The idea of stability of equilibria for discrete systems

Math 201, Spring 22
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The below graph shows the function $f$ for the dynamical system $m_{t+1} = f(m_t)$.

What are the equilibria of the dynamical system? (They are integers; enter them in increasing order.)
$E=$

For each equilibrium, cobweb (by hand or with optional applet) starting with initial conditions just above and below the equilibrium to determine its stability. 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.

Stability of equilibria:

Feedback from applet
Points on diagonal:
Points on function:

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

An equilibrium is stable if, when you start near (but not exactly on) the equilibrium, the solution moves toward the equilibrium (or at least doesn't move away). On the other hand, an equilibrium is unstable if, when you start near (but not exactly on) the equilibrium, the solution moves away from the equilibrium.
If you start cobwebbing near the equilibrium and the trajectory moves toward the equilibrium, then you can conclude that the equilibrium is stable. If, no matter how close you start to the equilibrium, the trajectory moves away from the equilibrium, then you can conclude that the equilibrium is unstable.

It's probably quicker to cobweb this one by hand, but if you like, you can use the applet to cobweb. (The applet is not graded.) It will tell you if the points are the right place but it doesn't affect your score. The applet only lets you cobweb from one initial condition at a time.

Helpful pages
- The idea of stability of equilibria for discrete dynamical systems
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The below graph shows the function $g$ for the dynamical system $q_{t+1} = g(q_t)$.

What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
$E =$

For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. 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.

Stability of equilibria:

Feedback from applet
Points on diagonal:
Points on function:

Need help?Hide help

Hint
You'll want to cobweb four times, above and below each equilibrium. The applet is there only to help you cobweb, if you like. You can also just cobweb by hand. The applet does not affect the number of points awarded for this question.
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The below graph shows the function $f$ for the dynamical system $r_{n+1} = f(r_n)$.

What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
$E =$

For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. 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.

Stability of equilibria:

Feedback from applet
Points on diagonal:
Points on function:

Need help?Hide help

Hint
Determining the stability of the upper equilibrium is a little tricky. You need to cobweb quite carefully to see if it is moving away or moving toward the equilibrium. Although the applet isn't graded, it could be useful to use the applet to cobweb around the upper equilibrium to see what's going on.
Of course, you might just be tempted to guess until the computer tells you what the right answer is (since there are only two possibilities for each equilibrium, it won't take you long to randomly get the right answer). Since that option won't be available to you on an exam, we strongly encourage you to use this opportunity to figure out how to determine stability yourself.

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The below graph shows the function $g$ for the dynamical system $h_{n+1} = g(h_n)$.

What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
$E =$

For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. 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.

Stability of equilibria:

Need help?Hide help

Hint
We didn't include a cobwebbing applet here. To prepare you for an exam, you'd best practice doing some cobwebbing by hand. This problem will give you quite a bit of practice, as there are a lot of equilibria.
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The below graph shows the function $f$ and the diagonal for the dynamical system \begin{align*} p_{n+1} &= f(p_n)\\ p_0 &= p_0. \end{align*}
1. What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
  $E= $
2. For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. 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.
  Stability of equilibria:
3. The basin of attraction of a stable equilibrium is the set of initial conditions $p_0$ for which the solution $p_n$ tends to the equilibrium for large time $n$. For this example, there is one stable equilibrium, it is $E = $
  . For that stable equilibrium, determine its basin of attraction.
  The basin of attraction is all values $p_0$ satisfying
  
  $< p_0 < $
  
  Need help?Hide help
  
  Hint
  To determine the basin of attraction, try cobwebbing from different initial conditions. If the trajectory approaches the equilibrium, then the initial condition is in that equilibrium's basin of attraction. You should find that for all initial conditions near the stable equilibrium, the trajectories approach the equilibrium. But, suddenly, when you get too far away from the equilibrium, the trajectories change their behavior.
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The population of fish $f_t$ in a lake in year $t$ evolve according to the dynamical system \begin{align*} f_{t+1}=g(f_{t}), \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $g$ is graphed by the thick curve. The thin line is the diagonal $f_{t+1}=f_t$.
1. Identify all equilibria of the system. Estimate their values in increasing order.
  $E = $
  
  Getting to nearest 100 is fine. They are all close to multiples of 50.
2. Determine the stability of the equilibria. (On an exam, you'd want to be sure to justify your answer by explaining the method you used.) Only values $f_t \ge 0$ make sense, so for one equilibrium, you need to check just on one side. Enter stability in same order as the equilibria.
  Stability of equilibria:
  
  Need help?Hide help
  
  Hint
  On an exam, you'd want to explain that you cobwebbed starting near the equilibrium the equilibrium and the solution moved toward the equilibrium so it was stable or the solution moved away from the equilibrium so it was unstable. Also, show your cobwebbing.
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3. (Basins of attraction) For what values of the initial population size $f_0$ will the fish population disappear? This tricky. Be sure to check what happens for very large $f_0$.
  The population will disappear if $f_0 < $
  or if $f_0 > $
  
  (Getting to nearest 100 is fine.)
  
  For what values of $f_0$ will the population stay at a relatively large number (we'll call it a healthy population.)
  We will have a healthy population of fish if
  $< f_0 <$
  
  When we have this healthy population, approximately how many fish are there?
  
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  Hint
  The tricky part is the large values of $f_t$. If you start with a large value of $f_0$ (say from all the way on the right of the graph), what is $f_1=g(f_0)$? Then, what happens to $f_2$, $f_3$, etc.? Are you witnessing the effects of overcrowding?
  If, when you start with this very large value of $f_0$, the trajectory approach an equilibrium, then that value of $f_0$ is in the basin of attraction of this equilibrium. If, as you decrease the initial condition $f_0$, the trajectory changes its behavior and approaches the other stable equilibrium, then you've crossed over to the second stable equilibrium's basin of attraction. You need to find the value of $f_0$ where the behavior of the trajectories switch from going to one stable equilibrium to the other stable equilibrium.
  
  Hide help

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Math 201, Spring 22