Logistic Map: Period Doubling Route to Chaos | Nonlinear Dynamics

[00:00:00]The thing that I'll be talking about today is the  logistic map and it's really simple--it's like shockingly simple. So this is just 1d, we  say x n plus one is r that'll be a parameter we could vary x n times 1 minus x n this was  created to be like a discrete population dynamics model similar to the logistic equation  if you remember the logistic equation that was a differential equation but it had a right  hand side that looked something like this where there was like a carrying capacity for  a population but here the space of x we're normalizing so that it just goes from zero and  one so we're talking about the interval zero to one what is r if you were to plot this you can  plot x n plus one so this is a bit like if you remember the graphical method for one dimensional  differential equations well here we're plotting the map there's zero there's one so here's x  and everything has to be between zero and one we're only allowing between zero and one where  does this peak it actually peaks at the halfway mark so one half it's a parabola goes up from zero  and then it goes back down to zero symmetrically and it peaks at r over four we want the map to  go from zero to one to zero and 1. so we want x n plus 1 to be somewhere inside the unit interval  and for that to happen we need that this peak be 1 or below so it's conventional to let  the parameter be something less than four but greater than zero and physically this is  supposed to represent the reproductive capacity of the population it was first in a paper by robert  may 1976 in nature it has some interesting name simple mathematical models with very complicated  dynamics and it's now become a classic he talks about first order and he called them  difference equations and maybe that's a term you've heard or used in addition to iterated  map or i think recursion relations are also used so he wrote this right and that's what  we're looking at he called the parameter a and referred to time the discrete time as t we're going to refer to the discrete time as  little n and use r as our bifurcation parameter robert may uh was a i think trained in physics but  then was in biology departments and was sort of a father of uh ecology so it's become this classic  this is where the logistic map was introduced in some sense it's the simplest non-linear map  because all it has is a quadratic non-linearity xn plus 1 equals if we were to write it out it's r  x n minus r x n squared if it was just r x n then it'd be a linear map easier to analyze but it's  non-linear what happens what's the big deal with this map you can pick an initial condition x  naught which has to be somewhere in this interval between zero and one and then you  would write you know x one equals r x naught one minus x naught and then you'd  say well x two whatever you get for x one right so this is it's really simple to code up as a for  loop and then plot it r x one 1 minus x1 and so on and then you can just see what happens so we'll  do that for some initial conditions i'll sketch some here but then we'll even see it numerically  it's you could pretty easily find numerical tools online for the logistic map because it's it's so  well known the parameter is going to play a big role bifurcations are very important here if we  were to look at what happens with r equals 0.5 along this axis the horizontal axis this will  be the discrete time n so it'll be times zero and the interval over here will be i'll be  plotting x n has to be between 0 and 1. it turns out for any initial condition for 0.5 you would  get something like this so here's our initial condition and then this will just get lower and  lower and lower and really the map only spits out these points x n goes to zero for any x naught so  any initial condition as n goes to infinity so x equals 0 is a fixed point of the map you might  put it that way at least for this value of r now sometimes people will connect the dots but  that's just to guide your eye the map only spits out points but if i wanted to guide your eye  i would draw straight segments between these and you can get an idea of what's happening right  and you could demonstrate that it goes to zero we could think of this also in terms of the  cobweb plot if you remember the cobweb plot where we write x n x n plus one we draw the  diagonal 45 diagonal line let's call this one that's one and this is zero zero for r 0.5  this thing rises to only the level of what r over four so that's 1 8. so it's not going to  be very high it's going to be it never actually other than at the origin it does not intersect  that 45 degree angle line so according to the cobweb plot construction if we started with some  initial condition we would go up to the curve and that tells us where to go we go from there  horizontally to this 45 degree line and then this would take us down to this point i kind of have  to zoom in here oh look at that look what i can do and then we would go across to here and  you can kind of see the cobweb plot is just taking us into the origin and this would  this will happen no matter where we start so i'm just using the same rules for the cobweb  plot that i did before when we were talking about the lorenz map so that's for r equals  0.5 what about for something a bit bigger so i'll look at r equals 2.8 this is what we get  for the the orbit if we start somewhere i'll start at some other initial condition remember this only  goes from between zero and one if i start say here the next iterate gets a little bit further up and  then i go up somewhere but then i go down and that just sort of dampens out until i reach a fixed  point and again the map only spits out the points we could draw lines to connect it to guide your  eye and you can see hopefully what's happening it looks like it overshoots but then it eventually  settles down to a fixed point that's non-zero x n goes to x star as n goes to infinity and x star  is not zero so there's some non-zero fixed point that shows up and it's the same x star for any  initial condition and remember the the range for r we said it's conventionally taken to be from  zero to four and i'm picking these strange values what about r equals 3.3 we do the same kind of  diagram so this is xn it goes between zero and one here's what we would find start with some  initial condition what we would get is that this will go up and eventually it'll settle down  on a cycle so it's going between two points over and over again so this is like the discrete  map version of a limit cycle instead of being a fixed point we settle onto a a cycle you know it  jumps between the same two points now as n goes to infinity and this is independent of the initial  condition so you take any initial condition it'll it'll settle on this we call this a period two  orbit or you could call it a period two cycle and it's also stable it is the attractor it's the  attractor because an open set but actually every initial condition ends up there and you get this  repeating so x n plus two equals x n as n goes to infinity right if this is x and then it's just  going between the two so that's interesting what if we turn this knob a little bit higher r equals  3.5 we get something that settles on a a cycle but it settles on something where it goes down  and then up and then down and then higher and then it repeats that down up kinda down up it's a lot  easier if i connect points and maybe even zoom in so it's repeating but it's not repeating every  two iterates it's repeating every four iterates so we would call this a period four cycle so  that's interesting what happened to three we had in this case we had period doubling and period  doubling will be sort of one of the main phenomena that goes on here you can find these numerically  some of them you can find analytically but think of just you tune r and you look for the value of  r where a stable two to the n cycle first occurs as we turn this knob even further we'll go from a  period 4 cycle to a period 8 period 16 and so on and i'll call the special value of r i'll call it  rn it's the value of r where a stable 2 to the n cycle first occurs so it turns out if you notice  what we had up here at least for r 2.8 we didn't have one it was just a fixed point for our 3.3  we did have a stable two cycle so that means two to the one so r1 that happens at three and that  can be shown so a period two cycle is born r2 occurs at so that would mean a two to the  two cycle or four cycle three point four nine if we keep continuing this r3 is at 3.54409  so that's a period 8 cycle 2 to the 3 is 8.

