Cycles and fixed points

Okay kiddo, let's talk about cycles and fixed points. Imagine you're playing on a swing set. When someone pushes you, you go back and forth, or back and forth and higher and higher, right? That's kind of what a cycle is.

In math, a cycle is something that repeats itself over and over again. It's like a pattern that moves in a loop. We can see cycles in different parts of life, like the water cycle or the seasons.

Now, what about fixed points? Well, let's use the example of a spinning top. When a top spins, it moves in a circle around its pointy tip, but that tip stays in the same spot, right? That point is called a fixed point.

In math, a fixed point is basically a number that doesn't change when we use an equation or rule on it. It's like a point that's stuck in place. We can find fixed points in different equations or formulas that we use in math class.

Sometimes, cycles and fixed points can work together. For example, let's say we have a function that moves things in a circle. If we start at a fixed point, the function will keep moving around in a loop, and we'll end up back at that same fixed point again. That's a cycle with a fixed point!

So, in summary, cycles and fixed points both have to do with how things move or stay the same. A cycle is something that repeats in a loop, and a fixed point is something that doesn't change when we use an equation or rule on it. Sometimes, they can work together to create a cycle with a fixed point.