Chaotic Dynamical Systems

This video introduces chaotic dynamical systems, which exhibit sensitive dependence on initial conditions. These systems are ubiquitous in natural and engineering systems, from turbulent fluids to the motion of objects in the solar system. Here, we discuss how to recognize chaos and how to numerically integrate these systems.

@eigensteve on Twitter

This video was produced at the University of Washington

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0:00 Overview of Chaotic Dynamics
8:49 Example: Planetary Dynamics
14:31 Example: Double Pendulum
19:12 Flow map Jacobian and Lyapunov Exponents
26:33 Symplectic Integration for Chaotic Hamiltonian Dynamics
33:41 Examples of Chaos in Fluid Turbulence
37:16 Synchrony and Order in Dynamics

[juliogodel]

A whole series of 60hrs is necessary here :) Thank you!

[StankyPickle]

Yes! More videos on Sympletic and Variational integrators please! 🙏

[ajred]

I’m a high schooler taking calculus 3, and I love your videos. In college I am planning on combining computer science and physics, and your videos solidified this idea 😊

[hoseinzahedifar]

Thank you very much for this interesting lecture...

Homework (Excercise):
In time 27:14: you should write down double pendulum equations of motion (from the Euler-Lagrange or Hamiltonian equations), and use the (classic) RK4 integration scheme for the problem to compute the trajectory of the double pendulum system. Utilizing the trajectory, calculate the system's energy and plot it versus time.

[javadrahmannezhad]

Amazing definition of chaotic systems with very interesting example videos. Thank you for the energy you put into your lectures.

[PeterHavelock]

Fascinating lecture. I was one of the last people to walk across the Millennium Bridge before the police stopped people crossing. I did not see anyone all all fours but it was certainly alarmingly wobbly.

[xskrat]

A few screen-shots have been saved to the directory: But a great wrapper on this series! Thank you!

[michaeljmcguffin]

This is some of the best content on youtube. Thank you for making these!!

[mintakan]

This was a great overview video. I was blown away by the metronomes synchronizing. Was scratching my head, on what would be the math, that can explain the eventual trajectory. Also, lots of other examples, relevant to space flight, such the (chaotic) dance amongst planetary bodies.

[rushabhyeshwante]

Thank you for explaining chaos with videos.

[sarahcorvidae]

This was a great journey - I enjoyed this course and learned a lot! Thank you and looking forward to more!

[TranquilSeaOfMath]

Really nice video. That 3rd generation double pendulum was amazing!

[raktimpal]

Learnt a lot. Thank you for the great lecture. 🙏

[iuriblancos]

Amazing! Thank you for sharing such a valuable material on youtube.

[YangmeiLin-humg]

Thank you so much for these fantastic lectures! 🎊🎊🎊

[pablo_CFO]

I really need a complete series about Lagrangian Coherent Structures.
Do it as a Christmas miracle 😅

[joehopfield]

As an old gymnast the double pendulum and controlling chaos with minimal energy is ... everything.

[alexbraun]

Merry Christmas...ANd thank you so much for your clear and concise lectures

[indiablackwell]

OMG, at 33:00 I actually horrified for a second that Steve was doing a commercial

[AirAdventurer]

Fascinating video! 1) Interesting to see you mention Jerry Marsden. 2) Have you ever communicated with Jim Yorke? He is a co-author of the paper that introduced the term "chaotic differential equation" and gave a lecture at UWM involving chaos in the double-pendulum. 3) Settle a bet: have you ever learned the Lebesgue integral in your studies? (I'll explain the "bet" later if you like.)