Math Insight

Nonlinear oscillation problems

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Total points: 10
  1. The Fitzhugh-Nagumo equations \begin{align*} \diff{v}{t} &= 1.1 v \left(v - 0.3\right) \left(1 - v\right) - w + I\\ \diff{w}{t} &= 0.04 \left(0.5 v - w\right) \end{align*} model the (normalized) voltage $v$ of a neuron and a recovery variable $w$. The goal for this problem is is to show that, for the above chosen parameters, the model will fire repetitively when we set the external current to $I=0.2$.
    1. First, calculate the nullclines of the model.

      $v$-nullcline:

      $w$-nullcline:

      Plot the nullclines on the following phase plane.

      Feedback from applet
      Step 1: nullclines:
      Step 2: equilibria locations:
      Step 2: equilibria types:
      Step 2: number of equilibria:
      Step 3: correct directions:
      Step 3: different regions:
      Step 4: correct directions:
      Step 4: different branches:
      Step 5: bottom boundary:
      Step 5: inner boundary location:
      Step 5: inner boundary size:
      Step 5: left boundary:
      Step 5: right boundary:
      Step 5: top boundary:

      When the step slider is at 1, you can drag the points to move the nullclines. Click each curve to change which nullcline it represents.

      (You won't be able to get full credit from the applet for now. Instructions for the remaining steps of the applet are below.

    2. Write down an equation for equilibria in terms of the voltage $v$.

      This equation should be a cubic equation for $v$. Use a computer program (or use the cubic formula if you are adventurous!) to solve for the three roots of the equation.

      $v_{ss} =$

      Separate your answers by commas. Include at least 5 significant digits in your response.

      Of these answers, how many are real?
      . Hence, the system has only
      equilbrium, which is
      $(v_{ss}, w_{ss}) =$

      Include at least 5 significant digits in your response.

      Plot the equilibrium on the above phase plane. It should confirm the value that you calculated. (Change the slider to step=2. Specify the number of equilibria with the $n_e$ slider.

    3. Calculate the Jacobian of the system.


      $J(v,w)=$




      Calculate the Jacobian at the equilibrium.



      $J＿=$




      Calculate the eigenvalues of the Jacobian evaluated at the steady state.
      $\lambda =$

      Separate answers by commas, include at least 5 significant digits in your response.

      Classify the equilibrium. It is a

      (one word per blank). Label the equilibrium in the above phase plane.

    4. Complete the above phase plane by drawing direction vectors in each region of the phase plane divided by the nullclines and on each segment of the nullclines. (Use steps 3 and 4 in the applet.)
    5. Use the Poincare-Bendixson theorem to prove the existence of a stable limit cycle. In the above applet, draw the annulus by changing step to 5 and moving the red points so that the red region satisfies the conditions of the Poincare-Bendixson theorem.

  2. Consider the system of differential equations: \begin{align*} \diff{ w }{t} &= - w^{2} - 5 y\\ \diff{ y }{t} &= c \left(- 4 w^{2} + 12 w - 5 y + 15\right), \end{align*} where $c$ is a positive parameter.
    1. What are the equations for the nullclines?

      $w$-nullcline:

      $y$-nullcline:

      Draw the nullclines on the phase plane. Use the thick sold blue line for the $w$-nullcline and the thin dashed green line for the $y$-nullcline.

      Feedback from applet
      step 1: nullclines:
      step 2: equilibrium locations:
    2. Since for this example, you can remove the parameter $c$ from the nullclines, the equilibria won't depend on $c$. Independent of the value of $c$, what are the equilibria?

      $(w_{eq}^1, y_{eq}^1) =$

      $(w_{eq}^2, y_{eq}^2) =$

      (Order the equilibria by their first component.) Draw the equilibria on the above phase plane. (Increase step to 2, increase $n_e$ to reveal red points for the equilibria, and move the equilibria to the correct locations.)

