Baker map
This simulation computes a class of generalized baker maps defined in the
unit square as follows:
xn+1 = xn / a
+ [b yn] / b
yn+1 = b
yn mod 1
where [y] = y - y mod 1
is the integer part of y, i.e., the largest integre not larger than y.
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You can select below the integer parameters a and b, as
well as the initial conditions for the x and y.
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Click and drag the mouse to select a part phase-space to be zoomed in.
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If Distance is selected a popup window will show the evolution
of the distance between the points of the orbits starting from (x0,y0)
and (x0+δx,y0+δy).
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For information on other elements, put over them the mouse pointer to
get a tooltip.
Activities
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Start the simulation by pressing Start. By default the orbit
will be erased after the first 100 steps to skip the transitory until
the system is on the attractor (well, very very close to it).
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Use different initial values for x and y. Is there any
difference in the displayed section? Discuss your answer.
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Use the mouse to select smaller and smaller parts of the attractor
section. The increased resolution will show that the attractor has
structure at all magnification levels: it is a fractal, much like a
Cantor set.
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Notice that at higher magnifications you will have to wait longer to
get a number of solution points high enough to see the attractor.
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Press Whole space to recover the original magnification.
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Change the values of the integers a and b and discuss
the shortcomings of computing this kind of maps with finite binary
precision (for instance, when b is even).
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You can run the example baker of Dynamics
Solver for a much faster simulation, which allow exploring the
Cantor set structure of the attractor.
This is an English translation of the Basque original for a course
on mechanics, oscillations and waves.
It requires Java
1.5 or newer and was created by Juan
M. Aguirregabiria with Easy
Java Simulations (Ejs) by Francisco
Esquembre. I thank Wolfgang Christian and Francisco Esquembre for
their help.