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

2. 2 Strange Attractors Lecture 12

3. GEM2505M 3 Today’s Lecture Strange Attractors Lorenz Equations Henon Map The Story Many physical systems are dissipative. Hence one would expect their attractors to be simple. This turns out to be untrue when so-called strange attractors were discovered. What is strange about a strange attractor?

4. GEM2505M 4 Higher Dimensions One cannot expect that all chaos-like phenomena can fully be described by one-dimensional systems. It is therefore useful, to look at higher dimensions. After one comes two …. Will two be enough? That depends! When the dimensionality is low, one generally deals with the dynamics of a single point. However, single units (think of cells, atoms, molecules etc.) can interact giving rise to many interesting collective phenomena. Systems built up of many individual units can have fairly high dimensions.

5. GEM2505M 5 Most systems in the real world must include some kind of friction. That is to say they loose (dissipate) energy. Dissipative Systems A common assumption was that in dissipative systems the final state would be rather simple. A point or some regular motion for example. Note: There are also conservative systems where there is no energy loss. Conservative systems too can have very interesting dynamical properties. However, simple dissipative systems have been discovered where the final state is anything but simple with chaotic dynamics and fractal structures. Assumption: Simple Fact: Can be chaotic

6. GEM2505M 6 Strange attractor is the name for the final state of a dissipative system that displays chaotic dynamics. Strange Attractors In strange attractors, chaos and fractals come together nicely illustrating clearly how chaos is a dynamical property and fractal a geometric property. Since the system is dissipative, the size of any area must shrink. Consequently, in the limit of an infinite number of iterations any area becomes infinitely small. Hence a strange attractor is an object with no area or volume!

7. GEM2505M 7 The Lorenz Attractor Definition The Lorenz equations are defined as: Parameters used by Lorenz The famous Lorenz attractor

8. GEM2505M 8 The Lorenz Attractor The Derivation The derivation of the equations is beyond the scope of this course. However: The equations are based on a model for the cylindrical fluid convection that appears on top of a heated plate It is hence not a model of the actual airflow

9. GEM2505M 9 The Lorenz Attractor Meaning of the Variables Roughly: xrelates to streamfunction that characterizes fluid flow yis proportional to the temperature difference between the upwards and downwards moving parts of a roll zdescribes the nonlinearity of the temperature difference along the roll

10. GEM2505M 10 The Lorenz Attractor Key Properties 1.There are only two nonlinearities xy and xz. 2.There is a symmetry (x,y) -> (-x,-y) (hence if x(t), y(t), z(t) is a solution, then –x(t),-y(t),z(t) is a solution too). 3.The Lorenz equations are dissipative. In fact, volumes shrink exponentially fast. 4.There are no repelling fixed points or orbits (this would contradict that all volumes contract).

11. GEM2505M 11 The Lorenz Attractor Dynamics Basically, the trajectory goes through the following two steps repeated ad infinitum: 1.Spiral outward 2.Move over to the other side Rather than stretch and fold this is: stretch-split-merge

12. GEM2505M 12 The Lorenz Attractor Dynamics 1.Spiral outward2.Move over to the other side

13. GEM2505M 13 The Lorenz Attractor Bifurcations http://risa.is.tokushima-u.ac.jp/~tetsushi/chen/chenbif/node8.html Since there are three parameters, bifurcations can in principle occur in many different ways. One example is given to the left.

14. GEM2505M 14 The Lorenz Attractor Sensitive Dependence Two nearby trajectories can stay close for quite a long time (depending on where one starts). However, at some point they strongly diverge. Trajectories still very close Trajectories far apart

15. GEM2505M 15 The Lorenz Attractor Sole p 11

16. GEM2505M 16 The Lorenz Attractor Fractal For the type of equations like the Lorenz equations, there is a theorem (called the uniqueness theorem) which states that its solutions are unique and hence that trajectories can never intersect. In the Lorenz attractor, we see the two sheets ‘merging’ but a real merge is not possible due to the above theorem. What we really get is a fractal.

