[HN Gopher] Strange Attractors
___________________________________________________________________
Strange Attractors
Author : T-A
Score : 87 points
Date : 2022-11-23 09:51 UTC (13 hours ago)
(HTM) web link (syntopia.github.io)
(TXT) w3m dump (syntopia.github.io)
| IIAOPSW wrote:
| Quick intro for those who don't know Chaos theory or what a
| strange attractor is.
|
| In the most general abstract sense suppose you have a system
| which is in some state and there is a deterministic rule which
| updates it to the next state. For now we can just represent the
| states and transitions as a bunch of circles and arrows
| connecting them. Draw it on a whiteboard if you must. Since
| transitions are deterministic, each state only transitions to a
| single other state. There might be two states that lead to one,
| but never one that leads to two. In general then, no matter which
| state you start in, it must either eventually lead to some closed
| loop of the same states over and over again or it runs off the
| page in an infinite succession of states. The first option is
| what we would call an "attractor" and the latter which could be
| aptly called a "repulsor". There's a third possibility I didn't
| mention (which is hard to conceptualize drawing things as a 2d
| diagram of states and transitions). You can have the state run
| off in an infinite succession of states, but rather than running
| off the page to infinity, it goes around and around visiting
| similar states over and over but never technically visiting the
| same state twice. Its a strange attractor.
|
| Now to be less general. If your system states are some continuous
| parameter, instead of circles and arrows it makes sense to
| represent the nth state as x_n and to represent the transition
| rule to the next state as a difference equation x_{n+1} = f(x_n).
| If your system has more than one defining parameter, do the same
| thing but with vector valued quantities. Lastly, if your
| transitions themselves are continuous rather than discrete,
| replace the difference equation with a differential equation
| dx/dt = f(x). No matter which of these you are using to represent
| system states and define how that system updates, the same basic
| patterns as before are still the only possibility. The solution
| to your equation either converges to some fixed point, diverges
| to infinity (usually exponential growth), or the curve stays
| bounded within some region of state space getting arbitrarily
| close to forming a loop but always just missing itself.
| ouid wrote:
| you've described a computer, it seems to me that you mean to
| put everything in some kind of normed vector space.
| pineconewarrior wrote:
| I had no idea what a strange attractor was before reading this
| and seeing this 3d demo. Between the two, I feel like I
| understand it quite well now.
|
| Thanks for the writeup!
| briga wrote:
| Great work on the visualization, looks fantastic! Impressive how
| far web graphics have advanced.
| q_revert wrote:
| despite having a phd in nonlinear dynamics, I struggle to wrap my
| head around most of this stuff... but I always found is deeply
| fascinating.
|
| In one system I studied we had a sequence of period doublings,
| which followed one of the feigenbaum constants[1]
|
| seeing this happen in the lab, and subsequently seeing it
| numerically, and uncovering the analytics behind it was a worthy
| use of 6 years
|
| --- [1] https://en.wikipedia.org/wiki/Feigenbaum_constants
| daniel-thompson wrote:
| This reminds me of Chaoscope^. It reached EOL a long time ago and
| the binaries might have succumbed to bitrot by now, but it was a
| lot of fun back in the day and produced some beautiful
| visualizations based on strange attractors.
|
| ^ http://www.chaoscope.org/index.htm
| gatane wrote:
| Is there a book about this? This isnt covered that deep in
| Dif.Eq. books...
| rikroots wrote:
| Not a book, but Paul Bourke's website - Fractals, Chaos, Self-
| Similarity - has a wealth of information.
| http://paulbourke.net/fractals/
| lanstin wrote:
| The Topology of Chaos, https://www.amazon.com/Topology-Chaos-
| Alice-Stretch-Squeezel...
|
| is a good modern book on it.
|
| It is a fun problem because it stems from continuous diff eq
| but the answers come from topological analysis around the fixed
| sets, and the ways fixed sets can change as continuous
| parameters change.
| peapicker wrote:
| "Strange Attractors: Creating Patterns in Chaos" by Julien C.
| Sprott.
