GEM2505M

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

2. Strange Attractors Lecture 12

3. Today’s Lecture 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? • Strange Attractors • Lorenz Equations • Henon Map

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

6. Strange Attractors Strange attractor is the name for the final state of a dissipative system that displays chaotic dynamics. 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! In strange attractors, chaos and fractals come together nicely illustrating clearly how chaos is a dynamical property and fractal a geometric property.

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

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. The Lorenz Attractor Meaning of the Variables Roughly: x relates to streamfunction that characterizes fluid flow y is proportional to the temperature difference between the upwards and downwards moving parts of a roll z describes the nonlinearity of the temperature difference along the roll

10. The Lorenz Attractor Key Properties • There are only two nonlinearitiesxy and xz. • 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). • The Lorenz equations are dissipative. In fact, volumes shrink exponentially fast. • There are no repelling fixed points or orbits (this would contradict that all volumes contract).

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

12. The Lorenz Attractor Dynamics • Spiral outward • Move over to the other side

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

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

15. The Lorenz Attractor Sole p 11

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. 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. 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. The Henon Map Definition 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 simplest way to extend the logistic map would be to just add linear terms. Logistic Map

20. The Henon Map Reasoning Attempt to simulate the stretching and folding in the Lorenz system. Start Stretch and Fold Squeeze Reflect Combining the three transformations yields the Henon map.

21. The Henon Map Key Properties • 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 Note: Henon map = discrete, Lorenz system = continuous

22. The Henon Map “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. 0.4 -0.4 1.5 -1.5 a = 1.4, b = 0.3

23. The Henon Map 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. a = 1.4, b = 0.3

24. The Henon Map n = 0 n = 1 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 = 2 n = 3 n = 5 n = 10 a = 1.4, b = 0.3

25. The Henon Map Sensitive Dependence (x0,y0) = (0,0) (x0,y0) = (0.000001,0) Difference between the trajectories. a = 1.4, b = 0.3

26. The Henon Map Bifurcations 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. b = 0.3

27. The Henon Map Bifurcations 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. b = 0.3

28. The Henon Map Fractal a = 1.4, b = 0.3 0.193068 The orbit of the Henon attractor has a fractal structure 0.19294 0.60782 0.60811

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

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

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

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

34. References