Transcript Taming Chaos

Taming Chaos
GEM2505M
Frederick H. Willeboordse
[email protected]
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Strange Attractors
Lecture 12
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Today’s Lecture
The Story
Strange Attractors
Lorenz Equations
Henon Map
Many physical systems are
dissipative. Hence one would
expect their attractors to be
simple. This turns out to be
untrue when so-called strange
attractors were discovered.
What is strange about a strange
attractor?
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Higher Dimensions
One cannot expect that all chaos-like phenomena can fully
be described by one-dimensional systems.
It is therefore useful, to look at higher dimensions. After
one comes two ….
Will two be enough? That depends!
When the dimensionality is low, one generally deals with the
dynamics of a single point. However, single units (think of
cells, atoms, molecules etc.) can interact giving rise to many
interesting collective phenomena. Systems built up of many
individual units can have fairly high dimensions.
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Dissipative Systems
Most systems in the real world must include some kind of
friction. That is to say they loose (dissipate) energy.
Assumption:
Simple
A common assumption was that in dissipative
systems the final state would be rather simple. A
point or some regular motion for example.
Fact:
Can be chaotic
However, simple dissipative systems have been
discovered where the final state is anything but
simple with chaotic dynamics and fractal
structures.
Note: There are also conservative systems where there is
no energy loss. Conservative systems too can have very
interesting dynamical properties.
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Strange Attractors
Strange attractor is the name for the final state of a
dissipative system that displays chaotic dynamics.
Since the system is dissipative, the size of any
area must shrink. Consequently, in the limit of an
infinite number of iterations any area becomes
infinitely small. Hence a strange attractor is an
object with no area or volume!
In strange attractors, chaos and fractals come together nicely
illustrating clearly how chaos is a dynamical property and
fractal a geometric property.
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The Lorenz Attractor
Definition
The Lorenz equations are defined as:
The famous Lorenz attractor
Parameters used by Lorenz
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The Lorenz Attractor
The Derivation
The derivation of the equations is beyond the scope of this
course. However:
•
The equations are based on a model
for the cylindrical fluid convection
that appears on top of a heated plate
•
It is hence not a model of the actual
airflow
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The Lorenz Attractor
Meaning of the Variables
Roughly:
x relates to streamfunction that characterizes
fluid flow
y is proportional to the temperature difference
between the upwards and downwards moving
parts of a roll
z describes the nonlinearity of the temperature
difference along the roll
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The Lorenz Attractor
Key Properties
1. There are only two nonlinearities xy and xz.
2. There is a symmetry (x,y) -> (-x,-y) (hence if x(t),
y(t), z(t) is a solution, then –x(t),-y(t),z(t) is a
solution too).
3. The Lorenz equations are dissipative. In fact,
volumes shrink exponentially fast.
4. There are no repelling fixed points or orbits (this
would contradict that all volumes contract).
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The Lorenz Attractor
Dynamics
Basically, the trajectory goes
through the following two steps
repeated ad infinitum:
1. Spiral outward
2. Move over to the other side
Rather than stretch and fold this is: stretch-split-merge
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The Lorenz Attractor
Dynamics
1. Spiral outward
2. Move over to the other side
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The Lorenz Attractor
Bifurcations
Since there are three
parameters, bifurcations
can in principle occur in
many different ways.
One example is given to
the left.
http://risa.is.tokushima-u.ac.jp/~tetsushi/chen/chenbif/node8.html
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The Lorenz Attractor
Sensitive Dependence
Trajectories
still very
close
Trajectories
far apart
Two nearby trajectories can stay close for quite a long time
(depending on where one starts). However, at some point they
strongly diverge.
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The Lorenz Attractor
Sole p 11
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The Lorenz Attractor
Fractal
For the type of equations like the Lorenz equations, there is
a theorem (called the uniqueness theorem) which states that
its solutions are unique and hence that trajectories can never
intersect.
In the Lorenz attractor, we see the two
sheets ‘merging’ but a real merge is not
possible due to the above theorem. What
we really get is a fractal.
