Sharkovskii's theorem
Imagine you have a bunch of toys of different colors and sizes. You want to line them all up in a row, but you're not sure how to do it. First, let's start with the smallest toy. You could put it in any spot, so let's just put it at the beginning.
Next, you have a toy that's twice as big as the smallest one. You could put it next to the smallest toy, or you could skip a spot and put it in the third spot.
Now you have a toy that's three times as big as the smallest one. You could put it next to the second toy, or you could skip a spot and put it in the fourth spot.
As you keep adding bigger and bigger toys, you might start to notice a pattern. You can always choose a spot for the next toy based on the size of the previous toys. Sometimes you'll skip a spot, sometimes you won't.
This is basically what Sharkovskii's Theorem says, but instead of toys, we're talking about numbers. The numbers have to be positive, and we're looking at their behavior when they cycle through a function.
The theorem says that if you have a function that maps numbers from a certain range to themselves (like taking the square root of a number between 0 and 1), there is a specific ordering of periods (how long it takes for numbers in the range to cycle through the function) that can occur.
For example, if a number in the range has a period of 3, then there must also be numbers with periods of 3, 6, 9, 12, and so on.
The ordering of periods follows a pattern similar to the one we saw with the toys. It's based on the size of the periods that came before it.
This theorem is important because it helps us understand the chaotic behavior of certain systems, like weather patterns or population growth. It also has applications in physics and engineering.
Next, you have a toy that's twice as big as the smallest one. You could put it next to the smallest toy, or you could skip a spot and put it in the third spot.
Now you have a toy that's three times as big as the smallest one. You could put it next to the second toy, or you could skip a spot and put it in the fourth spot.
As you keep adding bigger and bigger toys, you might start to notice a pattern. You can always choose a spot for the next toy based on the size of the previous toys. Sometimes you'll skip a spot, sometimes you won't.
This is basically what Sharkovskii's Theorem says, but instead of toys, we're talking about numbers. The numbers have to be positive, and we're looking at their behavior when they cycle through a function.
The theorem says that if you have a function that maps numbers from a certain range to themselves (like taking the square root of a number between 0 and 1), there is a specific ordering of periods (how long it takes for numbers in the range to cycle through the function) that can occur.
For example, if a number in the range has a period of 3, then there must also be numbers with periods of 3, 6, 9, 12, and so on.
The ordering of periods follows a pattern similar to the one we saw with the toys. It's based on the size of the periods that came before it.
This theorem is important because it helps us understand the chaotic behavior of certain systems, like weather patterns or population growth. It also has applications in physics and engineering.