In case deterministic chaos isn’t enough you, this post adds in something extra: a little bit of randomness. Rather than making things more complicated, this actually makes them smoother. If you’ve read the What is Chaos? series, you know that finding periodic orbits is important to understand chaos. The randomness allows you to determine how many periodic orbits you need to make predictions.
This is the final post for my explanation of chaos theory to a popular audience. When the motion is chaotic, it is impossible to make long time predictions for a particular trajectory, but it is possible to make long time statistical predictions. I hope to explain the basic ideas of how we can calculate these long time averages for a strange attractor.
This is Part VII of our now eight part series on chaos. The final goal is in sight: since we cannot make predictions for a single trajectory if there’s chaos, we should try to make statistical predictions instead. We cannot even approach this less ambitious goal directly. Instead, we first divide the space the chaos moves through into qualitatively different regions, each of which is labeled by some symbol. We then record the motion as a list of symbols of the regions it goes through. Doing this allows us to finally get a good definition of ‘chaos’: motion where the number of qualitatively different behaviors increases exponentially in time. The ‘topological entropy’ helps us make that definition more precise. Next week we will finally reach our goal.
The next part of my seven part explanation of chaos theory to a popular audience. Chaos is a mechanism that allows deterministic objects to behave unpredictably. I will explain why this happens and what kinds of predictions we can make when something is chaotic. So far, we have discussed mostly phenomenology. I would present some model – either physical or mathematical – and then describe its behavior. Now, instead of just describing the behavior, it’s time to understand how this behavior arises. What is the basic mechanism creating the strange behavior of chaotic systems? Why do some systems exhibit sensitive dependence on initial conditions? How can similar initial conditions rapidly become dramatically different, without everything flying apart? By now you should have seen that these sorts of behavior do happen. But how do they happen? The underlying mechanism is the title of this section: stretch and fold.
The next part of my seven part explanation of chaos theory to a popular audience. Chaos is a mechanism that allows deterministic objects to behave unpredictably. I will explain why this happens and what kinds of predictions we can make when something is chaotic. So far, we have focused on systems that change in discrete time. Part V is about how to describe something that moves chaotically in continuous time.
The next part of my seven part explanation of chaos theory to a popular audience. Chaos is a mechanism that allows deterministic objects to behave unpredictably. I will explain why this happens and what kinds of predictions we can make when something is chaotic. Part IV is about strange attractors. If you allow something to wander chaotic for a while, what kinds of patterns does it make? Surprisingly, the answer is often a fractal.
The next part of my seven part explanation of chaos theory to a popular audience. Chaos is a mechanism that allows deterministic objects to behave unpredictably. I will explain why this happens and what kinds of predictions we can make when something is chaotic. Part III will be about Lyapunov exponents. If you search for a definition of what chaos theory is, the most common result is that chaos occurs whenever there is a positive Lyapunov exponent. While I would prefer defining chaos to be anything with positive topological entropy, Lyapunov exponents are extremely important. They explain what we mean when we say that something ‘behaves unpredictably’.
I chose the name The Chaostician for this blog. To live up to this name, this blog should be about chaos, in particular, what a physicist means when referring to ‘chaos theory’. Let’s begin with an example. Something known as the logistic map is (arguably) the simplest chaotic system. We will now describe and analyze it in detail.
I chose the name The Chaostician for this blog. To live up to this name, this blog should be about chaos, in particular, what a physicist means when referring to ‘chaos theory’. This seven part series is my explanation of chaos theory to a popular audience. Chaos is a mechanism that allows deterministic objects to behave unpredictably. I will explain why this happens and what kinds of predictions we can make when something is chaotic.
Entropy is a weird thing. It is used in multiple seemingly unrelated contexts. None of them have intuitive definitions. I hope to simply explain what entropy is in these contexts and why they all deserve to have the same word.