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Waterwheel
It turns out that a simple and
conceptually easy to understand
model exists for the Lorenz
equations: the waterwheel!
In a waterwheel, leaky cups are attached to a wheel and
water is steadily poured in exactly from the top.
The main parameter to be varied here is
the water flow (having a role similar to
the nonlinearity in the logistic map).
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Waterwheel
For very low flow rates, the wheel
just stands still since more water
will drain out of the cup than can
flow in.
When the flow is big enough so that the cups start filling
up, the wheel will turn regularly in one direction or the
other.
If one then increases the flow even further, chaotic
switching between rotational directions occurs.
Note: The (simplified) model equations for the waterwheel can be
transformed into the Lorenz equations.
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The Henon Map
Definition
It was introduced by the French astronomer Michel Henon
in 1976 as a simplification of the Lorenz equations.
However, it also is the extension of the logistic map into
two dimensions.
The simplest way to extend the logistic map would be to
just add linear terms.
Logistic Map
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The Henon Map
Reasoning
Attempt to simulate the stretching and folding in the Lorenz
system.
Start
Stretch and Fold
Squeeze
Reflect
Combining the three transformations yields the Henon map.
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The Henon Map
Key Properties
•
•
•
Invertible: In the Lorenz system, each point in
phase space has a unique trajectory associated
with it. This is completely different from the
logistic map!
Dissipative: It contracts areas. In fact it does so
at the same rate everywhere in phase space.
There is a strange attractor
Note: Henon map = discrete, Lorenz system = continuous
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The Henon Map
“The” Henon Attractor
0.4
-0.4
1.5
-1.5
This is how the Henon
attractor looks.
To me: kind of like a
paper clip …
and quite different
indeed from the Lorenz
attractor.
a = 1.4, b = 0.3
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The Henon Map
Basin of Attraction
The basin of attraction is
the set of all points in the
plane the end up on the
attractor.
Other points escape to
infinity.
a = 1.4, b = 0.3
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The Henon Map
Area transform
Any square near the
attractor will be mapped
onto the attractor.
In order to apply the
transformation, the square
is built up of 10,000
points to which the Henon
map is applied one by
one.
n=0
n=1
n=2
n=3
n=5
n = 10
a = 1.4, b = 0.3
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The Henon Map
Sensitive Dependence
(x0,y0) = (0,0)
(x0,y0) = (0.000001,0)
Difference between
the trajectories.
a = 1.4, b = 0.3
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The Henon Map
Bifurcations
Just as for the logistic
map, we can generate a
bifurcation diagram by
varying the nonlinearity.
There are two differences
though. 1) We need to
choose a value for b and
keep it fixed. 2) We need
to start from several initial
conditions since there are
multiple attractors.
b = 0.3
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The Henon Map
Bifurcations
Multiple attractors?? If
we zoom in to the area
on the left, we see that
there are two separate
bifurcation diagrams.
Depending on the initial
condition, the orbit will
either go the upper or
the lower bifurcation
diagram.
b = 0.3
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The Henon Map
Fractal
0.193068
a = 1.4, b = 0.3
0.19294
The orbit of the
Henon attractor
has a fractal
structure
0.60782
0.60811
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The Henon Map
A little math
Not in exam
Invertability
The Henon map
After moving
the left
to
The inverted Henon map
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The Henon Map
A little math
Not in exam
Area reduction
In general, a 2-dimensional map is area reducing if it’s the
absolute value of the determinant of it’s Jacobian matrix is
smaller than 1.
The map is area reducing if:
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The Henon Map
A little math
Not in exam
Area reduction
Applying this to the Henon map we obtain:
area reducing if |b| < 1
Hence we see that the Henon map is area reducing and
that this reduction is the same everywhere.
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Key Points of the Day
Dissipative Systems
Lorenz Equations
Henon Map
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Think about it!
Waterwheel
Waterwheel
Countryside,
Farm,
Life!
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References
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