|
| I am lucky enough to have a 1st edition print, but there is now
| a free PDF online: https://sprott.physics.wisc.edu/SA.HTM
| squaredot wrote:
| Maybe the most famous is the Lorenz attractor, arising from the
| Lorenz System (this is one of the simulations shown in the
| website). In Wikipedia you find a description of the system
|
| https://en.wikipedia.org/wiki/Lorenz_system
| bmitc wrote:
| * _Chaos and Dynamical Systems_ by David Feldman
|
| * _Exploring Chaos: Theory and Experiment_ by Brian Davies
|
| * _Chaos: From Theory to Applications_ by A.A. Tsonis
|
| * _Chaos and Fractals: An Elementary Introduction_ by David
| Feldman
|
| * _Nonlinear Dynamics And Chaos_ by Tufillaro, Abbott, Reilly
|
| * _Computers, pattern, chaos, and beauty: Graphics from an
| unseen world_ by Clifford Pickover
|
| * _A First Course In Chaotic Dynamical Systems_ by Robert
| Devaney
|
| Just some references from my "to read" collection.
| kkylin wrote:
| James Gleick's "Chaos" remains, in my view, a great popular
| book on the subject, and manages to convey many of the ideas at
| a conceptual level without getting technical. And if you enjoy
| that, you might also like Strogatz's "Sync."
|
| For slightly more technical treatment, Strogatz's Nonlinear
| Dynamics and Chaos is now a standard text. It isn't terribly
| technical and is quite well written and (in my view) easy to
| read for anyone with a background in vector calculus, diff eqs,
| & perhaps a little bit of linear algebra. (There are now other
| good options to, some more mathematical and some more
| application-oriented, but I still think Strogatz is a good
| place to start.)
| azepoi wrote:
| Can you expand on the last part? I mean the good other
| technical books for someone already a bit familiar who wants
| a deeper dive? Thanks for the pointers
| edbaskerville wrote:
| Second Strogatz, does an amazing job linking the math to the
| visual intuition--probably less rigorous than some texts, but
| the perfect balance for an intro. Took an amazing course
| based on that book with Charlie Doering, who has sadly
| passed. He could draw multicolor, qualitatively-correct 3D
| attractors freehand on a chalkboard while helpfully narrating
| the system dynamics.
| IdealeZahlen wrote:
| What kind of ODE solvers are used to simulate chaotic systems?
| They must be very accurate if even a small error can result in a
| completely different result.
| j7ake wrote:
| You can use the standard Euler method with very small delta T.
| pixelpoet wrote:
| Respectfully disagree; Euler's method is absolutely terrible
| because it's unconditionally unstable; far better is to use
| something like Leapfrog or velocity Verlet (both of which
| have 2nd order accuracy and better stability, for exactly the
| same number of derivative evaluations). Euler integration is
| essentially always the wrong tool for the job.
| karpierz wrote:
| Would RK4 work for something like this, or does it lack
| some stability properties?
| jordigh wrote:
| RK4 is about as unstable. The backwards Euler method is
| baby's first stable ODE solver. If you want to make RK4
| stable, you have to change it into the implicit RK4
| method.
|
| However, the Lorenz system isn't stiff, which is usually
| what you use stable solvers for, nor any of the other
| systems here, I believe. RK4 should be fine, or even
| normal Euler with a reasonable step size. The chaos you
| see in these systems is not due to numerical inaccuracy.
| They're inherently chaotic. That's what makes chaos a
| subject worth studying.
| IIAOPSW wrote:
| If you want to see the folly of RK4, give it something
| like a ball bouncing and watch as it bounces slightly
| higher each time. Subtle at first, but trust me its
| there. The comment above you is right. If you have
| conservation of energy and want to keep it that way, use
| verlet.
|
| Edit: before anyone calls me out on this, the same trick
| also works when there's no discontinuity in the force
| function vis a vis collision with the floor. Planetary
| motion also drifts out of simple orbits. I just picked
| the ball bouncing because its a more amusing visual.
___________________________________________________________________
(page generated 2022-11-23 23:01 UTC)