Consider the function f defined by ,
where a>0. Given any initial point ,
it is clear that the dynamical system
will have orbit
Introduce a slight perturbation to the initial
condition, say, . The resulting orbit is:
So, after k iterations, these two orbits that started at
neighbouring points and have separated
by . Clearly, if 0<a<1, all orbits converge to zero.
For a=1, all points are fixed points of the map (f is the
identity map). For a>1, orbits diverge to infinity and,
choosing k sufficiently large, neighbouring initial points
have orbits that become arbitrarily far apart.
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