Conversely given a discrete dynamical system you can build a /higher-dimensional/ continuous system with a Poincare section that gives the discrete system back! So all the chaos you see in discrete evolutionary models will happen in higher dimensional continuous models too.
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(There are a *lot* of species so probably the chaos is there in the world around us ;) )
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The basic way to build a continuous system from a discrete is the /mapping torus/ -- if your discrete system is f: X-> X, you take X x [0,1]_t, glue Xx0 to Xx1 using f, and flow in the t direction! This is a basic mathematical construction that more folks should know!pic.twitter.com/PRYKID7mvA
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Now [0,1] with 0 glued to 1 is the circle, which sits in the plane. So if f: R -> R was your discrete 1d dynamical system (evolutionary model) then i can build a 3d continuous dynamical system with the same amount of chaos as your 1 d system ;) Thus Poincare-Bendixon fails in 3d!
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In general the relation between dimension and `complexity' is amazing. An incredible theorem of Kozlovski et al shows that Axiom A systems -- ones which behave topologically like nondeterminstic finite state machine -- comprise most discrete dynamical systems in dim 1! Annals '07
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On the other hand there *exist* 2d discrete dynamical systems which are computationally universal in the sense that one can encode a Turing machine into their dynamics. I should dig up the reference -- ask if you are curious! So complexity goes up fast with dimension :-)
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Finally, there are some crazy ideas of Tao to show nonsolvability of Navier-Stokes using universality of the dynamics of water ^_^ https://arxiv.org/abs/1707.02389 https://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-an-averaged-three-dimensional-navier-stokes-equation/ …
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Hilariously using symplectic geometry one can show that there are Turing-universal fluid flows in dimensions above 17 O_o :D :P https://arxiv.org/abs/1911.01963
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So the moral is: Laura acutely observed an amazing thing, coming from an important theorem, but the discrete/continuous dichotomy is an illusion if you give yourself more dimensions to play with. And the world is very high dimensional!
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Maybe interested:
@vgr@tarunchitra@betanalpha@AdamMarblestone1 reply 0 retweets 1 likeShow this thread
Fascinating. Thanks for the tag. I was aware of the poincare section technique, but [0, 1] gluing technique and beyond stuff was new to me.
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Replying to @vgr @eigenstate and
That's amazing. But do people actually do anything like this discrete->(equally chaotic continuous) construction in the evolutionary dynamics models?
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Replying to @AdamMarblestone @vgr and
I also wonder if this kind of thing bears upon Wolfram's attempted discrete picture of physics...
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