[00:11:11]another period doubling at three point five six  four four and so on what is going on with this sequence it's looking like it's converging  to something so r to the five that would be where there's a 32 cycle is 3.568759 looks like  it's converging to something close to like 3.5 7. it actually is so if we were to look at what's  going on r n converges essentially geometrically to some point called r infinity which robert may  in the paper refer to as the accumulation point if we were to sketch the interval this is r we could  draw r1 and everything at this point this is let's call this the accumulation point everything's  going towards r infinity so here is say our n here's our n minus 1. so we have  that the length of that first gap between the values the next value rn plus 1  is a smaller gap and the gaps keep getting smaller and smaller and it's kind of hard  to draw but it all accumulates at that point or infinity if you look at the ratio of rn minus  r n minus one so that kind of the longer gap divided by the size of the smaller gap as n  goes to infinity the ratio of those gaps becomes a fixed number 4.6692 something something  something and we'll say more about what that number is right now let's just say it's a it's  a mystery it's sometimes called delta but that's what's going on r infinity is something really  close to 3.57 there's still plenty of room between r infinity and four so you might wonder what's  going on there actually a shocking amount of stuff happens so what was drawn what's been drawn  and it's now become a pretty famous diagram is to plot something like a bifurcation diagram  it's not technically a bifurcation diagram but it's often called a bifurcation diagram so we're  plotting r and what did we so what do we have up here we're plotting x n on the attractor so we're  plotting all the points on the attractor so what have we shown up above uh well for 0.5 we were  at zero so this only goes between zero and one 1.

[00:14:01]this is what's happening at 0.5 right we just we  were at 0. in fact we stay at 0 until the value 1 and at the point one zero still is a fixed point  it's an unstable fixed point so we might draw that as dashed what we really care about is where the  points are there's the branch that shoots off we looked at 2.8 and that's just a point it's just a  fixed point but then at three this bifurcates into two branches so if you remember where we were at  3.3 this isn't going to be completely to scale but we had two points there's actually a branch coming  off that fixed point that goes between those two points there's still an unstable point but  it's unstable so you wouldn't observe it so it's usually not depicted in this thing that  would be called the bifurcation diagram of the logistic map at 3.5 we had four points and that's  because somewhere in there was r2 and that's where there's a splitting again we call this the  period doubling and this period doubling keeps happening at each of those values of rn so  it's it's hard to depict so i zoom in here where were we 3.5 i think so there's more and more  splittings you could just think of there being little it looks like a tree with branches and  eventually there's a r infinity you hit r infinity and all these branchings just go into a mess  of chaos i'll just show what look like tons of little branchings and then we have chaos and i  don't even know how to draw it but it you know you'll just have points all over the  place not actually all over the place but somewhere and then here's the weird thing  there's a window later on there's several windows there's an infinite number of windows but there  are big windows of going from chaos back to periodic behavior this is depicting  chaos and then we have periodic window and the kind of biggest periodic window has  period three that shows some period three orbits those branch and then there's chaos  again and then there's windows this diagram is an intermingling of chaotic and periodic windows  this was surprising why would such a simple map in some sense the simplest non-linear map shows  such rich behavior but there it is it's real in fact this point where the  kind of biggest periodic window shows up occurs at one plus two square root  two which is about three point eight two and that's where the period three stable attractor  was born it turns out you can find attractors of any period so give me any integer for some  parameter value there will be an attractor that has that period because in this region  that i've called chaos there's actually periodic windows inside there they might  be infinitesimally wide but they're there there's a lot to unpack here and try to understand