We will consider the family of maps
We can see this is a homeomorphism of R because the derivative is everywhere in the interval [21,+∞). Another nice property of the map is that fc′(0)=1 for all c. Also f′ is increasing so that this is the only point where fc′ is zero.
# This function returns the map f_c.deff(c):m(x)=1/2*(e^x+x-c)returnm
1/2*x + 1/2*e^x - 1/2
# The identity mapidentity(x)=x
A bifurcation occurs at the value c=1. Here we plot some nearby values
# Plot of f_0.9plot(f(0.9),-1,1,aspect_ratio=1)+plot(identity,color="red")
# Plot of f_1plot(f(1),-1,1,aspect_ratio=1)+plot(identity,color="red")
# Plot of f_1.1plot(f(1.1),-1,1,aspect_ratio=1)+plot(identity,color="red")
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∈R, limn→+∞fcn(x)=+∞. That is, Ws(+∞)=R.
At the value c=1: The map f1 has a single fixed point, f1(0)=0. For values of x<0, we have limn→+∞f1n(x)=0. For values of x>0, we have limn→+∞f1n(x)=+∞. That is,
At values of c>1: The map fc has two fixed points, denote them by a and b with $a
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 fc. First, let us compute the fixed points.
Observe that the x value of a fixed point uniquely determines the c value: