Lyapunov Exponents & Sensitive Dependence on Initial Conditions

[@megan7258]

this is so well explained! as a high school student, this video makes me so excited to study this at uni

[@ProfessorRoss]

That's excellent, Megan! I also got interested in nonlinear dynamics and chaos in high school, mostly by reading the book "Chaos" by James Gleick www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0143113453

[@PunmasterSTP]

How have your studies been going?

[@megan7258]

@@PunmasterSTP great! I’m really enjoying all my classes. Especially seeing the applications of calculus in my physics courses

[@PunmasterSTP]

@@megan7258 That's awesome!

[@DoNotChooseBlank]

6 months ago he will be a giga chad in 7 years@@megan7258

[@CEO_Bogwater]

I've been using your playlist to study for my mechanics final for the last few days now. I heard you talking about how unreliable the weather is and thought to myself, "Yeah, it's hard to predict the weather at my university too!" Only to find out that you are a professor at the school I go to! That was a fun realization! Hello from VT Physics lol.

[@ProfessorRoss]

Ha! That's pretty funny. I like the name EigenVecna -- are you a Stranger Things fan? If you want to take the course, I'll be teaching it this fall 2023 as AOE 4514 / ESM 4114 Nonlinear Dynamics and Chaos. I'll probably be using a different book, so it won't be the same.

[@pajeetsingh]

Simple explanation. Might be because you have better understanding than others.

[@ProfessorRoss]

Thank you. I hope so. Lyapunov exponents are sick.

[@PunmasterSTP]

Lyapunov? More like "Loving these lectures, all!" 👍

[@brendawilliams8062]

Thankyou

[@victoraguiar3489]

Grreat lecture, Shane. I can't really catch how you obtained the 10^10, though. I am assuming that lambda and *a* are constants. Cheers,

[@404QAMAR]

Dear Prof I required MATLAB code for the Henon map for plotting it's maximum Lyapunov exponents in two and three dimensions

[@aidancoletta7153]

New to this idea and im tryna understand better: when you say that the lyapunov exponent is .9 for the Lorenz system, I'm a little confused. If you're closer to the trivial equilibrium point of the Lorenz system at , shouldn't the lyapunov exponent be negative for some initial conditions?

[@ProfessorRoss]

Great question. The Lyapunov exponent is independent of initial location; it's determined from following any initial condition for long enough. Even when you're near the equilibrium point at , the dynamics will still take the state away from zipping around, and eventually going onto the strange attractor, where it will wander chaotically forever. (I'm assuming we're talking about a parameter r in the regime where the strange attractor exists).

[@smftrsddvjiou6443]

What I always wounder, for lamda(t) so exponent as function of time, it should becomde negative at some point, as the trajectories come closer together. Or ? So this concept is really only valid for initial delta almost zero ?

[@jagg_Z]

thanks so much

[@ProfessorRoss]

You're welcome!

[@natorgas]

Hello, is it possible to calculate the largest lyapunov exponent for a system that is chaotic but doesn't have an attractor?

[@ProfessorRoss]

Excellent question. The answer is yes. For example, the dynamics of the solar system, or any set of bodies interacting by gravity, can be chaotic, and if you calculate the spectrum of the Lyapunov exponents, the largest will be positive, but there is no attractor- motion can go on forever, but never settles down to a lower dimensional surface in the state space.

[@natorgas]

@@ProfessorRoss Thanks for the answer! So that means that it will keep growing forever? Since there are no boundaries in phase space(which before were the attractor)?

[@brendawilliams8062]

That

[@DoNotChooseBlank]

this looks like math in a whole new level 💀