    3. Calculate the Jacobian matrix at an arbitrary point $(w,y)$.


      $J(w,y)=$




    4. Calculate the Jacobian matrix at the first equilibrium $(w_{eq}^1, y_{eq}^1) = ＿$.


      $J＿=$




      What is the determinant of the Jacobian at this equilibrium?
      $\det J＿=$

      What is the trace of the Jacobian at this equilibrium?
      $\tr J＿=$

      Given that $c \gt 0$, the equilibrium $＿$ is stable for
      and unstable for
      . (In each blank, enter an inequality involving $c$, assuming $c \gt 0$. In the case where the stability doesn't change for all $c \gt 0$, enter “all c” and “no c” in the appropriate blanks.)

    5. Calculate the Jacobian matrix at the second equilibrium $(w_{eq}^2, y_{eq}^2) = ＿$.


      $J＿=$




      What is the determinant of the Jacobian at this equilibrium?
      $\det J＿=$

      What is the trace of the Jacobian at this equilibrium?
      $\tr J＿=$

      Given that $c \gt 0$, the equilibrium $＿$ is stable for
      and unstable for
      . (In each blank, enter an inequality involving $c$, assuming $c \gt 0$. In the case where the stability doesn't change for all $c \gt 0$, enter “all c” and “no c” in the appropriate blanks.)

    6. One of the equilibria, $(w_{eq}, y_{eq}) =$
      , undergoes a bifurcation at $c = $
      .

      When $c = ＿$, the eigenvalues of the Jacobian $J＿$ are $\lambda = $
      . (Separate answers by a comma. If rounding, include at least 5 significant digits.)

      Since these eigenvalues are
      and the equilibrium changes stability at $c = ＿$, the equilibrium undergoes a
      bifurcation at $c = ＿$.

    7. Based on the above results, the ＿ bifurcation theorem tells us that the system must have a
      around the equilibrium $＿$ near $c=＿$.

      There are two possible types of ＿ bifurcations.

      • If the system undergoes a supercritical ＿ bifurcation at $c=＿$, then there will be a small
        limit cycle surrounding the equilibrium when $c$ is
           $＿$.
      • If the system undergoes a subcritical ＿ bifurcation at $c=＿$, then there will be a small
        limit cycle surrounding the equilibrium when $c$ is
           $＿$.
    8. Simulate the system to determine which of the two options from the ＿ bifurcation theorem actually occurs. It turns out that the bifurcation is a

      bifurcation.

    9. Draw a bifurcation diagram summarizing your conclusions.

      Feedback from applet
      equilibrium location:
      equilibrium stability:
      Hopf bifurcation location:
      Hopf bifurcation type:
      limit cycle stability:
      limt cycle location:

      Drag the blue points so that the blue line segments trace out how the $w$-component of the equilibria change with $c$. To specify how the stability of each equilibrium changes with $c$, click each segment of the equilibrium lines to switch between solid (stable equilibrium) and dashed (unstable equilibrium).

      To specify the Hopf bifurcation and limit cycles, increase step to 2. Drag the red point to the location of the Hopf bifurcation. Clicking the point changes between supercritical and subcritical Hopf bifurcations. Drag the maroon points so that the thick maroon curve approximates the minimum and maximum values of a limit cycle for each value of $c$. The actual size and shape of these limit cycle curves is not important. It just must have the correct shape relative to the equilibria and the Hopf bifurcation point.

  3. Consider the system of differential equations: \begin{align*} \diff{ v }{t} &= - 3 v^{2} + 16 v - 5 w - 1\\ \diff{ w }{t} &= s \left(10 v - 5 w - 1\right), \end{align*} where $s$ is a positive parameter.
    1. What are the equations for the nullclines?

      $v$-nullcline:

      $w$-nullcline:

      Draw the nullclines on the phase plane. Use the thick sold blue line for the $v$-nullcline and the thin dashed green line for the $w$-nullcline.

