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

2. 2 The Logistic Map Lecture 7

3. GEM2505M 3 Today’s Lecture Population Dynamics The Logistic Map Cobwebs Bifurcation Diagram

4. GEM2505M 4 N i+1 = p N i Consider a population N from year to year. This can be described by: This year’s Population (the discrete version of Malthus’ Law) Population Dynamics Next year’s population ? What do you think this means? 1.The population grows until it reaches equilibrium 2.Population Explosion 3.Population Extinction 4.It depends on p

5. GEM2505M 5 N i+1 = p N i Answer: It depends on p. If p is positive, this means that the population will grow ever bigger. Population Dynamics p = 1 nothing happens p > 1 population explosion p < 1 extinction

6. GEM2505M 6 Of course an ever growing population is not realistic. Hence Verhulst added the term -bN 2 to Malthus’ Law. We can do that here too and obtain: N i+1 = p N i -bN i 2 This is called the logistic map (though usually it’s written in a different way). Population Dynamics With some math it can be expressed as:

7. GEM2505M 7 One may feel a bit uncomfortable with the discrete approach and consider a more continuous description. Where: This Greek D called delta means “change in” Size of the population Time Death rate Birth rate Population Dynamics Not in exam

8. GEM2505M 8 Regardless of the exact value of r, if it is larger than 0, exponential growth will inevitable lead to the exhaustion of all available resources not matter how small the organism! As a differential equation we have: Population Dynamics Growth Not in exam

9. GEM2505M 9 Population Dynamics Exponential Growth Not in exam

10. GEM2505M 10 In the most simple case, growth will stop accelerating, start decelerating and approach the carrying capacity of the environment. When it has reached the carrying capacity, the growth and death rates should roughly be equal. This can be incorporated into the equation by adding the term on the right hand side. = carrying capacity Population Dynamics Limiting growth Not in exam

11. GEM2505M 11 Thus we obtain: Or as a differential equation: Population Dynamics Limiting growth Not in exam

12. GEM2505M 12 The result is an s-shape like growth curve. Population Dynamics Not in exam Limiting growth Acceleration Inflection Point Deceleration Steady State

13. GEM2505M 13 We’ll now show that the logistic map is just a discrete version of: This can easily be transformed by considering the right hand side as a square. Population Dynamics Not in exam Back to the discrete case

14. GEM2505M 14 Then, it is clear that the right hand side can be written as a square: First, lets write a bit simpler Population Dynamics Not in exam Towards the logistic map

15. GEM2505M 15 Population Dynamics Not in exam Towards the logistic map

16. GEM2505M 16 Population Dynamics Not in exam Towards the logistic map

17. GEM2505M 17 Population Dynamics Towards the logistic map From previous slide. Not in exam

18. GEM2505M 18 Population Dynamics The logistic map If we make a function plot of the logistic map we obtain: ? What do you think this means? 1.Nothing 2.We can read of x n in this graph 3.We can read of x n+1 in this graph 4.It’s upside down

19. GEM2505M 19 Population Dynamics The logistic map Answer: With the help of such a plot, we can graphically determine the value of x n+1.

20. GEM2505M 20 Cobwebs Let’s take a = 2.0. I.e. Value of x n+1 Value of x n+2 Value of x n+1 Value of x n 1) 2) 3) flip over blue point => green point

21. GEM2505M 21 Cobwebs 2.0

22. GEM2505M 22 Sensitive Dependence 2.0

23. GEM2505M 23 Parameter dependence  = 0.5  = 1.0  = 1.5

24. GEM2505M 24 Fixed Points This point will not change when applying the function map. As we can see from the cobweb, wherever we start, we’ll eventually end up at this point for this value of . This kind of a fixed point is called an attracting fixed point.

25. GEM2505M 25 Fixed Points This point will not change when applying the function map. As we can see from the cobweb, wherever we start, we’ll basically never end up at this point for this value of . This kind of a fixed point is called a repelling fixed point.

26. GEM2505M 26 Fixed Points This point will not change when applying the function map. Again the fixed point is repelling, but this time we see a regular period 2 orbit. How does this work?

27. GEM2505M 27 Fixed Points Compositions Just like for the original function, we can plot higher compositions.

28. GEM2505M 28 Compositions – Parameter Dep. 2 nd Composition

29. GEM2505M 29 Compositions – Parameter Dep. 1 st Composition 2 nd Composition 3 rd Composition

30. GEM2505M 30 Compositions – Parameter Dep. The period 1 fixed point is also a fixed point of the second composition. But something special happened. It changed from being attracting to being repelling. Also there are two new fixed points. They are attracting and the time series alternates between them.

31. GEM2505M 31 Transients As can be seen from the cobweb, it usually takes a few time steps until the sequence settles down to a some kind of a stable pattern.. This time is called the transient time and the part of the sequence that falls into this time the transient. Transient. Note: exactly where the transient ends is somewhat arbitrary.

32. GEM2505M 32 Plot 200 successive values of x for every value of  Bifurcation Diagram In a bifurcation diagram, the possible values of x are plotted versus the parameter.

33. GEM2505M 33 Bifurcation Diagram ? How many fixed points are there for  =2? 1.16 2.Infinitely many 3.Zero 4.1 At the end of the bifurcation cascade, we have the maximum amount of chaos

34. GEM2505M 34 Bifurcation Diagram Answer: Infinitely many! Sixth iterate

35. GEM2505M 35 Let's enlarge this area What's so special about this? Let's have a closer look. Bifurcation Diagram

36. GEM2505M 36 Let's try this somewhere else... Hey! This looks almost the same! Bifurcation Diagram

37. GEM2505M 37 Now let's enlarge this area Hard to see, isn't it? Let's enlarge a much smaller area! The area we enlarged before. Bifurcation Diagram

38. GEM2505M 38 The same again! Bifurcation Diagram

39. GEM2505M 39 Indeed, the logistic map repeats itself over and over again at ever smaller scales What's more, this behaviour was found to be universal! Yes, there's a fractal hidden in here. Bifurcation Diagram

40. GEM2505M 40 Simple Map. Amazing Properties! Key Points of the Day

41. GEM2505M 41 What is the map of nature? Think about it! Map, Directions, Lost, Chaos!

42. GEM2505M 42 References http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson4/Demo1.html http://www.expm.t.u-tokyo.ac.jp/~kanamaru/Chaos/e/