17. GEM2505M 17 Waterwheel It turns out that a simple and conceptually easy to understand model exists for the Lorenz equations: the waterwheel! In a waterwheel, leaky cups are attached to a wheel and water is steadily poured in exactly from the top. The main parameter to be varied here is the water flow (having a role similar to the nonlinearity in the logistic map).

18. GEM2505M 18 Waterwheel For very low flow rates, the wheel just stands still since more water will drain out of the cup than can flow in. When the flow is big enough so that the cups start filling up, the wheel will turn regularly in one direction or the other. If one then increases the flow even further, chaotic switching between rotational directions occurs. Note: The (simplified) model equations for the waterwheel can be transformed into the Lorenz equations.

19. GEM2505M 19 It was introduced by the French astronomer Michel Henon in 1976 as a simplification of the Lorenz equations. However, it also is the extension of the logistic map into two dimensions. The Henon Map The simplest way to extend the logistic map would be to just add linear terms. Definition Logistic Map

20. GEM2505M 20 The Henon Map Attempt to simulate the stretching and folding in the Lorenz system. Reasoning StartStretch and FoldSqueezeReflect Combining the three transformations yields the Henon map.

21. GEM2505M 21 The Henon Map Key Properties Note: Henon map = discrete, Lorenz system = continuous Invertible: In the Lorenz system, each point in phase space has a unique trajectory associated with it. This is completely different from the logistic map! Dissipative: It contracts areas. In fact it does so at the same rate everywhere in phase space. There is a strange attractor

22. GEM2505M 22 The Henon Map a = 1.4, b = 0.3 “The” Henon Attractor This is how the Henon attractor looks. To me: kind of like a paper clip … and quite different indeed from the Lorenz attractor. -1.5 1.5 -0.4 0.4

23. GEM2505M 23 The Henon Map a = 1.4, b = 0.3 Basin of Attraction The basin of attraction is the set of all points in the plane the end up on the attractor. Other points escape to infinity.

24. GEM2505M 24 The Henon Map a = 1.4, b = 0.3 Area transform Any square near the attractor will be mapped onto the attractor. In order to apply the transformation, the square is built up of 10,000 points to which the Henon map is applied one by one. n = 0n = 1 n = 2n = 3 n = 5n = 10

25. GEM2505M 25 The Henon Map a = 1.4, b = 0.3 Sensitive Dependence (x 0,y 0 ) = (0,0) (x 0,y 0 ) = (0.000001,0) Difference between the trajectories.

26. GEM2505M 26 The Henon Map Bifurcations b = 0.3 Just as for the logistic map, we can generate a bifurcation diagram by varying the nonlinearity. There are two differences though. 1) We need to choose a value for b and keep it fixed. 2) We need to start from several initial conditions since there are multiple attractors.

27. GEM2505M 27 The Henon Map Bifurcations b = 0.3 Multiple attractors?? If we zoom in to the area on the left, we see that there are two separate bifurcation diagrams. Depending on the initial condition, the orbit will either go the upper or the lower bifurcation diagram.

28. GEM2505M 28 The Henon Map a = 1.4, b = 0.3 Fractal 0.607820.60811 0.19294 0.193068 The orbit of the Henon attractor has a fractal structure

29. GEM2505M 29 The Henon Map A little math Invertability Not in exam The Henon map The inverted Henon map After moving to the left

30. GEM2505M 30 The Henon Map A little math Area reduction Not in exam In general, a 2-dimensional map is area reducing if it’s the absolute value of the determinant of it’s Jacobian matrix is smaller than 1. The map is area reducing if:

31. GEM2505M 31 The Henon Map A little math Area reduction Not in exam Applying this to the Henon map we obtain: Hence we see that the Henon map is area reducing and that this reduction is the same everywhere. area reducing if |b| < 1

32. GEM2505M 32 Dissipative Systems Lorenz Equations Henon Map Key Points of the Day

33. GEM2505M 33 Waterwheel Think about it! Waterwheel Countryside, Farm, Life!

34. GEM2505M 34 References