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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.
Since for any x, it follows that the
Lyapunov exponent of the linear map is the same
for any orbit, namely . The
Lyapunov exponent is negative if 0<a<1, in which
case all orbits converge to zero. When a>1,
the Lyapunov exponent is positive and the distance
between neighbouring orbits becomes arbitrarily large.
Hence, is related to the sensitivity
of orbits to perturbations. For ,
neighbouring orbits remain close. However, when
a>1, even though , the
system is not chaotic. This is due to the fact that the
system is unbounded for a>1. (Boundedness is
another requirement for chaos; this is better
illustrated by another example.)
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Next: Example II: The
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