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Orbits of Maps

Forward orbits of maps are generated in the same manner as generating the forward orbit of a flow. Position the pointer within the 2-D Image window; click \framebox{SELECT} to set an initial point and \framebox{ADJUST} to iterate the map forward. Continue the iteration with \framebox{ADJUST}. You can also use the Orbits and Selected Point panels as with vector fields. You will see that for the default parameter values almost every initial condition is mapped asymptotically towards an invariant curve which winds around the torus.

Since this map is a diffeomorphism (for small values of the nonlinearity a), the inverse of the map exists, and therefore we can compute backward orbits. Click \framebox{MENU} followed by \framebox{ADJUST} a few times and convince yourself that almost every initial condition is backwards asymptotic to a period five orbit for the default parameter values. (Figure [*])



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1998-11-02