What was that ?

1. What was that?

3. The Mandelbrot Set is a beautiful example of Chaos Theory. Chaos theory is a discipline of pure mathematics that has many applications in medicine, physics, engineering , biology, psychology etc. Chaotic behavior can be witnessed in nature. Mathematicians attempt to explain such behavior through rigid analysis of mathematical models.

4. More specifically, Chaos theory studies the behaviors of dynamical systems. These “dynamical systems” exhibit sensitivity to initial conditions . This sensitivity is commonly referred to as the butterfly effect.

5. When a butterfly flaps its wings in one part of the world it can cause a hurricane  in another part of the world.   • Author Unknown

6. Edward Lorenz Contributions 1917-2008 Mathematician & Meteorologist • Coined the term “Butterfly Effect” • Discovered that even a tiny alteration in initial conditions (.506127 to .506), can transform a long term forecast. • Butterfly effect is based on his 1972 paper “ Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

7. It is just the concept that small events can have large, widespread consequences.

8. Is the “Butterfly Effect” REAL?

9. Yes or No?

10. YES! The event that a butterfly flaps its wings, represents a small change in the initial condition of the system. This small change causes a chain of events that lead to huge large-scale alterations. In other words, If the butterfly decided NOT TO flap its wings, the end result may have been different!

11. RECAP: Chaotic behavior can be witnessed in nature. Mathematicians attempt to explain such behavior through rigid analysis of mathematical models.

12. Let’s take a look at one such model….

13. Colonial Growth of Single Cell Organisms - By Rivkah Singh

14. FactIf a Meteorite carrying an alien colony Of single cell organisms crashes on earth, It will only take 63.1 Days to cover the entire surface of the United States. Assuming that the 48 states have an area of 1019 square mm and in n days the area of the mold doubles to 2n times its starting area.

15. QUESTION How long will it take for the organisms to inhabit the entire planet? Assume the total area of the surface of the earth is 5 x 1020 mm2 , and the organisms double in size every day . Or rather, in n days they have grown to 2n times their beginning area.

16. Let’s assume that the mold colony begins as a one-square mm blob and it’s area doubles in size every day. Since we know that the entire surface of the earth is 5 x 1020 mm2,   Log(2n ) = log (5 x 1020 ) N x log 2 = log 5 + 20 log 10 N = (log5 + 20log10)/log2 = 0.6989 + 20 Thus, n = = 68.7  Therefore, it will take a mere 68.7 days for the mold to inhabit the entire surface of the Earth!

17. If our alien mold is harmful to human life, it would appear that we are doomed. However, Are we certain that the world is doomed in 68.7 days?

18. Why?

19. Are you sure?

20. Let’s take a look at some basic Wikipedia definitions Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect. Butterfly effect- In chaos theory, the butterfly effect is the sensitive dependence on initial conditions, where a small change at one place in a deterministic nonlinear system can result in large differences to a later state..

21. In other words, If our system is both non-linear and dynamic, we can study it even further………………………………………………….

22. Our alien colony exhibits exponential growth. Days Area 63.1 64.1 2 x 65.1 4 x 66.1 8 x 67.1 16 x 68.1 32 x 69.1 64 x

23. Single Cell Colonial Growth is an example of a dynamical system Single Cell Organisms have the following characteristics: • 1.They reproduce within regular intervals. These intervals are commonly refereed to as generations. • 2.They exhibit fecundity. Fecundity are the fixed rates of birth and death Dynamical systems have the following requirements: • 1. The state of the system must be defined. • 2. It must include a set rule of change. This rule is referred to as a dynamic. (by dynamic it is inferred that it includes all sorts of deterministic change)

24. OK, so our alien system is both nonlinear and dynamical….. Now what?

25. Now we find a formula……

26. Since we know that the colonial growth of single cell organisms is an example of a dynamical system we can set up a formula In order to set up a formula, we must first decide on the variable of the populace after n generations and the fecundity Where: Pn = the quantity of cells inherent within the population after the nth generation F = fecundity : f= 1 +( birthrate) – (death rate) Thus Pn+1 = f * Pn Is our formula that will determine the size of our populace for every generation

27. With a formula, we can make a mathematical model.The best way to develop a mathematical model is with the use of computers

28. This model was developed with software called Geogebra GeoGebra Tube

29. Pn+1 = *Pn is equivalent to the equation of a straight line y=a*x. This line is called the compounding line. The addition of two lines f(x) and g(x)(the 45 degree line) enable us to find the graphical iteration more easily.

30. Single Cell growth exhibiting exponential growth, Decay, and bifurcation

34. Time series for periodic cycling

35. Are we doomed yet ? Are We Doomed Yet?

36. The tent map is the simplest dynamical system. Function[If[x ≤ 0.5, a x, a - a x], 0, 1

37. How is the Tent Map related to our example of colonial growth? It is related because if we assume our population is governed by the equation: Xn+1 = Xn + b*Xn – d*Xn Where b = births, d = deaths, and n=generation . Then after time, when space is limited it starts to control the birth rate. Xn+1 =Xn+b*(1-Xn) – d*Xn Here 1 = the total space or the universal set and Xn is the space already occupied by the previous generation. Thus, 1-Xn is the fraction of space still available. So if we measure time in units of 1/d (rescale time so that d=1). We have our tent. Which is: Xn+1=a*Xn If Xn is less than 0.5 Xn+1=a*(1-Xn) otherwise

38. When a <1, 0 is an attractor.

39. An example of a few iterations when a>1

40. a=1.99 Chaos

41. A=1.6

42. a = 1.2 The a values between 1 and 2 never exactly repeat. Hence, there is deterministic chaos..

43. We can therefore conclude that since our time series acts erratically and chaotic. It exhibits deterministic chaos. Hence, we are not necessarily “doomed” . We are saved by the knowledge that small differences in our initial conditions (however minute) give us diverging outcomes; therefore, prediction as precise as ours in the beginning of this presentation is nearly impossible. We can safely assume that even if our system is deterministic – it does not automatically make it predictable.