Math Insight

Stability of equilibria of a differential equation

Math 1241, Fall 2019
Name:
ID #:
Due date: Nov. 29, 2019, 11:59 p.m.
Table/group #:
Group members:
Total points: 3
  1. Consider the dynamical system \begin{align*} \diff{ z }{t} &= f(z) \end{align*} where the function $f$ is graphed below. All zeros of the function are integers.
    Feedback from applet
    Directions:
    Equilibria:
    Number of equilibria:
    Stability of equilibria:
    1. Estimate the value of all equilibria and indicate their location on the graph.

      Equilibria: $z_e=$

      Enter in increasing order, separated by commas.

    2. The equilibria divide the state space of $z$ into four intervals. Draw a horizontal arrow on the graph to show the direction of change of $z$ in each interval. Based on these arrows, determine the stability of each equilibrium. Indicate the stability of the equilibrium on the graph by using an open symbol for unstable and a closed symbol for stable.

      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.

    3. At each stable equilibrium, what is the sign of the slope $f'(z)$?

      At each unstable equilibrium, what is the sign of the slope $f'(z)$?
    4. Summarize your results in a “phase line” plot. Below is a horizontal $z$-axis, which represents the state space of the state variable $z(t)$. At each stable equilibrium, draw a filled in circle. At each unstable equilibrium, draw an open circle. In the intervals between equilibria, draw the “vector field,” representing the evolution of the system, i.e., draw leftward or rightward arrows that show the direction of motion.
      Feedback from applet
      equilibria:
      number of equilibria:
      stability of equilibria:
      vector field:
    5. Now, create a plot of the solution, i.e., a graph of $z$ versus $t$. First, plot the equilibrium solutions, which will be horizontal lines since the $z(t)$ is constant at an equilibrium. (Use the $n_e$ slider to reveal the horizontal lines one at a line.) Draw stable equilibria as solid lines and unstable equilibria as dashed lines. (Click a line to switch between dashed and solid.) Then, plot approximate solutions to the differential equation with initial conditions $z(0) = -7.5, -6.5, 0, 3, 4.5$. (Use the $n_c$ slider to reveal the curves one at a time. Click the curves to change their shape.)
      Feedback from applet
      Equilibria:
      Final values of curves:
      Initial conditions of curves:
      Speed profiles of curves:
      Stability of equilibria:

  2. For the dynamical system \begin{align*} \diff{ v }{t} &= \left(10 v^{2} + 8 v\right) e^{- v}, \end{align*} find all equilibria and analytically determine their stability (i.e., use the stability theorem). Using this information, draw a phase line diagram with equilibria and vector field. Use solid circles for stable equilibria and open circles for unstable equilibria.

    Equilibria:

    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.

    Feedback from applet
    equilibria:
    number of equilibria:
    stability of equilibria:
    vector field:

  3. Consider the dynamical system \begin{align*} \diff{ s }{t} &= 3 \left(s + 3\right) \left(s - 1\right) \left(s - 2\right)\\ s(0) & = s_0, \end{align*} where $s_0$ is an initial condition. Find all equilibria and analytically determine their stability. Use this information to draw a phase line diagram, including equilibria and vector field (using solid circles for stable equilibria and open circles of unstable equilibria). Then, sketch a graph of the solutions corresponding to the equilibria (using solid lines for stable equilibria and dashed lines for unstable equilibria) and solutions for initial conditions $s_0 = -7.5, 0.0, 1.5, 7.0$.

    Equilibria: $s_e=$

    Entering in increasing order.

    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.

    The phase line:

    Feedback from applet
    equilibria:
    number of equilibria:
    stability of equilibria:
    vector field:

    A plot of solutions, including equilibria, versus time:

    Feedback from applet
    Equilibria:
    Final values of curves:
    Initial conditions of curves:
    Stability of equilibria: