= f(w_n)\\ w_0 &= 3, \end{align*} where $f(w) = - 0.08 w^{3} + 0.56 w^{2} + 0.44 w - 1.2$, the equilibria are $E=-1$, $E=3$, and $E=5$. For each equilibrium, determine the stability. In each blank, enter stable if an equilibrium is stable or unstable if it is unstable.

    Stability of equilibria $E=-1$:

    Stability of equilibria $E=3$:

    Stability of equilibria $E=5$:

  2. Consider the dynamical system \begin{align*} u_{ t+1} - u_t &= b u_t\left(1-\frac{ u_t }{ 9000 }\right) \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? Enter them in increasing order.

    Equilibrium 1: $u = $

    Equilibrium 2: $u = $

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

    Equilibrium 1 is stable for
    $\lt b \lt$

    Equilibrium 2 is stable for
    $\lt b \lt$

  3. For the discrete dynamical system \begin{align*} x_{ n+1 } - x_n &= - 0.07 x_{n}^{3} + 1.75 x_{n}\\ x_0 &= -9, \end{align*} determine the equilibria and their stability.

    Equilibria =
    (If more than one equilibrium, enter in increasing order, separated by commas.)

    Stability of equilibria:

    Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

    For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

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Math 201, Spring 2015