x = f(x,y) (1) y = g(x,y)represents a discrete system with two degrees of freedom. This

dx -- = f(x,y) dt dy -- = g(x,y) dtWhy? Because this is solved by the following iteration

x = f(x,y)*dt + x (2) y = g(x,y)*dt + ywith 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

dx -- = f(x,y,z) dt dy -- = g(x,y,z) dt dz -- = g(x,y,z) dtAnyway 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 dtthese are solved by the iteration

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