Dynamical Systems And Chaos: Pattern Formation Summary Part 1

Dynamical Systems And Chaos: Pattern Formation Summary Part 1


so let’s summarize unit 9 as usual I’ll begin with a topical summary of what we covered in this unit and then speak a little bit more broadly and generally about the key themes and ideas so this unit was on pattern formation and began by pointing out that we’ve seen that dynamical systems are capable of exhibiting chaos that’s unpredictable a periodic behavior butterfly effect but there’s a lot more to the story dynamical systems can do much more than produce chaos or disorder they can also produce pattern structure organization and so on and in this unit we looked at one particular example of a pattern forming dynamical system a class of systems known as reaction-diffusion systems so first we talked about diffusion and I tried actually to demonstrate it by dropping a few drops of food coloring in a glass of water diffusion is just the tendency of a substance to spread out as a result of random motion so the random ink molecules or food coloring molecules they bounce around and eventually they’ll spread themselves out evenly throughout the glass of water another way to say this is that chemicals tend to move from regions of high concentration to low concentration in the net result of this is that diffusion tends to smooth out any differences in concentration leading to a homogeneous or uniform distribution of the food coloring or whatever it is that’s diffusing so diffusion is described mathematically by the diffusion equation here it is in this equation D is the diffusion constant it’s related to how fast the substance diffuses different substances will diffuse at different rates in different media often this is something one could measure experimentally this is the rate of change of concentration and this is the laplacian this is a certain type of spatial derivative of the system so i should say that you here is the concentration of a chemical and it’s a function of x and y we might have say on this sheet of paper different concentrations at different points on the page so any event this describes the diffusion of substances once you know d in the initial conditions in the equilibrium state when this stops changing if I set this equal to zero then I’m in effect setting this quantity equal to zero and that has the effect of picking out the most boring function possible it’ll just be a flat level distribution if the boundary conditions allow it if there’s more say there’s some chemicals flowing in on this side and out on this side when we would just get a smooth distribution and interpolation between those two values so again the main point is that diffusion leads to boring functions functions that are as homogeneous as possible alright but things get more interesting in reaction diffusion systems so in this we now have two different types of chemicals often called U and V or a and B the details may vary and you will be the concentration of one chemical as a function of X and Y so different concentrations at different points in space same story for V and both of these chemicals diffuse they spread out across the surface but they also interact with each other so their equation the equation of motion what determines their values the rule the dynamical system are these two equations so this is just diffusion for you but then this says there’s something some added term that’s an interaction typically between U and V so its diffusion plus something else that depends on you on V same thing for the V equation V will diffuse plus there’s some other interaction a different interaction of function G of but it’s a function of U and V so its diffusion plus something else this is a deterministic dynamical system there’s no element of chance there’s no stochasticity in the rules and it’s a spatially extended in amical system so if we give the initial condition initial values of U and V everywhere in the lattice or uh the rectangle it doesn’t have to be a lattice and we specify what the conditions are at the edges then that determines the the future values everywhere today is piece of paper another point that’s important is that this rule is local and let me explain what I mean by that this use of local is not an entirely standard one but I think it’s justified the idea is that if I want to determine say the next value of you so I know the present value of U and V at some particular point on on on our surface if I want to figure out the next value using something like oil errs method all I need to know is the value of you here the value of V here and then the derivatives of U and V also at this point so what I mean by local is I don’t if I want to know the next value of you how you changes at this point I don’t need to know the value of you over here so that these equations are all local and that they’re in any given equation x and y is constant it keeps a reference to the same point so again maybe another way to say this is there aren’t any long-range interactions here the next value of U is determined by the present value of you and just it’s a little bit about its curvature given by the laplacian at that point we have a local deterministic spatially extended dynamical system so in reaction-diffusion systems or specifically activator inhibitor systems the following set up typically is the case you is usually an activator it’s something that catalyzes its own growth so the presence of view gives rise to more you so rabbit growth growing exponentially growing rabbits or an exam this rabbits we don’t usually talk about rabbits catalyzing their own growth but in a sense that’s what they do v is some sort of an inhibitor it’s something that typically would also grow in the presence of you but it would also inhibit you so it prevents you from growing too much and in the example I gave these were foxes foxes grow in the presence of you foxes eat rabbits so they’re get to be more or foxes but they but the Fox is inhibit the rabbits they they prevent them from growing and growing so if we have these two things and that V diffuses faster than you this can lead to stable spatial structures and we saw some examples of that I’ll show you one again in a second the particular shapes determine depend on a number of different things the relative diffusion rates how much faster one diffuses than the other and also the geometry of the system and in some sense the initial values as well and of course it would also depend on the particular functions you choose for you excuse me for F and G and there are lots of different possibilities and one can do three components u and V and W so there are lots of different models here and the mathematics of analyzing them analytically gets pretty complex pretty quickly but in the additional reading section for this unit I have some suggestions for places you can go to learn more okay so these are reaction-diffusion systems one particular type of them here’s some results we experimented with the excellent program at experiment REM digitel here’s the URL there’s a link to this on the links program page and this was the white paw setting so this makes cheetah spots there it is that doesn’t quit doesn’t show up that well in black and white it’s a more compelling picture on screen and here the diffusion rate of you is 3.5 they might be hard to see and be with 16 and that program specifies what the f and g functions are as well the main point is that even though we have a diffusive system where things should be spreading out it shouldn’t you should expect to have a higher density of say you here than here and have that be a permanent situation it should diffuse away that’s what diffusion does it smooths things out but if we have reaction diffusion systems where things interact in the way I describe one can get a variety of different stable spatial structures so these systems form patterns stable spatial structures sort of seemingly out of nowhere this isn’t just a mathematical result there are lots of physical systems that do this I showed a little bit of a video of a belousov sabot in ski experiment taking a place on a petri dish here’s a link to that YouTube video this link is also in the additional resources section this video is by Stephen Morris of the University of Toronto who will be talking to next week so this looks a little different than the cheetah spots but it’s the same general thing reaction-diffusion we get these sort of propagating wave fronts that move out and then the wave fronts collide with each other and interact in interesting ways all right so to summarize once more pattern formation there’s more to the study of dynamical systems than just chaos lots more it’s very often the case that simple spatially extended dynamical systems with local rules like these reaction-diffusion systems are capable of producing stable global patterns and structures and this begins to give some insight into how patterns might emerge in an otherwise structureless world the reaction to fusion systems that we study here are just one of many many examples of pattern forming systems there are lots of different classes of models the partial differential equations of reaction diffusion systems is just one of them but in general there’s a certain creativity to these simple dynamical systems that can not only produce chaos but can also produce these interesting structures

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