# Math Insight

### Two-dimensional nonlinear problems

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Total points: 10
1. Consider the quadratic system of differential equations: \begin{align*} \diff{ w }{t} &= - w + y^{2} - 3 y + 2\\ \diff{ y }{t} &= - w - y^{2} + 2 y + 2. \end{align*}
1. What is the equation for the $w$-nullcline:

What is the equation for the $y$-nullcline:

2. The system has two equilibria. Calculate them.

The equilibria are $(w_1,y_1)=$
and $(w_2,y_2) =$

(Enter the equilibria in increasing order when ordered by their $y$-component.)

3. On the below phase plane, plot the $w$-nullcline with the thick solid blue curve and the $y$-nullcline with the thin dashed green curve.

Plot the equilibria on the phase plane. (Increase step to 2, increase $n_e$ to reveal equilibria and move them to the correct locations.)

Feedback from applet
step 1: nullclines:
step 2: equilbrium classifications:
step 2: equilibrium locations:
step 3: number of nullcline vectors:
step 3: number of region vectors:
step 3: vector directions in regions:
step 3: vector directions on nullclines:
step 3: vector locations in regions:
step 3: vector locations on nullclines:

(We will classify the equilibria below. Until that point, the equilibrium classification on the phase plane will be marked incorrect.)

4. The nullclines should have divided the phase plane into six regions. In each region, determine the qualitative direction that solutions must move and sketch a direction vector in that region. (Increase step to 3 and increase $n_{rv}$ to reveal vectors for the regions. Move the vectors into the different regions and point them in the correct direction.)
5. Each nullcline should be divided in to three segments by the other nullcline. Plot direction vectors on each nullcline segment indicating the direction that solutions must cross the nullcline. (Increase increase $n_{nv}$ to reveal vectors for the nullcline segments. Move the vectors into the different segments and point them in the correct direction.)
6. Calculate the Jacobian matrix of the system

$D\vc{f}(w,y)=$
7. Evaluate the Jacobian matrix at the first equilibrium $(w_1,y_1)=＿$

$D\vc{f}＿ =$

The eigenvalues are $\lambda_1 =$
and $\lambda_2=$

(Keep at least 5 significant digits when rounding.)

Classify the equilibrium $(w_1,y_1)=＿$:
(enter words)
On the above applet, show the equilibrium classification by clicking on the equilibrium until the correct label appears. (SD=saddle, SN=stable node, UN=unstable node, SS=stable spiral, US=unstable spiral, C=center).

8. Evaluate the Jacobian matrix at the second equilibrium $(w_2,y_2)=＿$

$D\vc{f}＿ =$

The eigenvalues are $\lambda_1 =$
and $\lambda_2=$

(Keep at least 5 significant digits when rounding.)

Classify the equilibrium $(w_2,y_2)=＿$:

On the above applet, show the equilibrium classification by clicking on the equilibrium until the correct label appears.

9. Sketch a plausible solution of the trajectory $(w(t), y(t))$ on the phase plane for the initial condition $(w(0), y(0)) = (1.1,2.0)$. (Any trajectory is fine as long as it is consistent with the direction vectors you have determined. It does not need to match the actual solution of the system.)
10. Sketch the solutions $w(t)$ and $y(t)$ versus time. These curves should correspond to the choice you made in part (i). (Sketching them on the same axis or different axes is fine.)

2. Consider the quadratic system of differential equations: \begin{align*} \diff{ x }{t} &= - u + 2 x^{2} - 3 x - 1\\ \diff{ u }{t} &= - u + 2 x + 2. \end{align*}
1. What is the equation for the $x$-nullcline:

What is the equation for the $u$-nullcline:

2. The system has two equilibria. Calculate them.

The equilibria are $(x_1,u_1)=$
and $(x_2,u_2) =$

(Enter the equilibria in increasing order when ordered by their $x$-component.)

3. On the below phase plane, plot the $x$-nullcline with the thick solid blue curve and the $u$-nullcline with the thin dashed green curve.

Plot the equilibria on the phase plane. (Increase step to 2, increase $n_e$ to reveal equilibria and move them to the correct locations.)

Feedback from applet
step 1: nullclines:
step 2: equilbrium classifications:
step 2: equilibrium locations:
step 3: number of nullcline vectors:
step 3: number of region vectors:
step 3: vector directions in regions:
step 3: vector directions on nullclines:
step 3: vector locations in regions:
step 3: vector locations on nullclines:

(We will classify the equilibria below. Until that point, the equilibrium classification on the phase plane will be marked incorrect.)

4. The nullclines should have divided the phase plane into six regions. In each region, determine the qualitative direction that solutions must move and sketch a direction vector in that region. (Increase step to 3 and increase $n_{rv}$ to reveal vectors for the regions. Move the vectors into the different regions and point them in the correct direction.)
5. Each nullcline should be divided in to three segments by the other nullcline. Plot direction vectors on each nullcline segment indicating the direction that solutions must cross the nullcline. (Increase increase $n_{nv}$ to reveal vectors for the nullcline segments. Move the vectors into the different segments and point them in the correct direction.)
6. Calculate the Jacobian matrix of the system

$D\vc{f}(x,u)=$
7. Evaluate the Jacobian matrix at the first equilibrium $(x_1,u_1)=＿$

$D\vc{f}＿ =$

The eigenvalues are $\lambda_1 =$
and $\lambda_2=$

(Keep at least 5 significant digits when rounding.)

Classify the equilibrium $(x_1,u_1)=＿$:
(enter words)
On the above applet, show the equilibrium classification by clicking on the equilibrium until the correct label appears. (SD=saddle, SN=stable node, UN=unstable node, SS=stable spiral, US=unstable spiral, C=center).

8. Evaluate the Jacobian matrix at the second equilibrium $(x_2,u_2)=＿$

$D\vc{f}＿ =$

The eigenvalues are $\lambda_1 =$
and $\lambda_2=$

(Keep at least 5 significant digits when rounding.)

Classify the equilibrium $(x_2,u_2)=＿$:

On the above applet, show the equilibrium classification by clicking on the equilibrium until the correct label appears.

9. Sketch a plausible solution of the trajectory $(x(t), u(t))$ on the phase plane for the initial condition $(x(0), u(0)) = (0.2,-2.7)$. (Any trajectory is fine as long as it is consistent with the direction vectors you have determined. It does not need to match the actual solution of the system.)
10. Sketch the solutions $x(t)$ and $u(t)$ versus time. These curves should correspond to the choice you made in part (i). (Sketching them on the same axis or different axes is fine.)