Bifurcation diagram

Do you know how when you keep adding more and more seed to a flower pot, the plant grows bigger and bigger? If you keep adding seeds at a certain speed, the plant will keep growing until it reaches a point where it cannot grow anymore. This is similar to what happens in a bifurcation diagram.

A bifurcation diagram is a way to display the behavior of a system over time as a parameter changes. Think of a system as a plant pot and the parameter as the seeds you add to the pot. As you add more seeds to the plant pot, the bifurcation diagram shows how the behavior of the system changes.

For example, let's look at a pendulum swinging back and forth. The parameter could be the length of the string holding the pendulum. As the length of the string changes, the bifurcation diagram will show how the behavior of the pendulum changes - it might start swinging at a different speed, or in a different direction.

The bifurcation diagram is usually represented as a graph where the parameter is plotted on the horizontal axis, and the behavior of the system is plotted on the vertical axis. Every point on the graph represents a possible behavior of the system.

As you move along the horizontal axis, you might notice that the graph starts to split into different branches, like a tree with many branches. This is where the term "bifurcation" comes from. Each branch represents a different behavior of the system, and the point where each branch starts is called a "bifurcation point."

So, to summarize, a bifurcation diagram is like a graph that shows how a system's behavior changes as a parameter is changed - like adding more seeds to a flower pot. And just like how a plant pot can only handle so many seeds before it can't grow anymore, a system can only handle so much change before its behavior drastically alters.