# 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.
BACK to Dynamical Systems Tutorial
Maintained by Giuseppe Zito:Giuseppe.Zito@cern.ch