Physics 213 9-8-03 THOUGHT QUESTIONS AND OBSERVATIONS - An interesting historical aside: the logistic map for r=4 was known in the 1940s to generate random-like digits and was used by several of the early computer pioneers at Los Alamos (where the first big computer was built and used in the US, as part of the bomb effort) as a cheap easy source of random numbers. These pioneers also knew that this was not a good random number generator since whenever the value x[i] came close to zero, to 1, to the fixed point 1-1/r, there would be long-lived correlations. Can you see why? - For the parameter value r=4 and for other values of r close to 4, there are "complicated" dynamics in that there is no simple predictable pattern to the time series, e.g., no approximate periodicities, no exponential decay, etc. Given that such a simple model produces such a complicated time behavior, could it be the case that all observed time dependences have the same kind of mechanism, that some kind of logistic map explains why the stock market, the weather, and brain EEGs (electroencephalograms) are so irregular? The answer is no, that there are different kinds of dynamical complexities and that, for example, not all forms of chaos are equally complex. We will quantify this later in the course when we discuss fractals and fractal dimensions but a quick heuristic justification is to think of EMBEDDING a time series into higher-dimensional spaces. For example, let's say we have an empirical time series x[i] for i=1,2,.... Then we could a sequence form a sequence of two-dimensional vectors like this: X[1] = ( x(1), x(2) ), X[2] = ( x(2), x(3) ) , X[3] = ( x(3), x(4) ) , ... and we can see if there is a pattern of points on the plane. If these data come from a logistic map or any one-dimensional map, then the Nth vector will have the form: X[N] = ( x[N], f(x[N]) ) , and you should be able to see that all the 2-vectors will collapse on to the quadratic curve (x,f(x)) rather than being sprinkled over the plane. This idea generalizes to higher-dimensional spaces although one needs to turn to a computer algorithm since it is difficult to visualize structures in 3d, 4d, etc spaces. If you want to try this yourself, you can create a sequence of two vectors out of some time series in Mathematica by using the Table function like this: vec2 = Table[ { x[[i]], x[[i+1]] }, {i, Length[x]-1} ] ListPlot[ vec2 ] ; (* best not to connect points with segments *) As two examples, let's see what happens to random numbers generated by Mathematica: (* 100 random uniform numbers in [0,1) *) data1 = Table[ Random[], {100} ] ; (* look at the time series directly *) ListPlot[ data1 ] ; ListPlot[ Table[ { data1[[i]], data1[[i + 1]] }, {i, Length[data1] - 2}] ] ; (* Now try logistic map *) data2 = plotlogistic[ 4, 0.1, 100] ; ListPlot[ data2 ] ; ListPlot[ Table[ { data2[[i]], data2[[i + 1]] }, {i, Length[data2] - 1}] ] ; While the going is hot, let's see what the effect of observational noise is on the logistic map: noisydata = Table[ data2[[i]] + Random[Real, {-0.05, 0.05}], {i, Length[data2]}] ; ListPlot[ noisydata ] ; (* looks random *) ListPlot[ Table[ { noisydata[[i]], noisydata[[i + 1]] }, {i, Length[noisydata] - 1}] ] ; Finally, go get fancy, here is a plot of a 3d embedding of a r=4 logistic map g1 = Show[ Graphics3D[ Table[ Point[ {data2[[i]], data2[[i+1]], data2[[i+2]] } ] , {i, Length[data2]-2} ] ] PlotRange -> All ] Show[ g1, ViewPoint -> {1.443, 3.025, 0.464} ] ; You can right-click on a 3d image like g1, select menu options Input/3d Viewpoint Selector, rotate the view in real time, then paste the viewpoint into wherever the text cursor is located to get a new viewpoint to plot. REVIEW: LINEAR STABILITY CRITERION FOR A K-CYCLE - Consider a 1d map x[n+1] = f(x[n]), and assume we somehow know a periodic k-cycle of this map: p[1], p[2], ..., p[k] , such that p[1] = f(p[k]) . Then I claim that this k-cycle is linearly stable if and only if the following condition holds: | f'(p[1]) f'(p[2]) ... f'(p[k]) | < 1 , i.e., the product of the magnitudes of the derivatives of the map function, each evaluated at a distinct point of the cycle, must be less than 1. - Proof: this is just a neat application of the chain-rule of calculus, applied repeatedly to the function f(x) composed with itself k times: f^(k)(x) = f(f(f(....