Transcript The Clockwork Syndrome - Frederick H. Willeboordse

Taming Chaos
GEM2505M
Frederick H. Willeboordse
[email protected]
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The Clockwork Syndrome
Lecture 1
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Today’s Lecture
What does Dynamics mean?
Symbolic Dynamics
Iterative Maps
Everything is Determined
The World of Newton
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What does Dynamics Mean?
In this context, dynamics refers to:
The way objects move over time.
The way a system evolves.
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.
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What does Dynamics Mean?
Coordinate system
1 - Dimension
2 - Dimensions
2
1
-2
-1
0
1
2
We look at how the red
dot changes position on
the line.
-2
-1
0
1
2
We look at how the red
dot changes position in
the plane.
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What does Dynamics Mean?
Co-ordinate system
1 - Dimension
2 - Dimensions
2
1
-2
-1
0
1
2
The variable x can
describe the position of
the point.
-2
-1
0
1
2
Here, since there are two
directions, we need two
variables x and y to
describe the position of
the point.
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What does Dynamics Mean?
1 - Dimension
Over time, the position of the red dot changes.
Time n = 1
-2
-1
0
1
2
-2
-1
0
1
2
-2
-1
0
1
2
Time n = 2
Time n = 3
Hence we can say:
x1= -1
x2= 1
x3= 0
These values are the states of the variable xn
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What does Dynamics Mean?
1 - Dimension
Value of xn
Now, there’s a nice way to draw this in a graph:
2
1
0
-1
0
1
2
3
4
5
6
7
Time n
-2
For clarity, the dots are often connected by a line.
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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:
2
1
-2
-1
0
1
2
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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.
2
1
-2
-1
0
1
2
0
1
2
2
1
-2
-1
Again we can draw a line which maps out the
trajectory. Note there’s no time axis in this example.
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Symbolic Dynamics
What does this mean? Well exactly what the two words
indicate: The evolution of a system of symbols
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
The Rules: a → ab
b → bb
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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.
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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:
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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.
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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 x2
Now you know mathematicians like to be precise and
in order to do so one needs some decent notation.
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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) = x2
Applying the map again, the next step is:
Step 2. f(x2) = x4
This is good but we have not really expressed that we are
dealing with “the first step” and the “second step” in this
notation.
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Iterative Maps
Before, we saw that it is quite convenient to use a subscript
of the variable to indicate the time.
Initial value of x: x0
First value of x: x1
Second value of x: x2
Thus we get:
Called the initial condition.
Called the first iterate.
Called the second iterate.
Step 1. x1 = f(x0 ) = x02
Step 2. x2 = f(x1 ) = x12
Step 3. x3 = f(x2 ) = x22
The sequence of successive time steps x0,x1,x2, … is
called the orbit.
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Iterative Maps
Lastly, we can generalize this and write:
xn+1 = f(xn ) = xn2
What will x∞ be for the above
map if we start with:
x0 = 1.5 → x∞ = ∞
x0 = 0.5 → x∞ = 0
Note: ∞ is the symbol for infinity.
x0 = 1.5
x0 = 0.5
?
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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 nonlinear
-2
xn+1
xn+1
2
2
1
1
-1
0
linear
1
2
xn
-2
-1
0
1
2
xn
non-linear
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What does Dynamics Mean?
(Non-)linear
Examples:
xn+1 = axn
xn+1 = xn2
xn+1
-2
-1
xn+1
2
2
1
1
0
linear
1
2
xn
-2
-1
0
1
2
xn
non-linear
Btw. in this particular non-linear case, you can see that
when starting from 0 or 1, xn+1 remains at 0 or 1. Such
a point is called a fixed point.
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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(a x) = a f(x)
In higher dimensions, this also includes e.g. rotations.
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optional
What does Dynamics Mean?
How about this?
xn+1
xn+1 = 1+ xn
2
1
-2
-1
0
1
2
xn
linear?
The map f(x) = 1+ x does not seem to be linear:
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?
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What does Dynamics Mean?
optional
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:
xn+1 = a xn + b
Now let us introduce the
transformation:
xn = c x’n + d
If we insert this
into (1) we get:
(1)
c x’n+1 + d = a c x’n + a d + b
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What does Dynamics Mean?
optional
Moving the d to the right we obtain:
c x’n+1 = c a x’n + (a d + b – d)
So if we set
d=ad+b
(2)
which is possible as long
as a is not equal to 1. Equation (2) becomes:
x’n+1= a x’n
which is a linear equation.
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Everything is Determined
Philosophical Determinism
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.
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.
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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
da Caprio
Vinci
1452-1519
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.
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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.
1643-1727
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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.
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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.
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The World of Newton
Keppler
AU stands for the
Astronomical Unit:
1.5 x 108 km
One of the great successes of Newton’s theories. An
explanation of Keppler’s laws.
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The World of Newton
The End of Newton’s world
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!
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The World of Newton
The End of Newton’s world?
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.
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Key Points of the Day
Evolving systems states by
rules/recipes.
Determinism
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Think about it!
Is the world a
clockwork?
Clock,
Work,
Salary,
The 5 Cs are mine!
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