Stability of equilibria of a differential equation

Math 2241, Spring 2023
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Due date: April 26, 2023, 11:59 p.m.
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Total points: 1

Consider the dynamical system \begin{align*} \diff{ z }{t} &= f(z) \end{align*} where the function $f$ is graphed below. All zeros of the function are integers.

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Equilibria:
Number of equilibria:
Stability of equilibria:
1. Estimate the value of all equilibria and indicate their location on the graph.
  Equilibria: $z_e=$
  
  Enter in increasing order, separated by commas.
  
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  Hint
  Online, specify number of equilibria with $n_e$. Then, drag the red points to the locations on the graph corresponding to the equilibria. (Although the points have two coordinates, $(x,y)$, the values of the equilibria are just their first coordinates.)
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2. The equilibria divide the state space of $z$ into four intervals. Draw a horizontal arrow on the graph to show the direction of change of $z$ in each interval. Based on these arrows, determine the stability of each equilibrium. Indicate the stability of the equilibrium on the graph by using an open symbol for unstable and a closed symbol for stable.
  
  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.
  
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  Hint
  Online, click “show directions” to reveal line segments between the equilibria. Then, click each line segment until the arrow points in the direction determined by the sign of $\diff{ z }{t}$ (i.e., the sign of $f$.)
  
  An equilibrium will be stable if the arrows point toward it and unstable if the arrows point away from it. Online, click an equilibrium to switch its representation from a closed to an open symbol.
  
  In the stability answer blank, if, for example, there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable.
  
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3. At each stable equilibrium, what is the sign of the slope $f'(z)$?
  
  At each unstable equilibrium, what is the sign of the slope $f'(z)$?
4. Summarize your results in a “phase line” plot. Below is a horizontal $z$-axis, which represents the state space of the state variable $z(t)$. At each stable equilibrium, draw a filled in circle. At each unstable equilibrium, draw an open circle. In the intervals between equilibria, draw the “vector field,” representing the evolution of the system, i.e., draw leftward or rightward arrows that show the direction of motion.
  
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  equilibria:
  number of equilibria:
  stability of equilibria:
  vector field:
  
  Need help?Hide help
  
  Hint
  Online, specify the number of equilibria with $n_e$, then drag the red circles to the locations of the equilibria. Click the red circles to change between open and closed symbols. Click the line segments between the equilibria to display the vector field. Click again to change the direction of the vector field.
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5. Now, create a plot of the solution, i.e., a graph of $z$ versus $t$. First, plot the equilibrium solutions (as they are constant, they will be horizontal lines). Draw stable equilibria as solid lines and unstable equilibria as dashed lines. Then, plot approximate solutions to the differential equation with initial conditions $z(0) = -7.5, -6.5, 0, 3, 4.5$.
  
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  Equilibria:
  Final values of curves:
  Initial conditions of curves:
  Speed profiles of curves:
  Stability of equilibria:
  
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  Hint
  Online, specify the number of equilibria with $n_e$. Draw the pink points so the lines are in the correct locations. Click to change between solid (stable equilibrium) and dash (unstable equilibrium) lines.
  To draw additional solutions that are not constant, increase $n_c$ to the desired number of solutions. Drag the point on the vertical axis to specify the initial condition. Click each curve to alternate its speed profile between speeding up, speeding up then slowing down, and slowing down. Finally, move the second point on each curve to specify the final value of that solution.
  
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For the dynamical system \begin{align*} \diff{ v }{t} &= \left(10 v^{2} + 8 v\right) e^{- v}, \end{align*} find all equilibria and analytically determine their stability (i.e., use the stability theorem). Using this information, draw a phase line diagram with equilibria and vector field. Use solid circles for stable equilibria and open circles for unstable equilibria.
Equilibria:

Enter 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.

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equilibria:
number of equilibria:
stability of equilibria:
vector field:

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Hint
If rounding, include at least 5 significant digits.
Online, for the phase line, drag the slider $n_e$ 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.

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Consider the dynamical system \begin{align*} \diff{ w }{t} &= 3 \left(w + 1\right) \left(w + 0\right) \left(w - 1\right)\\ w(0) & = w_0, \end{align*} where $w_0$ is an initial condition. Find all equilibria and analytically determine their stability. Use this information to draw a phase line diagram, including equilibria and vector field (using solid circles for stable equilibria and open circles of unstable equilibria). Then, sketch a graph of the solutions corresponding to the equilibria (using solid lines for stable equilibria and dashed lines for unstable equilibria) and solutions for initial conditions $w_0 = -5.5, -0.5, 0.5, 3.5$.
Equilibria: $w_e=$

Entering in increasing order.

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.

The phase line:

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equilibria:
number of equilibria:
stability of equilibria:
vector field:

A plot of solutions, including equilibria, versus time:

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Equilibria:
Final values of curves:
Initial conditions of curves:
Stability of equilibria:

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Hint
Online, for the phase line, specify the number of equilibria with $n_e$, then drag the red circles to the locations of the equilibria. Click the red circles to change between open and closed symbols. Click the line segments between the equilibria to display the vector field. Click again to change the direction of the vector field.
Online, for the plot versus time, specify the number of equilibra with $n_e$. Since equilibrium solutions are constant, they appear has horizontal lines. Drag the pink points to the locations of the equilibria, and click to change between solid and dashed lines. Drag $n_c$ to specify the number of additional solution curves, drag the points on the vertical axis to specify initial conditions. Click the curves to specify their speed profile, which will alternate among speeding up, speeding up then slowing down, and slowing down. Drag the right points to determine the ending values of the curves.

Since for this example, we don't have a plot of the function determining the speed $\diff{ w }{t}$, it's hard to know whether or not the solution is first speeding up before slowing down, so it doesn't matter what speed profile you use, as long as the curve starts and ends in the right places.

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Math 2241, Spring 2023

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Math 2241, Spring 2023