# Math Insight

### Problem set: Determining stability by cobwebbing linear approximations around equilibria

Math 1241, Fall 2020
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Due date: Oct. 23, 2020, 11:59 p.m.
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Total points: 3
1. The following questions concern the model of equation (1) from cobwebbing linear approximation page: \begin{align*} z_{n+1} = a z_n, \quad \text{for $n=0,1,2,\ldots$} \end{align*} where $a$ is a parameter. The cobwebbing linear approximation page will give you instructions for answering these questions via cobwebbing.
1. What condition on the parameter $a$ will ensure there is exactly one equilibrium? $a \ne$

Given that condition, what is the equilibrium? $E=$

The value of this equilibrium
depend on the value of the parameter $a$.

Under what conditions are there more than one equilibrium? $a =$

Are there any conditions for which there are no equilibria?

2. How does the stability of the equilibria depend on the value of the parameter $a$?
The equilibrium is stable for
$\lt a \lt$

The equilibrium is unstable for $a \gt$
or $a \lt$

In both these cases, the one equilibrium is $E=$

We have two special cases that are missed by the above two conditions.

• For $a=$
, the solution spirals around the one equilibrium at $E=$
. Since the solution
get further away from the equilibrium, we still say the equilibrium is
.
• For $a=$
, there are an infinite number of equilibria. However, since the solution
move for any initial condition, the solution
get further away from any equilibrium. Therefore, we still say each equilibrium is
.
3. How does the behavior of the dynamical system depend on the sign of $a$?
When $a$ is negative, the solution
the equilibrium.
When $a$ is positive, the solution
the equlibrium.

2. The following questions concern the model of equation (2) from cobwebbing linear approximation page. \begin{align*} y_{n+1} = b y_n + c, \quad \text{for $n=0,1,2,\ldots$} \end{align*} where $b$ and $c$ are parameters. The cobwebbing linear approximation page will give you instructions for answering these questions via cobwebbing.
1. What condition on the parameter $b$ will ensure there is exactly one equilibrium? $b \ne$

Given that condition, what is the equilibrium? $E=$

The value of this equilibrium
depend on the value of the parameter $b$.
The value of this equilibrium
depend on the value of the parameter $c$.

Under what conditions are there more than one equilibrium? $b =$
and $c=$

Under what conditions are there no equilibria? $b =$
and $c \ne$

2. How does the stability of the equilibria depend on the values of the parameter $b$ and $c$?
The equilibrium is stable for
$\lt b \lt$

The equilibrium is unstable for $b \gt$
or $b \lt$

The stability of the equilibrium
depend on the value of $c$.
In these cases, the one equilibrium is $E=$

We have some special cases that are missed by the above conditions.

• For $b=$
, the solution spirals around the one equilibrium at $E=$
. Since the solution
get further away from the equilibrium, we still say the equilibrium is
.
• For $b=$
and $c=$
, there are an infinite number of equilibria. However, since the solution
move for any initial condition, the solution
get further away from any equilibrium. Therefore, we still say each equilibrium is
.
• Since for $b=$
and $c \ne$
, there are no equilibria, it doesn't make sense to discuss stability.
3. How does the behavior of the dynamical system depend on the sign of $b$? On the sign of $c$?
When $b$ is negative, the solution
the equilibrium.
When $b$ is positive, the solution
the equilibrium.
The behavior of the solution
depend on the sign of $c$

3. The following questions concern the model of equation (3) from cobwebbing linear approximation page. \begin{align*} x_{n+1} = f(x_n) \quad \text{for $n=0,1,2,\ldots$} \end{align*} for function $f(x)$. The cobwebbing linear approximation page will give you instructions for answering these questions via cobwebbing.
1. Is it possible for a nonlinear system $x_{n+1}=f(x_n)$ to have zero equilibria?
One equilibrium?
Two or more equilibria?
Graphically, how can you tell how many equilibria there are? You look for the number of
between the graph of
and the graph of the
.
2. Assume that the derivative of some function $f(x)$ at $x=c$ is $f'(c)=m$. Given that information, write down the formula for the linear approximation (or tangent line) of $f(x)$ calculated at $x=c$.
$L_c(x)=$
3. If we want to analyze the behavior of the dynamical system near an equilibrium $x=E$, we should look at the tangent line to $f$ evaluated at $x=$
. If we assume that $f'(E)=a$, write down the formula for this linear approximation.
$L_E(x)=$
4. Since we can approximate $f$ by its tangent line near the equilibrium, what conditions on the slope $f'(E)=a$ of the tangent line will give you a stable equilibrium?
The equilibrium $E$ will be stable if
$\lt f'(E) \lt$
.

What conditions on the slope of the tangent line will give you an unstable equilibrium?
The equilibrium $E$ will be unstable if $f'(E) \gt$
or $f'(E) \lt$
.

To reiterate, the critical quantity is the slope of the tangent line or the
of $f$ evaluated at $x=$
.