Bifurcations in homeomorphisms of $\mathbb R$.

We will consider the family of maps $$f_c(x) = \frac{1}{2} (e^x + x - c).$$ We can see this is a homeomorphism of $\mathbb R$ because the derivative is everywhere in the interval $[\frac{1}{2},+\infty)$. Another nice property of the map is that $f_c'(0)=1$ for all $c$. Also $f'$ is increasing so that this is the only point where $f'_c$ is zero.

A bifurcation occurs at the value $c=1$. Here we plot some nearby values

A bifurcation is a sudden change in the dynamics as we change the parameters of a family of dynamical systems. In this case, a bifurcation occurs at the value $c=1$:

• For values of $c<1$: For every $x \in \mathbb R$, $\lim_{n \to +\infty} f_c^n(x)=+\infty$. That is, $W^s(+\infty)=\mathbb R$.

• At the value $c=1$: The map $f_1$ has a single fixed point, $f_1(0)=0$. For values of $x<0$, we have $\lim_{n \to +\infty} f_1^n(x)=0$. For values of $x>0$, we have $\lim_{n \to +\infty} f_1^n(x)=+\infty$. That is, $$W^s(0)=(-\infty,0] \quad \text{and} \quad W^s(+\infty)=(0,+\infty).$$

• At values of $c>1$: The map $f_c$ has two fixed points, denote them by $a$ and $b$ with $a<b$. The point $a$ is an attracting fixed point while $b$ is repelling. We have $$W^s(a)=(-\infty,b) \quad \text{and} \quad W^s(+\infty)=(b,+\infty).$$

Visualizing the maps through a vector field.

We can visualize this bifurcation in the $(x,c)$ plane, where dynamics in the horizontal line of height $c$ represent the action of $f_c$. First, let us compute the fixed points.

Observe that the $x$ value of a fixed point uniquely determines the $c$ value:

Since $c$ is a parameter, it is constant under iteration. We define the map $$F(x,c) = \big(f_c(x), c\big).$$

We can visualize $F$ as a vector field. At each point $(x,c)$, we join $(x,c)$ to its image $F(x,c)$ by a displacement vector with value $F(x,c)-(x,c)$. We just compute this to be: