1 GEM2505M Frederick H. Willeboordse Taming Chaos

2 The Clockwork Syndrome Lecture 1

GEM2505M 3 Today’s Lecture What does Dynamics mean? Symbolic Dynamics Iterative Maps Everything is Determined The World of Newton

GEM2505M 4 What does Dynamics Mean? We could look at how an actual ball moves e.g. but often we’re just dealing with points in a plane that are representative for the system we’re dealing with. The way objects move over time. The way a system evolves. In this context, dynamics refers to:

GEM2505M 5 What does Dynamics Mean? Coordinate system 1 - Dimension2 - Dimensions We look at how the red dot changes position on the line. We look at how the red dot changes position in the plane

GEM2505M 6 What does Dynamics Mean? Co-ordinate system 1 - Dimension2 - Dimensions The variable x can describe the position of the point. Here, since there are two directions, we need two variables x and y to describe the position of the point

GEM2505M 7 What does Dynamics Mean? 1 - Dimension Over time, the position of the red dot changes Time n = 1 Time n = 2 Time n = 3 Hence we can say:x 1 = -1x 2 = 1x 3 = 0 These values are the states of the variable x n

GEM2505M 8 What does Dynamics Mean? 1 - Dimension Now, there’s a nice way to draw this in a graph: For clarity, the dots are often connected by a line Time n Value of x n

GEM2505M 9 What does Dynamics Mean? 2 - Dimensions In two dimensions, a point in a plane can be described by a vector If the red dot is at this location at time n=1 we write:

GEM2505M 10 What does Dynamics Mean? 2 - Dimensions If the red dot moves to this location at time n=2 we write: If the red dot moves to this location at time n=3 we write: Old location of red dot. Again we can draw a line which maps out the trajectory. Note there’s no time axis in this example.

GEM2505M 11 Symbolic Dynamics Let us say we have a system that can be described by a string of symbols Over time, these symbols change according to a fixed set of rules Say our symbols are a and b What does this mean? Well exactly what the two words indicate: The evolution of a system of symbols The Rules: a → ab b → bb

GEM2505M 12 Symbolic Dynamics Lets say we start with the string b and then apply the rules over and over again. The Rules: a → ab b → bb 1. b → bb 2. bb → bbbb 3. bbbb → bbbbbbbb Well that’s a bit boring, but that’s all there is too it. Note, that the next “state” is entirely predictable. Also note that the rule is like a little program.

GEM2505M 13 Symbolic Dynamics So let’s start with the string ba instead and see what happens The Rules: a → ab b → bb 1. ba → bbab 2. bbab → bbbbabbb 3. bbbbabbbb → bbbbbbbbabbbbbbbbb As you can see, the key is to apply a rule over an over again. Of course, one can make systems with more interesting rules. We’ll get back to that in a later lecture. But:

GEM2505M 14 Symbolic Dynamics Do you know of anything really useful that is based entirely on symbolic dynamics? ? The computer! All it does is repeatedly apply rules to strings of zeros and ones.

GEM2505M 15 Iterative Maps Iteration just means repetition here. So an iterative map is a map that you apply over and over again. What’s a map? A map is just a rule for changing the value of a variable. E.g. the rule ‘square’ changes x to x 2 Now you know mathematicians like to be precise and in order to do so one needs some decent notation.

GEM2505M 16 Iterative Maps So we call the rule “f” and write the variable to which we apply it between brackets. If we start with x, the first step is: Step 1. f(x) = x 2 Applying the map again, the next step is: Step 2. f(x 2 ) = x 4 This is good but we have not really expressed that we are dealing with “the first step” and the “second step” in this notation.

GEM2505M 17 Iterative Maps Before, we saw that it is quite convenient to use a subscript of the variable to indicate the time. Thus we get: Initial value of x: First value of x: Second value of x: x0x1x2x0x1x2 Step 1. x 1 = f(x 0 ) = x 0 2 Step 2. x 2 = f(x 1 ) = x 1 2 Step 3. x 3 = f(x 2 ) = x 2 2 Called the first iterate. Called the second iterate. Called the initial condition. The sequence of successive time steps x 0,x 1,x 2, … is called the orbit.

GEM2505M 18 Iterative Maps Lastly, we can generalize this and write: x n+1 = f(x n ) = x n 2 What will x ∞ be for the above map if we start with: x 0 = 0.5 x 0 = 1.5 Note: ∞ is the symbol for infinity. ? x 0 = 1.5 → x ∞ = ∞ x 0 = 0.5 → x ∞ = 0

GEM2505M 19 What does Dynamics Mean? (Non-)linear A map changes a variable as we have seen. In one dimension, if these changes are on a line, we call the map linear. If they are on a curve, we call then non- linear linear non-linear xnxn x n+1 xnxn

GEM2505M 20 What does Dynamics Mean? (Non-)linear Examples: linear non-linear x n+1 =  x n x n+1 = x n 2 Btw. in this particular non-linear case, you can see that when starting from 0 or 1, x n+1 remains at 0 or 1. Such a point is called a fixed point. xnxn x n+1 xnxn

GEM2505M 21 What does Dynamics Mean? (Non-)linear Technically speaking, a map is linear if the following two conditions are fulfilled: f(x + y) = f(x) + f(y) f(  x) =  f(x) In higher dimensions, this also includes e.g. rotations.

GEM2505M 22 x n+1 =  x n What does Dynamics Mean? How about this? The map f(x) = 1+ x does not seem to be linear: linear? x n+1 xnxn f(a + b) = 1 + a + b = f(a) + f(b) = 2 + a + b That’s a bit strange, if you look at this as the motion of an object, it is clearly on a line! Shouldn’t that be considered linear? optional

GEM2505M 23 What does Dynamics Mean? In physics one is often free to translate and scale. If this is the case we’re fine since every line can be transformed into a linear form: In general, for motion on a line we have: optional x n+1 = a x n + b x n = c x’ n + d Now let us introduce the transformation: If we insert this into (1) we get: (1) c x’ n+1 + d = a c x’ n + a d + b

GEM2505M 24 What does Dynamics Mean? So if we set optional which is possible as long c x’ n+1 = c a x’ n + (a d + b – d) d = a d + b x’ n+1 = a x’ n as a is not equal to 1. Equation (2) becomes: which is a linear equation. Moving the d to the right we obtain: (2)

GEM2505M 25 Everything is Determined As a philosophical belief about the material world, determinism can be traced as least as far back as the time of Ancient Greece, several thousand years ago. Philosophical Determinism Determinism is the philosophical belief that every event or action is the inevitable result of preceding events and actions. Thus, in principle at least, every event or action can be completely predicted in advance, if we know the rules.

GEM2505M 26 Everything is Determined Determinism gained a foothold in modern science around the year 1500 A.D. with the establishment of the idea that cause-and-effect rules govern all motion and structure on the material level Leonardo di Caprioda Vinci Leonardo da Vinci was one of the instrumental figures in the transition to the modern scientific approach through his brilliant explorations in science, art and engineering.

GEM2505M 27 The World of Newton Newton laid the foundation for differential and integral calculus. His work on optics and gravitation make him one of the greatest scientists the world has known. Isaac Newton In 1687 he published Philosophiae naturalis principia mathematica or Principia as it is usually known.

GEM2505M 28 The World of Newton Newton discovered a concise set of principles, expressible in only a few sentences, which he showed could predict the motion in an astonishingly wide variety of systems to a very high degree of accuracy. Newton demonstrated that his three laws of motion, combined through the process of logic, could accurately predict the orbits in time of the planets around the sun, the shapes of the paths of projectiles on earth, and the schedule of the ocean tides throughout the month and year, among other things.

GEM2505M 29 The World of Newton I.A body with no forces acting on it is either at rest or moves with constant speed in a straight line. II.The acceleration of a body is directly proportional to the net force acting on it and inversely proportional to its mass. III.Two bodies always exert equal and opposite forces on each other.

GEM2505M 30 The World of Newton AU stands for the Astronomical Unit: 1.5 x 108 km Keppler One of the great successes of Newton’s theories. An explanation of Keppler’s laws.

GEM2505M 31 The World of Newton In the light of the overwhelming successes of Newton’s and other theories, it is only ‘natural’ to think of the world as deterministic, orderly and predictive. However, if we forget about all that for a moment, one doesn’t need to be a rocket scientist to see that in fact the world is exceedingly complex. Indeed, often we even can’t predict the next day’s weather! The End of Newton’s world

GEM2505M 32 The World of Newton Sometimes, it is argued that the purely deterministic worldview came to an end with the introduction of quantum mechanics. However, quantum mechanics is a linear theory! Although laws are expressed in terms of probabilities, time irreversability and complexity cannot elegantly be incorporated. Of course, quantum mechanics is even more successful than the classical mechanics à la Newton leading to an even firmer (and I think wrong) believe that it can explain everything. The End of Newton’s world?

GEM2505M 33 Evolving systems states by rules/recipes. Determinism Key Points of the Day

GEM2505M 34 Is the world a clockwork? Think about it! Clock, Work, Salary, The 5 Cs are mine!