Chaos, Poincare sections and Lyapunov exponent

Lecture on Chaos, Poincare sections and Lyapunov exponent by Dr. Andrés Aragoneses (Eastern Washington University). Introduction to chaos through the double pendulum and the Duffing oscillator.

[@workerpowernow]

hi Andres- I was in your EM class in spring of 2017 at Carleton. I found your videos when looking for review materials for my classical mechanics class at university of washington where i'm in the PhD program. Thanks for the helpful content

[@physicswithandresaragonese250]

Hi Carl. Thank you. I'm glad you found them helpful. I'm sure you're doing very interesting physics at UW!, not so far from here, EWU.

[@NicholasPellegrino]

Thanks so much for the awesome video!

[@yourfather1518]

thanks a lot! it helps me a lot about my poor celestial dynamic and modern control theory

[@physicswithandresaragonese250]

For any dynamical system, you just compute momentum as a function of time, and position as a function of time. Then you plot momentum versus position.

[@raphaelziegler7839]

Thank you, this was very helpful and interesting!

[@nicolasalderete6176]

Thanks for your video! Very clear

[@Revan176]

I'm very interested in nonlinear mechanics and chaos theory, but I since I changed my attention to nonlinear Continuumsmechanics, there is a little uneasiness on my mind: In my studies much is said about nonlinear Partial diffenrential Equations having nonliniarities in the Spatial coordinates, but there are usually no considerations about chaotic properties of the equations. Every time chaos just appears in the temporal coordinate of the equation. And I wonder why. I came up with two explanations so far: the first one is, that there is a fundamental difference between boundary conditions and initial conditions, in a case that systems with just imitial conditions could lead into chaos, whereas systems with boundaryconditions can not. The second explanation was, that in material modells one requirement for the constitutive law is, that it matches an locality requirement, so that an action within that material is supposed to result in an reaction which appears in the same lokation of the action. Is one of these explanations correct?

[@Revan176]

Or maybe the explanation is that sensitivities in the boundary conditions, which would be chaotic behaviour in the spatial coordinates of an PDE is in fact what is called a stability problem. Does that makes sence?

[@physicswithandresaragonese250]

@@Revan176 Chaos means that a system is sensitive to the initial conditions, so it is related with time evolution and the non-possibility of making long term predictions. We will see chaos as a system evolves in time and its evolution can take different routes depending on very similar, but different, initial conditions. These systems have constraints, and boundary conditions, but still present chaos. For example, for most chaotic systems they are restricted to a limited region in phase space. They cover almost all of that limited space, but they never visit the same point in phase space, because that would mean the motion is going to be repeated and then periodic, predictable, non chaotic. Of course you can try to quantify the lack of structure in space, not in time, we use entropy for that, and we can also track the evolution of entropy as time evolves. This is not an easy problem, but doable.

[@mandavasairaghavendradines6582]

That's helpful!

[@misbahshahzadi959]

could you please provide the code to plot these poincare section?

[@dawid_dahl]

Thanks a lot for this very interesting video! I have a question. Is it possible for me to say that we will never be able to accurately predict the future because it would require so much computing power that the energy that would be required to power such a computer would create a black hole? If that would be the case, we will always be in for a surprise, even in the future when computers become more powerful. If so, life would always be exciting. Which would be nice.

[@physicswithandresaragonese250]

We will never be able to predict the future for different reasons. In theory we need infinite precision in our measurements. That is impossible from a classical and a quantum point of view. Also, We're part of the universe, so, we need to know ourselves with that infinite precision. But, also, quantum mechanically that's completely forbidden, there's no such thing as the actual position and velocity of a particle. You can have the information spread out an not need to create a black hole. Asa a matter of fact, the information that falls into a black hole is proportional to its surface, but in that field, we need a quantum description of gravity, which we don't have so far.

[@yogeshyadav698]

Isn't he Berlin From Money Heist

[@physicswithandresaragonese250]

I didn't know what you meant until I watch that show this year.