Chaos Theory and Predictability

1. Chaos Theory and Predictability Anthony R. Lupo Department of Soil, Environmental, and Atmospheric Sciences 302 E ABNR Building University of Missouri Columbia, MO 65211

2. Chaos Theory and Predictability • Some popular images…………..

3. Chaos Theory and Predictability • Any attempt at weather “forecasting is immoral and damaging to the character of a meteorologist” - Max Margules (1904) (1856 – 1920) • Margules work forms the foundation of modern Energetics analysis.

4. Chaos Theory and Predictability • “Chaotic” or non-linear dynamics Is perhaps one of the most important “discovery” or way of relating to and/or describing natural systems in the 20th century! • “Caoz” Chaos and order are opposites in the Greek language - like good versus evil. • Important in the sense that we’ll describe the behavior of “non-linear” systems!

5. Chaos Theory and Predictability • Physical systems can be classified as:  Deterministic  laws of motion are known and orderly (future can be directly determined from past) • Stochastic / random  no laws of motion, we can only use probability to predict the location of parcels, we cannot predict future states of the system without statistics. Only give probabilities!  

6. Chaos Theory and Predictability

7. Chaos Theory and Predictability • Chaotic systems  We know the laws of motion, but these systems exhibit “random” behavior, due to non-linear mechanisms. Their behavior may be irregular, and may be described statistically. • E. Lorenz and B. Saltzman  Chaos is “order without periodicity”.

8. Chaos Theory and Predictability • Classifying linear systems • If I have a linear set of equations represented as: (1) And ‘b’ is the vector to be determined. We’ll assume the solutions are non-trivial. • Q: What does that mean again for b? • A: b is not 0!

9. Chaos Theory and Predictability • Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots (source: Mathworld) (l) • Eigenvectors are a special set of vectors associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic vectors (vector ‘b’)

10. Chaos Theory and Predictability • Thus we can easily solve this problem since we can substitute this into the equations (1) from before and we get: Solve, and so, now the general solution is: • Values of ‘c’ are constants of course. The vectors b1 and b2 are called “eigenvectors” of the eigenvaluesl1 and l2.

11. Chaos Theory and Predictability • Particular Solution:

12. Chaos Theory and Predictability • One Dimensional Non-linear dynamics • We will examine this because it provides a nice basis for learning the topic and then applying to higher dimensional systems. •  However, this can provide useful analysis of atmospheric systems as well (time series analysis). Bengtssen (1985) Tellus – Blocking. Federov et al. (2003) BAMS for El Nino. Mokhov et al. (1998, 2000, 2004). Mokhov et al. (2004) for El Nino via SSTs (see also Mokhov and Smirnov, 2006), but also for temperatures in the stratosphere. Lupo et al (2006) temperature and precip records. Lupo and Kunz (2005), and Hussain et al. (2007) height fields, blocking.  

13. Chaos Theory and Predictability • First order dynamic system: • (Leibnitz notation is “x –dot”?) • If x is a real function, then the first derivative will represent a(n) (imaginary) “flow” or “velocity” along the x – axis. Thus, we will plot x versus “x – dot”

14. Chaos Theory and Predictability • Draw: • Then, the sign of “f(x)” determines the sign of the one – dimensional phase velocity. • Flow to the right (left): f(x) > 0 (f(x) < 0)

15. Chaos Theory and Predictability • Two Dimensional Non-linear dynamics • Note here that each equation has an ‘x’ and a ‘y’ in it. Thus, the first deriviatives of x and y, depend on x and y. This is an example of non-linearity. What if in the first equation ‘Ax’ was a constant? What kind of function would we have?  • Solutions to this are trajectories moving in the (x,y) phase plane.

16. Chaos Theory and Predictability • Coupled set: If the set of equations above are functions of x and y, or f(x,y). • Uncoupled set: If the set of equations above are functions of x and y separately.

17. Chaos Theory and Predictability • Definitions • Bifurcation point: In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied.

18. Chaos Theory and Predictability • Example: “pitchfork” bifurcation (subcritical) • Solution has three roots, x=0, x2 = r

19. Chaos Theory and Predictability • The devil is in the details?............

20. Chaos Theory and Predictability • An attractor is a set of states (points in the phase space), invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction

21. Chaos Theory and Predictability How we see it…….

22. Chaos Theory and Predictability Mathematics looks at Equation of Motion (NS) is space such that: • Closed or compact space such that  boundaries are closed and that within the space divergence = 0 • Complete set  div = 0 and all the “interesting” sequences of vectors in space, the support space solutions are zero.

23. Chaos Theory and Predictability • Ok, let’s look at a simple harmonic oscillator (pendulum):  • Where m = mass and k = Hooke’s constant. • When we divide through by mass, we get a Sturm – Liouville type equation.

24. Chaos Theory and Predictability • One way to solve this is to make the problem “self adjoint” or to set up a couplet of first order equations like so let:

25. Chaos Theory and Predictability • Then divide these two equations by each other to get: • What kind of figure is this?

26. Chaos Theory and Predictability

27. Chaos Theory and Predictability • A set of ellipses in the phase space.

28. Chaos Theory and Predictability • Here it is convenient that the origin is the center! • At the center, the “flow” is still, and since the first derivative of x is positive, we consider the “flow” to be anticyclonic (NH) “clockwise” around the origin. The eigenvalues are: • Now as the flow does not approach or repel from the center, we can classify this as “neutrally stable”.