      Feedback from applet
      step 1: nullclines:
      step 2: equilibrium locations:
    2. Since for this example, you can remove the parameter $s$ from the nullclines, the equilibria won't depend on $s$. Independent of the value of $s$, what are the equilibria?

      $(v_{eq}^1, w_{eq}^1) =$

      $(v_{eq}^2, w_{eq}^2) =$

      (Order the equilibria by their first component.) Draw the equilibria on the above phase plane. (Increase step to 2, increase $n_e$ to reveal red points for the equilibria, and move the equilibria to the correct locations.)

    3. Calculate the Jacobian matrix at an arbitrary point $(v,w)$.


      $J(v,w)=$




    4. Calculate the Jacobian matrix at the first equilibrium $(v_{eq}^1, w_{eq}^1) = ＿$.


      $J＿=$




      What is the determinant of the Jacobian at this equilibrium?
      $\det J＿=$

      What is the trace of the Jacobian at this equilibrium?
      $\tr J＿=$

      Given that $s \gt 0$, the equilibrium $＿$ is stable for
      and unstable for
      . (In each blank, enter an inequality involving $s$, assuming $s \gt 0$. In the case where the stability doesn't change for all $s \gt 0$, enter “all s” and “no s” in the appropriate blanks.)

    5. Calculate the Jacobian matrix at the second equilibrium $(v_{eq}^2, w_{eq}^2) = ＿$.


      $J＿=$




      What is the determinant of the Jacobian at this equilibrium?
      $\det J＿=$

      What is the trace of the Jacobian at this equilibrium?
      $\tr J＿=$

      Given that $s \gt 0$, the equilibrium $＿$ is stable for
      and unstable for
      . (In each blank, enter an inequality involving $s$, assuming $s \gt 0$. In the case where the stability doesn't change for all $s \gt 0$, enter “all s” and “no s” in the appropriate blanks.)

    6. One of the equilibria, $(v_{eq}, w_{eq}) =$
      , undergoes a bifurcation at $s = $
      .

      When $s = ＿$, the eigenvalues of the Jacobian $J＿$ are $\lambda = $
      . (Separate answers by a comma. If rounding, include at least 5 significant digits.)

      Since these eigenvalues are
      and the equilibrium changes stability at $s = ＿$, the equilibrium undergoes a
      bifurcation at $s = ＿$.

    7. Based on the above results, the ＿ bifurcation theorem tells us that the system must have a
      around the equilibrium $＿$ near $s=＿$.

      There are two possible types of ＿ bifurcations.

      • If the system undergoes a supercritical ＿ bifurcation at $s=＿$, then there will be a small
        limit cycle surrounding the equilibrium when $s$ is
           $＿$.
      • If the system undergoes a subcritical ＿ bifurcation at $s=＿$, then there will be a small
        limit cycle surrounding the equilibrium when $s$ is
           $＿$.
    8. Simulate the system to determine which of the two options from the ＿ bifurcation theorem actually occurs. It turns out that the bifurcation is a

      bifurcation.

    9. Draw a bifurcation diagram summarizing your conclusions.

      Feedback from applet
      equilibrium location:
      equilibrium stability:
      Hopf bifurcation location:
      Hopf bifurcation type:
      limit cycle stability:
      limt cycle location:

      Drag the blue points so that the blue line segments trace out how the $v$-component of the equilibria change with $s$. To specify how the stability of each equilibrium changes with $s$, click each segment of the equilibrium lines to switch between solid (stable equilibrium) and dashed (unstable equilibrium).

      To specify the Hopf bifurcation and limit cycles, increase step to 2. Drag the red point to the location of the Hopf bifurcation. Clicking the point changes between supercritical and subcritical Hopf bifurcations. Drag the maroon points so that the thick maroon curve approximates the minimum and maximum values of a limit cycle for each value of $s$. The actual size and shape of these limit cycle curves is not important. It just must have the correct shape relative to the equilibria and the Hopf bifurcation point.

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Math 5447, Fall 2022