Pendulum as a dynamical system
In fact the relationship between discrete maps and physical systems is
slightly more complex.
x = f(x,y) (1)
y = g(x,y)
represents a discrete system with two degrees of freedom. This isn't
connected to a continuous system with 2 degrees of freedom like those
described by the differential equation:
-- = f(x,y)
-- = g(x,y)
Because this is solved by the following iteration
x = f(x,y)*dt + x (2)
y = g(x,y)*dt + y
with dt being a constant chosen in such a way that the orbits generated from
(2) should be continuous. You can't have chaos for this kind of iterations
that can be thought as a subset of the more general iterations (1).
To have chaotic orbits in continuous physical systems, at least 3 degrees
of freedom are needed. Discrete iterations like (1) are then associated to
these physical systems through a mechanism called Poincare' map .
This is similar to cutting the 3D orbits with a plane and considering only
the intersections points. It is difficult to pass from the motion equation
to the discrete Poincare map, but we can say that a random planar map can
represent a 3D motion of a 3D continuous physical system described by the
set of differential equations:
-- = f(x,y,z)
-- = g(x,y,z)
-- = g(x,y,z)
Anyway we can study such physical systems directly, by solving them
numerically and then studying only the iteration produced by two of the
3 coordinates , for example x,y.
Overall properties like chaos and symmetry should be the same.
For example we can consider the driven pendulum:
-- = y
-- = (-1/q)*y - sin(x) + g*cos(z)
-- = w
these are solved by the iteration
x = y*dt + x
y = ((-1/q)*y-sin(x)+g*cos(z))*dt + y
z = w*dt + z
We can play with the value of dt getting orbits that became more and more
discrete. But chaos and simmetry are kept.
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