TITLE: Equilibrium points of a family of differential equations
QUESTION [1 upvotes]: Give an example of a family of differential equations $x'=f_a(x)$ for which there are no equilibrium points if $a<0$; a single equilibrium point if $a=0$ and no equilibrium points if $a>0$.

I have an  example of differential equations but it only has a single equilibrium point if $a=1/4$,2 equilibrium points if $a<1/4$ and 0 equilibrium points if $a>1/4$.
$x'=x^2+y, y'=x-y+a$, 
The example above can help to solve the first exercise? If yes, how?
I feel lost, any kind of help will be very appreciated.

REPLY [2 votes]: How about the family of differential equations on $\Bbb R$:
$\dot x = x^2 + a^2 = f_a(x); \tag 1$
for $a \ne 0$, 
$\dot x > 0, \; \forall x \in \Bbb R; \tag 2$
for $a = 0$, the equation (1) reduces to
$\dot x = x^2, \tag 3$
which has precisely one equilibrium at $x = 0$.
A similar example in two dimensions is given by
$f_a(x, y) = \begin{pmatrix} y^2 + a^2 \\ x^2 + a^2 \end{pmatrix}, \tag 4$
that is,
$\dot x = y^2 + a^2, \tag 5$
$\dot y = x^2 + a^2. \tag 6$
Not totally sure at the moment how your example system might help with your problem, though you might be able to modify it so that it has no zeroes when $a < 1/4$, whilst leaving the number of of equilibria for values of $a \ge 1/4$ undisturbed.
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