Bifurcation diagram of a sin map(plotting it)

[Oijl]

Homework Statement

Make a bifurcation diagram for the sin map x{sub t+1} = f(x{sub t}) where f(x) = rsin(pi*x).

Homework Equations

The Attempt at a Solution

A bifurcation diagram... I have read and understood what a bifurcation diagram is, and how it is constructed, but to construct one myself I think I find myself a bit muddled.

I am having trouble thinking up the actual formulas I'd have to write if I were plotting one of these...

I understand that this is pretty general question, but how would I set about writing out a bifurcation diagram, mathamatically, since I am needing to put it into a program that will graph it for me?

 

[anathema]

Hi there,

To construct a bifurcation diagram for the given sin map, we first need to understand the behavior of the map as the parameter r varies. This will give us an idea of how the system changes and bifurcates as we alter the value of r.

To begin, let's rewrite the map in the form x{sub t+1} = r*sin(pi*x{sub t}). Now, we can start with a certain initial condition for x{sub t} and iterate the map for different values of r. For each value of r, we can plot the resulting values of x{sub t+1} on the y-axis and the corresponding values of r on the x-axis. This will give us a graph of x{sub t+1} vs. r, which is known as a bifurcation diagram.

To make the diagram clearer, we can also plot the points with different colors or symbols for different values of r. This will help us to identify the regions where bifurcations occur. As we increase r, we will start to see multiple branches of x{sub t+1} appearing, indicating the presence of multiple stable states for the system. These branches will continue to split and form new branches as r increases, leading to a complex and intricate bifurcation diagram.

To summarize, constructing a bifurcation diagram for the given sin map would involve iterating the map for different values of r and plotting the resulting values of x{sub t+1} vs. r. This can be done using a computer program or by hand, depending on the complexity of the system. I hope this helps and happy graphing!