(x)))...) . I.e., the k-cycle is linearly stable if all fixed points of the map f^(k) are linearly stable. PROOF OF EXISTENCE OF CHAOS FOR r=4 IN THE LOGISTIC MAP - It is not too hard to see that one can construct maps for which all derivatives have magnitude > 1 no matter where the derivatives are evaluated, e.g., the tent map with constant slope 2 or -2. This may seem phony but there is a change of variables: x[i] = (1/2) (1 - Cos(Pi y[i])) , which converts the logistic map with r=4 into the tent map for y[i]. Such a transformation is known only for this special value of k. - Change of variable converts logistic map to tent map with slope 2 or -2 everywhere. From this, we can deduce that all fixed points and cycles of any order are unstable. There are infinitely many cycles but only countably infinitely many so "most" or "typical" initial states will not lie on a fixed point or cycle. (The real numbers are a far bigger set than any countable set.) Thus most initial conditions can not end up on a fixed point or on a cycle and yet the dynamics must be bounded since r <= 4 and 0 <= x <= 1. We conclude that the long-time dynamics must be bounded and nonperiodic and a little more thinking shows that tiny perturbations must grow exponentially rapidly with an exponent (Lyapunov exponent) with analytical value lambda = ln(2), where the 2 comes from the constant slope of the tent map. Thus we have argued loosely a result that holds rigorously, for r=4 most initial states evolve after transients to a chaotic orbits of a known degree of instability. - My short discussion is based on some classic beautiful thinking about the simplest possible maps. An elegant discussion is given in Chapter 2 of Ott's book "Chaos in Dynamical Systems" which gives more information than I do in my lecture and I encourage you to look at. This book is on reserve in the Math-Physics library. COEXISTENCE OF DENSE SET OF UNSTABLE PERIODIC ORBITS WITH CHAOS - Composing the triangle map multiple times and using the fact that the range of the triangle map is [0,1], the same as the domain, we conclude that there must be intersections of the 45 degree line with f^(p)(x), so we conclude that there are periodic orbits of all order but that they are all unstable by the arguments of the previous lecture. So we have bounded nonperiodic dynamics for most initial conditions. - The fact that there is a dense set of periodic orbits whose orbits approach the chaotic one arbitrarily closely as the orbit period (cycle length) gets longer and longer turns out to be a general feature of chaotic dynamics and is one of the nontrivial ways that chaotic dynamics differs from idealized pure noise, which would not have such structure. Of course, real experimental systems are perturbed and our experimental resolution is limited so the true infinity of unstable periodic orbits (UPOS) will not be detectable. These UPOs are the basis of one of the more interesting advances in chaos theory of the last ten years or so, that by observation of a time series, one can deduce a series of weak perturbations that can convert unpredictable chaotic behavior into periodic behavior. Success has been demonstrated for systems with few variables, but not yet for spatially extended continuous media like heart tissue or the weather. RESTING POINT: GOALS OF NONLINEAR DYNAMICS - We have now discussed one specific example, the logistic map, in considerable detail. With this example, I hope you appreciate more concretely the kinds of questions that people ask in the field of nonlinear dynamics: - What kinds of nontransient dynamical states can be found in dynamical systems? We will need to invent ways to classify the time series that these systems create. - As a parameter is varied, there are bifurcations from one kind of non-transient dynamics to other kinds of dynamics. What kinds of bifurcations can occur? - Can we not only classify various bifurcations but also predict the parameter values at which they occur? This is the topic of stability of dynamics. - What we have NOT done is developed any systematic insight about why complicated dynamics occurs or about what kinds of bifurcations can take place, and what happens in more general maps and flows of several variables. We will spend the rest of the semester answering these questions to various levels of depth and applying our insights to some empirical data.