Consider the nonlinear differential equation $$5\frac{d^2y}{dt^2}+4\left|\frac{dy}{dt}\right|\frac{dy}{dt}+4 y=0.$$ instead of the more common linear harmonic oscillator equation.

Note that the sign of the term $$\left|\frac{dy}{dt}\right|\frac{dy}{dt}$$ is the same as the sign of $\displaystyle \frac{dy}{dt}$ so, as with the standard harmonic equation, our equation models a damping force that is always directed opposite the direction of motion.

The difference between this model and others is the size of the damping for small and large velocities.

One of the many examples of situations for which this is a better model than linear damping is the drag on airplane tires from wet snow or slush. Drag from only four inches of slush was enough to cause the 1958 crash during take-off of the plane carrying the Manchester United soccer team. Currently large airplanes are allowed to take off and landin no more than one-half inch of wet snow or slush. [Stanley Stewart, Air Disasters, Barnes and Noble, 1986, as cited in Blanchard, Devaney, Hall, 2012.]

Below is a phase-plane plot for this differential equation. $v(t)=y'(t)$ and therefore, according to the formula for the differential equation, $$v'(t) = y''(t)=-\frac{4}{5}y-\frac{4}{5}\lvert v \rvert v.$$ If $y(t)$ is a solution to the differential equation then the curve $\big(y(t),y'(t)\big) = \big(y(t),v(t)\big)$ is tangent to the vector field, so one can find solutions by looking for curves that are tangent to the vector field.

Examine solutions using the phase plane below. Click on the phase plane to graph solutions $\big(y(t),v(t)\big)$; the clicked point is the initial point $\big(y(0),v(0)\big)$. You may change the range of $t$ that you wish to plot, then push the "Redraw ..." button to implement the change.

Then answer the questions below by writing your answers in the boxes below the questions. Remember to click the "Submit answers" button when you are done!

Set new bounds for $t$: min = , max = with min $\leq$ 0 $\leq$ max.\}

Redraw using the new bounds Clear the solution curves

Questions:

Are solutions to this system periodic? Explain your answer in terms of the phase plane plot.

If there are periodic solutions, choose a particular initial condition and estimate the period of the corresponding solution to two decimal places. Explain how you found this value.

Do all solutions have the same period? How can you decide that using solutions plotted in the phase plane?

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Write a paragraph summarizing the observations you made. Your summary should include a description of the behavior of the solutions for this system for different initial conditions, and compare similarities and differences between distinct solutions.

Adapted from Differential Equations, 4th Ed., Blanchard, Devaney, Hall, 2012.