29. Chaos Theory and Predictability Thus, the system behaves well close to certain “fixed points”, which are at least neutrally stable. • System is forever predictable in a dynamic sense, and well behaved. • we could move to an area where the behavior changes, a bifurcation point which is called a “separatrix”. • Beyond this, system is unpredictable, or less so, and can only use statistical methods. It’s unstable!

30. Chaos Theory and Predictability • Hopf’s Bifurcation: • Hopf (1942) demonstrated that systems of non-linear differential equations (of higher order that 2) can have peculiar behavior. • These type of systems can change behavior from one type of behavior (e.g., stable spiral to a stable limit cycle), this type is a supercritical Hopf bifurcation.

31. Chaos Theory and Predictability • Hopf’s Bifurcation: • Subcritical Hopf Bifurcations have a very different behavior and these we will explore in connection with Lorenz’s equations, which describe the atmosphere’s behavior in a simplistic way. With this type of behavior, the trajectories can “jump” to another attractor which may be a fixed point, limit cycle, or a “strange” attractor (chaotic attractor – occurs in 3 – D only!)

32. Chaos Theory and Predictability • Example of an elliptic equation in meteorology:

33. Chaos Theory and Predictability • Taken from Lupo et al. 2001 (MWR)

34. Chaos Theory and Predictability • Ok, let’s modify the equation above: • d is now the “damping constant”, so let’s “damp“ (“add energy to”) this expression d > 0 (d< 0). • Then the oscillator loses (gains) energy and the determinant of the quadratic solution is also less (greater) than zero! So trajectories spin toward (away from) the center. This is a(n) “(un)stable spiral”.

35. Chaos Theory and Predictability • Example: forced Pendulum (J. Hansen)

36. Chaos Theory and Predictability • Another Example: behavior of a temperature series for Des Moines, IA (taken from Birk et. al. 2010)

37. Chaos Theory and Predictability • Another Example: behavior of 500 hPa heights in the N. Hemi. (taken from Lupo et. al. 2007, Izvestia)

38. Chaos Theory and Predictability • Sensitive Dependence on Initial Conditions (SDIC – not a federal program ). • Start with the simple system : • A iterative-type equation used often to demonstrate population dynamics: • Experiment with k = 0.5, 1.0, 1.5, 1.6, 1.7, 2.0

39. Chaos Theory and Predictability • For each, use the following xn and graph side-by-side to compare the behavior of the system. • Xn = -0.5, Xn = -0.50001 • Try to find: “period 2” attractor or attracting point: behavior1  behavior2  behavior 1  behavior2, and a “period 4” attractor. • Period 2 behaves like the large-scale flow?

40. Chaos Theory and Predictability • Examine the initial conditions. One can be taken to be a “measurement” and the other, a “deviation” or “error”, whether it’s “generated” or real. It’s a point in the ball-park of the original. • Asside: Heisenberg’s Uncertainty Principle  All measurements are subject to a certain level of uncertainty.

41. Chaos Theory and Predictability • X = 0.5 X = 0.50001

42. Chaos Theory and Predictability • What’s the diff?

43. Chaos Theory and Predictability • The differences that emerge illustrate the concept of Sensitive Dependence on Initial Conditions (SDIC). This is an important concept in Dynamic systems. This is also the concept behind ENSEMBLE FORECASTING! • Toth, Z., and E. Kalnay, 1993: Ensemble forecasting at NCEP: The generation of perturbations. Bull.Amer. Meteor. Soc., 74, 2317 – 2330. • Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea.Rev., 125, 3297 – 3319. • Tracton, M.S., and E. Kalnay, 1993: Ensemble forecasting at the National Meteorological Center: Practical Aspects. Wea. and Forecasting, 8, 379 – 39

44. Chaos Theory and Predictability • The basic laws of geophysical fluid dynamics describe fluid motions, they are a highly non-linear set of differentials and/or differential equations. • e.g., • Given the proper set of initial and/or boundary conditions, perfect resolution, infinite computer power, and precise measurements, all future states of the atmosphere can be predicted forever!

45. Chaos Theory and Predictability • Given that this is not the case, these equations have an infinite set of solutions, thus anything in the phase space is possible. • In spite of this, the same “solutions” appear time and time again!

46. Chaos Theory and Predictability • Note: We will define Degrees of Freedom  here this will mean the number of coordinates in the phase space. • Advances in this area have involved taking expressions with an infinite number of degrees of freedom and replacing them with expressions of finite degrees of freedom. • For the equation of motion, whether we talk about math or meteorology, we usually examine the N-S equations in 2-D sense. Mathematically, this is one of the Million dollar problems to solve in 3-d (no “uniqueness” of solutions!).

47. Chaos Theory and Predictability • “Chaotic” Systems: • 1. A system that displays SDIC. • 2. Possesses “Fractal” dimensionality

48. Chaos Theory and Predictability • Fractal geometry – “self similar” • Norwegian Model L. Lemon

49. Chaos Theory and Predictability • Fractals: • Fractal geometry was developed by Benoit Mandelbrot (1983) in his book the Fractal Geometry of Nature. Fractal comes from “Fractus” – broken and irregular. • Fractals are precisely a defining characteristic of the strange attractor and distinguishes these from familiar attractors.

50. Chaos Theory and Predictability • 3. Dissipative system: • Lyapunov Exponents - defined as the average rates of exponential divergence or convergence of nearby trajectories. • They are also in a very real sense, they provide a quantitative measure of SDIC. Let’s introduce the concept using the simplest type of differential equation.