Consider the differential equation $$m\frac{d^2y}{dt^2}+(y^2-\alpha)\frac{dy}{dt}+ky=0$$ where the usual damping coefficient "$b$" is replaced with the factor $y^2-\alpha$.

Suppose $m=5$, $\alpha=2$ and $k=4$; that is, we'll focus on the equation: $$5\frac{d^2y}{dt^2}+(y^2-2)\frac{dy}{dt}+4 y=0,$$ a nonlinear, second-order, ordinary differential equation with constant coefficients.

Below is a phase-plane plot of the vector field $\big(y'(t),v'(t)\big)$ in the $y,v$-plane, where $v$ is defined by the equation $$y'(t) = v(t)$$ and therefore, according to the formula for the differential equation, $$v'(t) = y''(t) = -\frac{4}{5}y-\frac{1}{5}(y^2-2)v.$$ 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.

(The grapher seems to choke sometimes when $t<0$, perhaps because $\big(y(t),v(t)\big)$ has jumped too far away from the screen. You can fix this by setting "min" to be a small negative number like -0.1 .)

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:

Is it reasonable to interpret $y^2-2$ as some type of damping? Why or why not?

Provide a complete description of the long-term behavior of the solutions. Are the solutions periodic? If so, what does the period seem to be?

Explain why this equation is not a good model for something like a mass-spring system.

Give an example of some other type of physical or biological system that could be modeled by this equation.

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