Pendulum as a dynamical system

In fact the relationship between discrete maps and physical systems is slightly more complex. The iteration:

   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:

  dx  
  --  = f(x,y)
  dt

  dy  
  --  = g(x,y)
  dt

Why? 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:

 dx
 -- = f(x,y,z)
 dt

 dy
 -- = g(x,y,z)
 dt

 dz
 -- = g(x,y,z)
 dt

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:

 dx
 -- = y       
 dt

 dy
 -- = (-1/q)*y - sin(x) + g*cos(z)
 dt

 dz
 -- = w
 dt

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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Maintained by Giuseppe Zito:Giuseppe.Zito@cern.ch