GEM2505M

1. Taming Chaos GEM2505M Frederick H. Willeboordse frederik@chaos.nus.edu.sg

2. Quantifying the Dynamics Lecture 10

3. Important Notice! Special Q&A session. See announcement on web!

4. Today’s Lecture The Story We’ve seen how we can understand some of the main features of the bifurcation diagram. How can we quantify some of these features? • Lyapunov Exponents • Homoclinic Points • Intermittency • Fractals & The Logistic Map

5. Stability Thus far we considered the stability of a single fixed point. How about the stability of a period-k orbit? The most straightforward answers is that it is determined by the slope of the kth composition as we have seen before. But, in mathematical rather than graphical terms, what do we do? What is, e.g., the stability of a period 2 orbit?

6. Stability The second composition of the logistic map is given by: The slope is given by the derivative which is (using the chain rule): But what do we see? The term in brackets is just x1! And therefore, the slope of the second composition is given by:

7. Stability If it’s funny once, it’s funny twice! This same procedure works of course also for higher iterates and we can conclude that the stability of a period-n orbit is given by: Again the absolute value must be smaller than one in order for the orbit to be attracting. Since we know from the bifurcation diagram that x0 … xn-1 change for different nonlinearities, we can wonder whether there is a ‘most stable’ orbit.

8. Second composition Stability Super-stable orbits Indeed there is: Slope = 0 The smallest absolute value is of course 0. Hence an orbit which has a fixed point with a slope of 0 (a horizontal line) is the most stable orbit and therefore called super-stable. As is a product and a unequal to zero for all periodic orbits with a period larger than 1, we can immediately infer that a super-stable orbit contains the point x = 0. A super-stable orbit contains the point x = 0

9. Stability Super-stable orbits And as before, we can find these graphically right away by identifying where the periodic orbits in the bifurcation diagram intersect the x-axis. Points on super-stable orbits How many super-stable orbits are there?

10. Lyapunov Exponents Of course, if we talk about stability, we would like to have some kind of a number, a quantifier, that can tell us in a relative sense how stable an orbit is. As such one could think that the product of derivatives would provide such a quantifier. This is not really the case, however, since the number of terms in the product depends on the periodicity and in the case of a chaotic orbit would be infinite. One option would be to divide by the number of terms.

11. Lyapunov Exponents However … When considering sensitive dependence on initial conditions one can see that errors grow exponentially fast. Both axes linear Y-axis log, x-axis linear

12. Lyapunov Exponents Put differently …. Now we have a straight line. Both axes linear Both axes linear Take the log of the data Take the log of the data, anddivide by the x-value.

13. Lyapunov Exponents Put differently …we see that if something grows exponentially in time, then the log of that something divided by the time remains constant. Therefore we could argue that a reasonable quantifier for the stability l of an orbit is:

14. Lyapunov Exponents Since: We obtain: And if the orbit is not periodic, we should take the limit or more generally l is called the Lyapunov exponent of the orbit.

15. Lyapunov Exponents Not in exam The preceding few slides are plausible enough but do not really stress the fundamental connection between the Lyapunov exponent and the derivative or the growth of an error.In order to do so, let us choose a somewhat different approach. The difference between two initially nearby orbits can be expressed as:

16. Lyapunov Exponents Not in exam Dividing both sides by e we obtain: Note: This is the first derivative of the function f n for In other words:

17. Lyapunov Exponents Not in exam According to the chain rule we have: And consequently:

18. Lyapunov Exponents Not in exam Dropping the “approximate” and taking the log: Which, after reversing the order, taking the limit, dividing by n and changing the ln of a product to a sum of ln, again becomes:

19. Lyapunov Exponents Recall: Periodic orbits If we have a period-k orbit, the Lyapunov exponent becomes: a = 0.75 E.g. for period 2 we have: with x1 and x2 the two periodic points.

20. Lyapunov Exponents We have seen that for increasing a, the orbits bifurcate. What would the Lyapunov exponent be exactly at a bifurcation point? (e.g. a = 0.75) ? What would the Lyapunov exp. be? • Depends on a (not all bif. points have the same l) • 1 or minus 1 • 0 • 1/2k with k the periodicity just before the bif.

21. Lyapunov Exponents Super-stable orbits Super-stable orbits go through 0. Consequently, the Lyapunov exponent is given by: Points on super-stable orbits E.g. Why? Since the ln of 0 is minus infinity (and all the other terms are finite).

22. Lyapunov Exponents Versus a 1 2 3 4 Similarly to the bifurcation diagram, we can plot l versus a. • Second bifurcation • Period 4 super-stable orbit • Third bifurcation • Period 3 super-stable orbit

23. Lyapunov Exponents We just saw that the Lyapunov exponent of a super-stable orbit is minus infinity. Yet in the graph of the Lyapunov exponent versus the smallest exponent is around minus 2.5. ? Why would that be? • Our calculation is wrong • The graph is always wrong • The resolution of the graph is limited • There is no ‘infinity’ in the real world

24. Homoclinic Points Homoclinic points were discovered by Henry Poincaré in his studies of the solar system. In a similar form they also exist in the logistic map. Third Iterate a = 1.75 Plot exactly touches diagonal From the left, zigzags in to fixed point (cannot pass it) From the right, zigzags away from fixed point.

25. Homoclinic Points The point where the plot touches the diagonal is a so-called saddle point which is both attracting and repelling, depending on the side from which it is approached. Here, homoclinic points are all those points on the repelling side (i.e. right hand side) of the saddle that when iterated will eventually end up on the saddle via the attracting side. Note: Here we do not have stable and unstable manifolds since these require two or more dimensions.

26. Intermittency When the plot is very close to touching but does not actually touch the diagonal yet, a small channel is left. a = 1.7498 a = 1.7496 a = 1.7498 Every third time step is plotted. While passing through this channel, the x-values of the orbit do not change much leading to ‘laminar’ looking sections in the time series.

27. Intermittency Route to chaos Histogram a = 1.7496 Starting form the opening point of the period three window (a = 1.75), when decreasing the non-linearity a, the length of the laminar regions decreases from infinitely long to very short. Hence this is an alternative route to chaos as compared to the period-doubling route to chaos discussed previously.

28. Fractal Histogram for a = 2.0 Fractals in the logistic map The orbit of the logistic map at a = 2.0 is not fractal as canreadily be seen from the histogram to the right. However, there are fractal structures in the bifurcation diagram. For example the set of super-stable points.

29. Fractal accumulation pointa» 1.401155 Fractal dimension Another fractal may be at the accumulation point where the orbit is neither periodic nor chaotic. Some estimates are that D(a = a¥) = 0.538. Conceptually, how can one understand this? If one approaches the accumulation point from the chaotic side (starting at say a = 1.6), one can see that first there are two bands, then four, eight, etc. this is similar to the construction of the Cantor set. 1.60 a 1.37 Remove

30. Fractal or enlarged …

31. Fractal Relationship to theMandelbrot set x = 0.25 x = -2.0 The Logistic map can be written as: Which is exactly the real part of the iterative map used for the Mandelbrot set. Period three window

32. Key Points of the Day • Stability • Lyapunov Exponents • Intermittency

33. Think about it! • Is nature based on stability or instability? Stable, House, Cards, Unstable!

34. References http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson4/Demo1.html http://www.expm.t.u-tokyo.ac.jp/~kanamaru/Chaos/e/