If you set $a=-0.4$ and $b=2$, what are the equilibria?
$E = $
If you leave $a$ and $b$ as parameters, but require that $a \ne 0$, what are the equilibria? (They might depend on the values of $a$, $b$, and/or $c$.)
$E=$
Hint
Just repeat your calculation from last time, but use $a$ rather than $-0.4$ and $b$ rather than $2$. If you want an intermediate step, try repeating the calculation with $a=3$ and $b=2$.
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As long as $a \ne 0$, does the number of equilibria depend on the parameter $a$?
On $b$?
One special case that we have avoided so far is when $a=0$. Is your above formula for the equilibrium valid for when $a=0$?
Why or why not?
If $a=0$ does the change in the state variable $z$ in each time step depend on the value of $z$?
The change in $z$ at each time step is
.
What if, in addition to $a=0$, we also set $b=0$, then what is the change in $z$ at each time step?
. In that case, if we start with the initial condition $c=7$, then $z_1= $
, $z_2 =$
, $z_3 =$
. We can conclude that $z_t=7$
an equilibrium.
Was there anything special with the value $7$?
If $a=0$ and $b=0$, can you find any other equilibria?
In fact, if we start with
initial condition, the value of $z_t$
change with time step $t$. In this case, we can conclude that
number is an equilibrium for the dynamical system. How many equilibria are there?
(If for some reason, you need to enter the symbol $\infty$ online in an answer blank, you can type oo or the symbol ∞.)
Hint
The change in the state variable $z$ in a time step is $z_{n+1}-z_n$.
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Let's see if anything is different when $a = 0$ but $b \ne 0$. Recall when $a=0$, the change in $z$ at each time step is
. Now, if $b \ne 0$, we know that at every time step the value of $z$
change. Can we find any initial condition $c$ for which $z$ stays at that value, i.e. $z_1=c$?
. Therefore, if $a = 0$ and $b \ne 0$, the dynamical system has how many equilibria?
Hint
If you want to start with a concrete example, set $a=0$ and $b=1$ to see what you get.
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