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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 so

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.

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.