# Math Insight

### Bifurcation of single differential equation practice

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1. For the dynamical system \begin{align*} \diff{ u }{t} = f(u, \beta), \end{align*} where the function $f$ of $u$ also depends on a parameter $\beta$, a bifurcation diagram with respect to the parameter $\beta$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.

To better determine values from the diagram, you can click the “show point” box and move the point around to read off coordinates from different parts of the graph.

1. When $\beta= 0$, how many equilibria are there? Determine their values, rounded to the nearest integer, and their stability.

To report your answers, use the below applet to sketch the phase line for when $\beta= 0$, including equilibria and direction field. Use a solid circle for stable equilibria and an open circle for unstable equilibria. Place circles for equilibria at integer values, using the rounded values of equilibria determined from the bifurcation diagram.

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

Drag the slider labeled $n_e$ to specify the number of equilibria. Drag the red points to the location of the equilibria. You can click on a point to change it between open and solid. Click the segments between equilibria to show a direction field. Clicking the segment again changes the direction of the vectors.

2. When $\beta= 4$, how many equilibria are there? Determine their values, rounded to the nearest integer, and their stability.

To report your answers, use the below applet to sketch the phase line for when $\beta= 4$, including equilibria and direction field. Use a solid circle for stable equilibria and an open circle for unstable equilibria. Place circles for equilibria at integer values, using the rounded values of equilibria determined from the bifurcation diagram. Follow the above instructions to change the phase line applet.

Feedback from applet
equilibria:
number of equilibria:
stability of equilibria:
vector field:
3. When $\beta= 10$, how many equilibria are there? Determine their values, rounded to the nearest integer, and their stability.

To report your answers, use the below applet to sketch the phase line for when $\beta= 10$, including equilibria and direction field. Use a solid circle for stable equilibria and an open circle for unstable equilibria. Place circles for equilibria at integer values, using the rounded values of equilibria determined from the bifurcation diagram. Follow the above instructions to change the phase line applet.

Feedback from applet
equilibria:
number of equilibria:
stability of equilibria:
vector field:
4. Identify any bifurcation points.

Bifurcations points are at $\beta =$
. (If there are multiple bifurcation points, separate the values of $\beta$ by commas.)

2. For the dynamical system $\diff{ v }{t} = g(v,\alpha),$ the function $g$ of $v$ depends on a parameter $\alpha$, as shown in the graphs for $\alpha=-23, -11, -7, -3$, below. For values of $\alpha$ in between those shown, $g$ changes smoothly, so its graph will be somewhere in between the snapshots shown.

$\alpha=-23$

$\alpha=-7$

$\alpha=-11$

$\alpha=-3$

1. There is a bifurcation that occurs for a particular value of $\alpha$, which we'll denote by $\alpha^*$. What is the value of $\alpha^*$?

$\alpha^*=$

2. At the bifurcation point, the number of equilibria changes.

For $-23 \le \alpha < \alpha^*$, the number of equilibria =
.

For $\alpha^* < \alpha \le -3$, the number of equilibria =
.

3. Use the below applet to sketch a bifurcation diagram with respect to the parameter $\alpha$. The bifurcation diagram should represent now the number, location, and stability of the equilibria depend on the value of $\alpha$ for $-23 \le \alpha \le -3$.

Feedback from applet
bifurcation point:
branches of equilibria:
stability of equilibrium branches:

Move the red point to indicate the location of the bifurcation point. Move the two points on the adjoining curves to indicate how the equilibria change as a function of the parameter $\alpha$. Move two points on the line so that the line indicates how the third equilibrium changes a function of the parameter $\alpha$. You can click each curve or line to switch between a solid line (indicating stable equilibrium) and a dashed line (indicating unstable equilibrium).

3. For the dynamical system \begin{align*} \diff{ s }{t} = g(s, \beta), \end{align*} where the function $g$ of $s$ also depends on a parameter $\beta$, a bifurcation diagram with respect to the parameter $\beta$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.

To better determine values from the diagram, you can click the “show point” box and move the point around to read off coordinates from different parts of the graph.

1. When $\beta= -9$, how many equilibria are there? Determine their values, rounded to the nearest integer, and their stability.

Number of equilibria:

Rounded values of equilibria:
. (If there are more than one equilibrium, enter them 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. If there are no equilibria, enter none.

For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

2. When $\beta= 8$, how many equilibria are there? Determine their values, rounded to the nearest integer, and their stability.

Number of equilibria:

Rounded values of equilibria:
. (If there are more than one equilibrium, enter them 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. If there are no equilibria, enter none.

3. When $\beta= 10$, how many equilibria are there? Determine their values, rounded to the nearest integer, and their stability.

Number of equilibria:

Rounded values of equilibria:
. (If there are more than one equilibrium, enter them 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. If there are no equilibria, enter none.

4. Identify any bifurcation points.

Bifurcations points are at $\beta =$
. (If there are multiple bifurcation points, separate the values of $\beta$ by commas.)

4. For the dynamical system $\diff{ x }{t} = g(x,\beta),$ the function $f$ of $x$ depends on a parameter $\beta$, as shown in the phase lines for $\beta=-21, -12, -9, -6$, below.

$\beta=-21$

$\beta=-12$

$\beta=-9$

$\beta=-6$

1. There is a bifurcation that occurs for a particular value of $\beta$, which we'll denote by $\beta^*$. What is the value of $\beta^*$?

$\beta^*=$

2. At the bifurcation point, the number of equilibria changes.

For $-21 \le \beta < \beta^*$, the number of equilibria =
.

For $\beta^* < \beta \le -6$, the number of equilibria =
.

3. Use the below applet to sketch a bifurcation diagram with respect to the parameter $\beta$. The bifurcation diagram should represent now the number, location, and stability of the equilibria depend on the value of $\beta$ for $-21 \le \beta \le -6$.

Feedback from applet
bifurcation point:
branches of equilibria:
stability of equilibrium branches:

Move the red point to indicate the location of the bifurcation point. Move the two points on the adjoining curves to indicate how the equilibria change as a function of the parameter $\beta$. Move two points on the line so that the line indicates how the third equilibrium changes a function of the parameter $\beta$. You can click each curve or line to switch between a solid line (indicating stable equilibrium) and a dashed line (indicating unstable equilibrium).