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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

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.

pineconewarrior 4y ago
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!
ouid 4y ago
you've described a computer, it seems to me that you mean to put everything in some kind of normed vector space.
ok_computer 4y ago
Wrt: >>There might be two states that lead to one, but never one that leads to two There is constraint of uniqueness in deterministic differential equation solutions or chaotic maps so even if two flow trajectories appear to converge, for instance orbiting around the Lorenz equation butterfly solution, they are distinct and trace back to a given set of initial conditions. That’s the cool thing about dynamic systems, it’s just a set of parameters that cause something akin to numerical instability but is a property of the algebraic roots in that parameter range.
- IIAOPSW 4y ago
  Yes this is true you caught me. That's the one thing that doesn't quite fit the story when you move from my abstract graph / node story to diff eqs. That said, I'm pretty sure you still get "value collisions" in the case of iterated difference equations (at least for some maps). Uniqueness is unique to the continuous time case.
  - ok_computer 4y ago
    I had to think about this for a bit I've studied dynamical systems with chaos but am nowhere near proficient in the math. Around system fixed points there are stable and unstable manifolds. There can also be apparently (linearized) center maifolds where there is no growth or decay but that simply isn't true because it is so hard to for an eigenvalue to be exactly 0.000....0 (smooth time) or 1.000...0 for discrete maps. So you have to perform a center manifold reduction to inspect the fixed point behavior in higher order terms x^{1,2,3,..} y^{1,2,3...}, xy^{1,2,3,...}, x^{1,2,3,...}y, x^2*y^{1,2,3,...} etc. all linear combinations for the basis, in R^2 that gives so many combinations. Around these fixed points in unstable maps the manifolds themselves will intersect in other places than the fixed point which is confusing to me (homoclinic tangle: https://en.wikipedia.org/wiki/Homoclinic_connection https://en.wikipedia.org/wiki/Homoclinic_connection). But the chaotic map itself holds uniqueness of solution such that f(x_n): x_n->x_{n+1} and f^-1(x_{n+1}): x_{n+1}->x_n. This is key: deterministic systems follow uniqueness of solutions. You can trace time or steps backwards to a distinct t(0) or x_0. Without this the analysis does not work, then maybe dealing with probabilistic systems but these strange attractors arise from deterministic systems. I don't know much about probability. Anyway I think we're not necessarily disagreeing with each other just talking past certain really interesting points. What I think is the most conceptually simple way to visualize chaotic behavior that leads to strange attractors in R^n is the logistic map https://en.wikipedia.org/wiki/Logistic_map https://en.wikipedia.org/wiki/Logistic_map to visualize a R^1 system with one parameter that can cause bifurcation, phase doubling, then chaotic regions. The R^n analog to this would be a parameter set that arises in a fixed set {} instead of a fixed point x_o that the system will eventually evolve to. See double hinged pendulum as a simple chaotic system that evolves to a chaotic state. I think the system bifurcation is what you are alluding to where the roots split from one to two states and so forth. Anyway, dynamical systems is interesting and it would be amazing to see what further is applied from it in 200 years. Its a wild field because it quickly devolves into algebra, set theory, fractals and topology. To pick up on the front you need to understand so much deep theory that is over my head. Anyway cool stuff I love seeing these type articles here way more than the next best css framework or whatever.
gala8y 4y ago
I've been fascinated by NLDS and always wondered how could we use theory of chaos in, for example, psychology, treating states of mind as (strange) attractors. Would that ever be useful to use NLDS thinking in discussing mind/consciousness? edit NLDS = non-linear dynamical systems

pineconewarrior 4y ago

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!

ouid 4y ago

you've described a computer, it seems to me that you mean to put everything in some kind of normed vector space.

ok_computer 4y ago

Wrt: >>There might be two states that lead to one, but never one that leads to two There is constraint of uniqueness in deterministic differential equation solutions or chaotic maps so even if two flow trajectories appear to converge, for instance orbiting around the Lorenz equation butterfly solution, they are distinct and trace back to a given set of initial conditions. That’s the cool thing about dynamic systems, it’s just a set of parameters that cause something akin to numerical instability but is a property of the algebraic roots in that parameter range.

IIAOPSW 4y ago
Yes this is true you caught me. That's the one thing that doesn't quite fit the story when you move from my abstract graph / node story to diff eqs. That said, I'm pretty sure you still get "value collisions" in the case of iterated difference equations (at least for some maps). Uniqueness is unique to the continuous time case.
- ok_computer 4y ago
  I had to think about this for a bit I've studied dynamical systems with chaos but am nowhere near proficient in the math. Around system fixed points there are stable and unstable manifolds. There can also be apparently (linearized) center maifolds where there is no growth or decay but that simply isn't true because it is so hard to for an eigenvalue to be exactly 0.000....0 (smooth time) or 1.000...0 for discrete maps. So you have to perform a center manifold reduction to inspect the fixed point behavior in higher order terms x^{1,2,3,..} y^{1,2,3...}, xy^{1,2,3,...}, x^{1,2,3,...}y, x^2*y^{1,2,3,...} etc. all linear combinations for the basis, in R^2 that gives so many combinations. Around these fixed points in unstable maps the manifolds themselves will intersect in other places than the fixed point which is confusing to me (homoclinic tangle: https://en.wikipedia.org/wiki/Homoclinic_connection https://en.wikipedia.org/wiki/Homoclinic_connection). But the chaotic map itself holds uniqueness of solution such that f(x_n): x_n->x_{n+1} and f^-1(x_{n+1}): x_{n+1}->x_n. This is key: deterministic systems follow uniqueness of solutions. You can trace time or steps backwards to a distinct t(0) or x_0. Without this the analysis does not work, then maybe dealing with probabilistic systems but these strange attractors arise from deterministic systems. I don't know much about probability. Anyway I think we're not necessarily disagreeing with each other just talking past certain really interesting points. What I think is the most conceptually simple way to visualize chaotic behavior that leads to strange attractors in R^n is the logistic map https://en.wikipedia.org/wiki/Logistic_map https://en.wikipedia.org/wiki/Logistic_map to visualize a R^1 system with one parameter that can cause bifurcation, phase doubling, then chaotic regions. The R^n analog to this would be a parameter set that arises in a fixed set {} instead of a fixed point x_o that the system will eventually evolve to. See double hinged pendulum as a simple chaotic system that evolves to a chaotic state. I think the system bifurcation is what you are alluding to where the roots split from one to two states and so forth. Anyway, dynamical systems is interesting and it would be amazing to see what further is applied from it in 200 years. Its a wild field because it quickly devolves into algebra, set theory, fractals and topology. To pick up on the front you need to understand so much deep theory that is over my head. Anyway cool stuff I love seeing these type articles here way more than the next best css framework or whatever.

gala8y 4y ago

I've been fascinated by NLDS and always wondered how could we use theory of chaos in, for example, psychology, treating states of mind as (strange) attractors. Would that ever be useful to use NLDS thinking in discussing mind/consciousness? edit NLDS = non-linear dynamical systems