I am having trouble understanding the concept of bifurcation. Can someone help me understand $ 1b $ and $ 1c $ in this example? I understand to the point that in order to find the bifurcation, I need to solve $ ax + \sin x = 0, $ but I don't understand the next paragraph:
The bifurcation diagram are the solutions of $ \displaystyle a + \frac{\sin x}{x} = 0, $ which are plotted as the blue and red curves. It shows how as a departs from $ a = 0 $ and moves to $ |a| = 1, $ there are fewer and fewer rest points that such that sources and sinks cancel pairwise as $ |a| $ increases. For $ a > −1 $ near $ a = −1, $ there are only three rest points which collapse to one in a pitchfork bifurcation at $ x = 0 $ and $ a = −1. $ After $ |a| \ge 1 $ there is only one rest point at $ 0. $
At a bifurcation point, two or more of the solution curves parametrized by $a$ meet. Which means that there is a multiple solution. Which means that additionally to $f_a(x)=0$ one also has $f_a'(x)=0$ satisfied. Here that means that simultaneously $$ 0=ax+\sin(x)\\0=a+\cos(x) $$ Eliminating $a$ gives the equation $x=\tan(x)$ and every solution $x^*$ of that gives a bifurcation point with $a^*=-\cos(x^*)$.