Normally hyperbolic invariant manifold

Okay kiddo, let me explain something called the "normally hyperbolic invariant manifold" to you.

Imagine you're in a big swimming pool with a bunch of floaties. Some of the floaties are really bouncy and push you away if you get too close, while others are squishy and you can float right through them.

Now, let's say there's a really bouncy floatie in the pool and you want to avoid it. You could swim around it, but that would take a long time. Instead, you notice that there's a path that runs alongside the floatie where the water is calm and doesn't push you away. You swim along that path and make it safely to the other side without bouncing off the floatie.

This path is like a "manifold," which is just a fancy word for a path that follows the rules of a certain system. In this case, the rules are the bounciness of the floaties.

But what if the bouncy floatie is moving around the pool? You could keep following the original path, but it might not be safe anymore if the floatie moves into your way. So instead, you notice that there are lots of paths that are always safe to follow, no matter where the floatie is. These paths are like "invariant manifolds," because they stay the same even if the system (the pool and floaties) changes.

Now, let's add another layer. Imagine that the bouncy floatie is also wavy, so that it pushes you away in some directions but not others. It could be really hard to find a safe path to follow in this system, because the floatie is unpredictable.

But scientists have discovered that there is often one special path that is always safe, no matter how wavy and bouncy the floatie gets. This is the "normally hyperbolic invariant manifold," or NHIM for short. It's called "hyperbolic" because it bends and stretches in a certain way, and "normally" because it's the usual path that happens in many systems.

So, in summary, a normally hyperbolic invariant manifold is a special path that always stays safe to follow, even when the system it's in is changing and unpredictable. It's like finding a secret way around the pool where you never have to worry about bouncing off a floatie, no matter how bumpy the water gets. Isn't that cool?