Poincaré section
This simulation explores the biperiodic attractor of the following
dynamical system:
x' = x (a - b + z
+ d (1-z2)) - c y
y' = y (a
- b + z + d (1-z2)) + c x
z'
= a z - (x2 + y2 + z2)
where
each ' denotes a time derivative and, by default, a = 2.01, b
= 3, c = 0.25 and d = 0.2.
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You can select below the parameters a, b, c and d,
as well as the initial conditions for the dependent variables x,
y and z.
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The simulation will display a projection of the three-dimensional
orbit and the (x,z) coordinates of the solution points
at which y = 0.
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The point of view of the three-dimensional projection can be changed
with the mouse.
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The whole image can be moved with the mouse while pressing Ctrl.
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To change the zoom in the projections, press Shift
when moving up or down the mouse pointer.
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Put the mouse pointer over an element to get information about it.
Activities
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Press Start and, after a while, Clear, to see only the
asymptotic solution (i.e., the behavior on the attractor).
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To fully appreciate the bagel form of the attractor, uncheck Plane
and use the mouse to change the projection view. (You may also want to
set Ω = 0 to avoid the constant change of viewpoint.)
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Is the asymptotic solution periodic?
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Find the fixed points (i.e., the point (x,y,z) at
which x' = y' = z' = 0) and use you favorite
computing tool to discuss their stability.
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What happens for other parameter values?
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.